Merci Également À Bénédicte

Jiří Nathanaël D'

Aaraona

Assaf Antoine Abdelhak

Barbara

Bohdan, Clément, Clément, Côme, Cyril, Enzo, Fabio, Hiroshi, Houzhi, Junchao, Laure Benjamin

Luca

Marc Mi-Song Lucas

Simon, Sylvain, Yann, Yiyang Ziad Mohan

Keywords: random graphs, scaling limits, gaussian free field, Markov chains, mixing time

Remerciements Ce manuscrit résume un voyage de plusieurs années, qui m'aura permis d'explorer différentes questions des probabilités mais surtout mes goûts, mes capacités et mes limites. Voici quelques mots pour ceux qui y ont joué un rôle essentiel.

Je tiens en premier lieu à remercier Justin. Depuis le projet enthousiasmant que tu m'as proposé en janvier 2017, ton intuition et nos échanges m'ont ouvert de nombreux horizons mathématiques. Que ce soit pour me guider, me relire ou m'aider à m'organiser, tes conseils se sont révélés remarquablement efficaces. Je les garde précieusement pour la suite de ma carrière et pour pouvoir, j'espère, les transmettre à mon tour.

Je suis également redevable à Christina. J'ai beaucoup appris de ton expertise et de ton expérience depuis ce stage à Oxford en 2016, puis durant notre collaboration plus longue et plus riche que prévu.

verses tentatives d'expliquer l'objet de mes recherches : ces discussions m'ont beaucoup aidé à penser différement à mon travail, aussi lointaines puissent-elles paraître de celui-ci. Merci à mes ex-colocataires et mes amis Gros aux préoccupations gourmandes tout court.

Enfin, merci Ting-Chia pour tout ce que tu m'apportes chaque jour -tu es une inépuisable source d'encouragement et d'inspiration.

Résumé

Cette thèse est consacrée à l'étude de différents graphes aléatoires, définis par des propriétés locales (comme la distribution des degrés des sommets ou la probabilité que deux sommets donnés soient reliés par une arête), et dont on cherche à déterminer des caractéristiques globales, notamment leur géométrie et le comportement de marches aléatoires. Elle se compose de trois parties indépendantes. À chaque fois, le graphe aléatoire étudié admet un arbre comme limite locale, et une comparaison fine entre l'arbre et le graphe permet de transposer des propriétés du premier au second.

• La première partie (Chapitre 2) porte sur la limite d'échelle d'un graphe aléatoire critique, un modèle de configuration avec des degrés indépendants et distribués selon une même loi de puissance à queue lourde : elle a une variance, mais pas de troisième moment. Il était connu que les plus grandes composantes connexes de ce graphe ont une taille Θ(n a ) pour un graphe à n sommets, le paramère a dépendant du modèle. On montre que leur structure converge vers une version biaisée d'un arbre aléatoire continu particulier, l'arbre stable, auquel on rajoute un nombre fini de cycles.

• La seconde partie (Chapitre 3) est consacrée au champ libre gaussien sur des graphes aléatoires d-réguliers. On étudie la percolation du champ libre au-dessus d'un niveau fixé.

Si on baisse ce niveau en-dessous d'un certain seuil critique, on établit l'émergence avec grande probabilité d'une unique composante connexe géante englobant une proportion positive des sommets, entourée d'ilôts de taille logarithmique, tandis que seuls ces derniers survivent au-dessus du seuil critique. On montre alors que ce grand continent admet de remarquables similitudes avec la composante géante d'un graphe aléatoire célèbre, le modèle d'Erdős-Rényi.

• La troisième partie (Chapitre 4) traite de marches aléatoires sur des relèvements aléatoires ("random lifts") d'un graphe fini quelconque donné. Ces relèvements sont des graphes peu denses mais avec de bonnes propriétés de connectivité et une structure assez régulière, l'arbre associé étant périodique. On prouve que la marche aléatoire sur ces graphes admet un cutoff, c'est-à-dire qu'il y a une transition brusque entre le moment où le marcheur est encore localisé autour de son point de départ, et celui où on a définitivement perdu sa trace.

Les deux dernières parties ont été réalisées à Paris sous la direction de Justin Salez entre 2017 et 2021. La première est issue d'une collaboration avec Christina Goldschmidt (Oxford Statistics),

initiée en 2016 lors d'un stage de recherche et poursuivie jusqu'en 2020.

Mots-clés : graphes aléatoires, limites d'échelles, champ libre gaussien, chaînes de Markov, temps de mélange.

Introduction

For over half a century, a wide variety of random graphs have been studied. They were initially introduced as combinatorial and probabilistic objects. The randomness allows to bypass the difficulty to build explicit instances, when one wants to prove the existence of graphs satisfying a given property (P): it suffices to show that (P) is satisfied with positive probability by a graph chosen in a certain random way. We refer the interested reader to the book of Alon and Spencer [START_REF] Alon | The probabilistic method[END_REF] for pioneering examples of this probabilistic method, in the works of Szele and Erdős in the 1940s.

Nowadays, many random graphs play a central role in the modelling of real-world networks.

Popular examples include epidemic spreads, the Internet, interactions between proteins, social networks, and so on. The survey of Newman [START_REF] Newman | the structure and function of complex networks[END_REF], published in 2003, gives a taste of this diversity. Many applications have been developed since, so that an exhaustive state-of-the-art review would probably be very difficult.

To this inter-disciplinarity corresponds an intra-disciplinarity in probability: recent works on random graphs mix techniques from combinatorics, stochastic analysis, branching processes or ergodic theory for instance. On March 8, 2021 in the MathSciNet database (https:// mathscinet.ams.org/mathscinet/index.html), there were 12315 papers containing the words "random graphs" in the review text, 2531 of them dating from 2016 or later. 820 and 209 of them respectively had the MSC category 60G (Stochastic processes) as primary or secondary.

For the category 05 (Combinatorics), these numbers were 2738 and 713, for the category 37 (Dynamical systems and ergodic theory), 244 and 78.

Both this wide range of applications in various sciences and these numerous connections between different areas of mathematics make random graphs a rich field of study. This thesis illustrates the latter phenomenon. It consists of three parts on purely theoretical models of sparse random graphs, each related to different perspectives and research communities within probability. They are united however by the interaction of several techniques coming from these environments.

In this Introduction, we present some essential features of sparse random graphs, and set the results of this thesis in that context (Section 1.2 for Chapter 2, Section 1.3 for Chapter 3 and Section 1.4 for Chapter 4).

Sparse random graphs

Large sparse random graphs. Many probabilistic results on random graphs only hold asymptotically with respect to the "size" of the graph, which is often defined as its number of vertices or edges. In this Introduction, (G n ) n≥1 will denote a sequence of (possibly random) undirected graphs, with vertex set V n and edge set E n , and such that |V n | = n. This sequence is said to be sparse if |E n | = o(n 2 ). In other words, only a vanishing proportion of the potential n(n -1)/2 edges exist. In this Introduction, all graphs will satisfy |E n | = Θ(n).

One fundamental example of sparse random graph that will illustrate several of the points below is the Erdős-Rényi random graph, simultaneously introduced by Gilbert [START_REF] Gilbert | Random graphs[END_REF] and Erdős and Rényi [START_REF] Erdös | On random graphs i[END_REF] in 1959. For n ≥ 1 and c ∈ [0, n], ER(n, c/n) is the graph on n vertices such that for every pair of vertices x, y, there is an edge between x and y with probability c/n, independently of all other pairs of vertices. We set c to be a positive constant: the average number of edges is c/n × n(n -1)/2 = c(n -1)/2 = Θ(n) so that by the law of large numbers, ER(n, c/n) is sparse with high probability.

From local rules to global behaviour. Many models are defined by the local interactions between vertices. One knows typically the probability that two vertices share an edge, as in the Erdős-Rényi graph, or the distribution of the degrees of the vertices. A generic example in this case is the configuration model: one assigns (in a deterministic or random way) a degree d i to the i-th vertex of V n , for every 1 ≤ i ≤ n. Then G n is sampled uniformly at random among all graphs with this degree distribution (this requires that the sum of the d i 's be even, one can for instance replace d n by d n + 1 if this is not the case).

Then, one tries to infer properties on the global behaviour of G n as n → +∞. The questions investigated are often among the following.

• Size: what is the size of the largest of connected components? For instance, what is the probability that G n is connected as n → +∞?

• Connectivity: what are the diameter or the typical distance between vertices? How many edges should one remove to disconnect a component?

• Geometry: consider a connected component as a metric space embedded with the standard graph distance (the distance between two vertices being the length of the shortest path), rescaled by the typical distance. Does its distribution converge to some (compact) metric space? If yes, in which metric(s) does the convergence hold?

• Random walks: at what speed does the distribution of a random walker converge to the invariant distribution on a connected component of G n ?

Exploration and comparison with a branching process. A natural way to infer these global properties is to explore the graph neighbour by neighbour, using the random local con-struction rules. To keep track of this exploration, one records some statistics, for instance the sum of the degrees of the first k vertices visited, k = 1, 2, . . . Due to the sparsity, it is frequently the case that there are few or even no cycles on large parts of the graph. Moreover, the construction rules often entail that each exploration step is almost independent of the previous ones, or only depends on a simple parameter such as the number of vertices already discovered.

Hence the part of the graph that we explore can be compared to a branching process T . The latter turns out to be the local limit of G n , i.e. if x is a uniform vertex of G n and k ∈ N an arbitrary radius, the distribution of the ball B(x, k) converges to that of B(T, k). See [START_REF] Curien | Random graphs -the local convergence point of view[END_REF] for a very accessible lecture on this crucial notion. For instance, it is known that the local limit of ER(n, c/n) is a Galton-Watson tree with a Poisson reproduction law of parameter c.

The phase transition

Phase transition. Changing some parameters of the model can lead to spectacular modifications of the global picture of G n . There is often an abrupt phase transition, depending on whether a vertex discovered in the exploration leads in average to more or less than one new neighbour. Generally, if this average is below 1, a typical component that we explore will be close to a subcritical branching process, for instance a subcritical Galton-Watson tree, whose size has exponential moments. Hence, the largest connected components of the graph encompass only O(log n) vertices. This is the subcritical phase.

On the contrary, if we see more that one new neighbour in average, a connected component of G n has a positive chance to grow exponentially, until a positive proportion of the vertices have been discovered (which substantially affects the law of the number of new neighbours of each vertex). Then, there is a unique "giant" component of size Θ(n), with good connectivity properties. This is the supercritical phase. The limit case, when this number is close to 1, is generally harder to characterize. The possible existence and properties of a critical regime that would drastically differ from both the subcritical and supercritical ones usually offer thrilling research directions.

For instance, Erdős and Rényi [START_REF] Erdős | On the evolution of random graphs[END_REF] showed in 1960 that ER(n, c/n) undergoes such a phase transition. The supercritical regime corresponds to c > 1 and the subcritical regime to c < 1.

Random graphs at criticality. When the average number of new neighbours is 1, we might expect that the part of the graph that we explore is close to a critical branching process.

In the critical phase c = 1, Aldous [START_REF] Aldous | Brownian excursions, critical random graphs and the multiplicative coalescent[END_REF] established in a celebrated paper in 1997 that the largest connected components of ER(n, 1/n), whose number grows to infinity with n, are of size Θ(n 2/3 ). To do so, he built an exploration process, that roughly counts the number of vertices "seen" (i.e. we have explored one of their neighbours) but not yet explored. It converges to a slightly modified Brownian motion when suitably rescaled. This exponent 2/3 comes in fact from the fractal invariance of the Brownian motion: after n 2/3 vertices have been seen, each discovered vertex leads in average to (n -n 2/3 )/n = 1 -n -1/3 new vertices. If we want a connected component to survive, we should discover at least 1 new vertex at each explored vertex, so that there is a "deficit" of order n 2/3 × n -1/3 = n 1/3 vertices, that can be compensated by the Brownian fluctuations of order (n 2/3 ) 1/2 = n 1/3 of the stack of vertices seen but not yet explored.

Fifteen years later, Addario-Berry, Broutin and Goldschmidt [START_REF] Addario-Berry | The continuum limit of critical random graphs[END_REF] found a scaling limit for the structure of these largest components: when scaling the distances by a factor n -1/3 , they converge to a sequence of random trees, with a finite number of identifications of pairs of vertices to create cycles. These random trees are modifications of the Continuum Random Tree (CRT), a random tree which is coded by an excursion of the Brownian motion above 0. The CRT is an unavoidable bridge between random graphs and stochastic calculus, and has been thoroughly studied since the 1990s (see [START_REF] Aldous | The continuum random tree[END_REF][START_REF] Aldous | The continuum random tree. II. An overview[END_REF][START_REF] Aldous | The continuum random tree[END_REF][START_REF] Gall | Random trees and applications[END_REF]).

Universal scaling limits. This phase transition has been investigated on various other random graphs: configuration models, inhomogeneous random graphs, preferential attachment models, and so on (see for instance [START_REF] Bhamidi | Universality for critical heavy-tailed network models: Metric structure of maximal components[END_REF][START_REF] Broutin | Limits of multiplicative inhomogeneous random graphs and lévy trees: The continuum graphs[END_REF][START_REF] Dhara | Critical window for the configuration model: finite third moment degrees[END_REF][START_REF] Federico | Critical scaling limits of the random intersection graph[END_REF][START_REF] Nachmias | Critical percolation on random regular graphs[END_REF], we give a detailed bibliographical account in Section 2.2.5 of Chapter 2). The book of van der Hofstad [START_REF] Van Der Hofstad | Random graphs and complex networks: volume I[END_REF] gives a modern panorama of these models. It turns out that in most cases, one of the following two scenarii happens:

(I) the degree distribution of a vertex x ∈ V n chosen uniformly at random has a finite third moment. Then the largest connected components have size Θ(n 2/3 ), the diameter and typical distances between vertices are Θ(n 1/3 ).

(II) this degree distribution has a finite second moment but its third moment is infinite. If α+2) for some α ∈ (1, 2) and some slowly varying function L (for simplicity, think of it as a positive constant), then the largest connected components have size Θ(n α/(α+1) ). The diameter and typical distances between vertices are Θ(n (α-1)/(α+1) ).

P(deg(x) = k) ∼ k→+∞ L(k)k -(
When the degree distribution has infinite variance, there are enough "hubs" (vertices with arbitrarily large degrees as n → +∞) so that all pairs of hubs are connected, and every vertex is connected to a hub, hence distances in G n are drastically shrunk.

In both (I) and (II), one usually shows that the exploration process, when suitably rescaled, converges to a (possibly modified) Lévy process. The case (II) is often the hardest from a technical point of view: there are many large degrees that make some key functionals nonintegrable but not enough to simplify the structure of G n as in the infinite variance case.

In Chapter 2, we study a natural generic model that belongs to this case. It is a critical configuration model with i.i.d. degrees whose distribution ν satisfies (II). We prove that the rescaled largest connected components converge to modifications of a random tree coded by an α-stable Lévy process, the α-stable tree, with a finite number of vertex identifications to create cycles (Theorem 2.1.1). These limiting spaces are new objects for α ∈ (1, 2), whose study has recently been initiated in [START_REF] Goldschmidt | Stable graphs: distributions and line-breaking construction[END_REF]. This convergence generalizes what happens in the special case α = 2 (which belongs to (I) -the CRT is the α-stable tree for α = 2).

The originality and interest of our method lies in a measure-change that gives the law of the degrees met in the exploration w.r.t. a sequence of i.i.d. degrees distributed as ν, the size-bias of ν. It quantifies naturally an intuitive phenomenon, the progressive "degree depletion" of the vertices met during the exploration. In the latter, each new vertex is picked with a probability proportional to its degree, hence according to the size-bias of the distribution of the remaining degrees. But ν dominates stochastically ν, so that we tend to visit the vertices with a larger degree first. This measure-change converges in the limit, to express how our limiting spaces are obtained from stable trees. It would be interesting to look for a generalization of this procedure to other graphs.

Contrary to many works on similar models, our approach is independent from the multiplicative coalescent. The latter dynamic, already studied by Aldous in his seminal paper [START_REF] Aldous | Random walks on finite groups and rapidly mixing markov chains[END_REF], is suited to models with a monotonicity w.r.t. their parameters, like ER(n, p/n): edges are progressively added as one raises the value of p, and the probability that a link appears between two connected components is approximately proportional to the product of their sizes.

The Gaussian Free Field

The Gaussian Free Field on transient graphs. On an infinite transient graph G, one can define a centred Gaussian process (ϕ(x)) x∈G indexed by its vertices, such that its covariances are given by the Green function Γ G : for two vertices x, y of G, Cov(ϕ(x), ϕ(y)) = Γ G (x, y) = k≥0 P(X k = y), where (X k ) k≥0 is a Simple Random Walk (SRW) on G, starting at x. This process is called the Gaussian Free Field (GFF) on G. By this construction involving the Green function, the GFF is deeply connected to structural properties of G and to the behaviour of SRWs on G.

In the past two decades, several generalizations of the second Ray-Knight theorem have been found [START_REF] Eisenbaum | A rayknight theorem for symmetric markov processes[END_REF][START_REF] Le | Markov paths, loops and fields[END_REF]104,[START_REF] Sabot | Inverting ray-knight identity[END_REF], linking the distribution of the GFF with that of the local times of the random walk on G. In the same spirit, connections with random interlacements have been established [START_REF] Sznitman | Random interlacements and the gaussian free field[END_REF][START_REF] Sznitman | An isomorphism theorem for random interlacements[END_REF].

One way to study the GFF is to look at its level-sets. For h ∈ R, the level-set above h of ϕ, denoted E ≥h , is the subgraph of G induced by the set of vertices {x, ϕ(x) ≥ h}. Level-set percolation of the GFF has been investigated since the 1980s [START_REF] Bricmont | Percolation in strongly correlated systems: The massless gaussian field[END_REF][START_REF] Molchanov | Percolation in random fields[END_REF]. The last decade has witnessed blossoming literature on classical graphs, in particular Z d ( [START_REF] Drewitz | The Sign Clusters of the Massless Gaussian Free Field Percolate on {Z d , d ≥slant 3} (and more)[END_REF][START_REF] Duminil-Copin | Equality of critical parameters for percolation of gaussian free field level-sets[END_REF]104,[START_REF] Popov | On decoupling inequalities and percolation of excursion sets of the gaussian free field[END_REF][START_REF] Rodriguez | Phase transition and level-set percolation for the gaussian free field[END_REF]) and transient trees ( [START_REF] Abächerli | Level-set percolation of the Gaussian free field on regular graphs I: regular trees[END_REF][START_REF] Sznitman | Random interlacements and the gaussian free field[END_REF][START_REF] Sznitman | Coupling and an application to level-set percolation of the gaussian free field[END_REF]).

One central question is whether there exists (with positive probability) an infinite connected component in E ≥h , depending on h. In particular, is there a phase transition, i.e. does a nondegenerate threshold h G ∈ R exist, such that the answer is yes for h < h G and no for h > h G ? Compared to classical percolation models, such as Bernoulli bond percolation where all edges are independent, the GFF level-set percolation brings long-range correlations that may play an important role, in particular in graphs like Z d where the Green function decays only polynomially w.r.t. the distance between vertices. Very recently, GFF percolation has even been used to show a result on Bernoulli bond percolation [START_REF] Duminil-Copin | Existence of phase transition for percolation using the gaussian free field[END_REF].

GFF level-sets on finite graphs. If G n is connected, one can define an analogous field on G n , the zero-average Gaussian Free Field. Its covariance is given by the zero-average Green function Γ Gn : for x, y ∈ V n , Γ Gn (x, y) := +∞ 0 (P(X t = y) -π n (y)) dt, where (X t ) t≥0 is a continuous-time SRW on G n : its trajectory is that of a SRW, and the time intervals between two consecutive jumps are i.i.d. with law Exp [START_REF] Abächerli | Local picture and level-set percolation of the gaussian free field on a large discrete torus[END_REF]. π n is the invariant distribution of the SRW on G n .

If (G n ) n≥0 has a local limit G, a natural question is whether some characteristics of the GFF on G can be transferred to the zero-average GFF on G n . For instance, one might ask whether a phase transition for the existence of an infinite connected component of the level-set E ≥h in G corresponds to a phase transition for the emergence of a "macroscopic" component of size Θ(n) in the level-set E ≥h n above h in G n . For G = T d the d-regular tree, Sznitman [START_REF] Sznitman | Coupling and an application to level-set percolation of the gaussian free field[END_REF] showed that there exists a level-set percolation threshold h ∈ (0, ∞). Abächerli and Černý [START_REF] Abächerli | Level-set percolation of the Gaussian free field on regular graphs II: finite expanders[END_REF] proved that if G n is a typical d-regular graph, then for h > h (subcritical phase), the connected components of E ≥h n have size O(log n), and for h < h (supercritical phase), a positive proportion of the vertices are in mesoscopic components of size at least Θ(n a ), for some non explicit constant a > 0. The question on the existence of a macroscopic component (i.e. a = 1) was left open.

In Chapter 3, we give a positive answer for most regular graphs: if G n is a uniform random d-regular graph, then w.h.p. E ≥h n has a unique giant component

C (n) 1 of size Θ(n). More pre- cisely, |C (n) 1 | = η(h)n(1 + o(1)
), where η(h) > 0 is the probability that the connected component C h of a given vertex of T d in the level-set above h is infinite (Theorem 3.1.1).

We also show that C

(n) 1 shares several "global" properties with the giant component of ER(n, p/n) for p > 1 (hence in the supercritical regime), concerning the diameter, the typical distance between vertices, the core and the kernel (Theorem 3.1.2). The local limit of C (n) 1 is C h conditioned to be infinite.

Our proofs rely on an annealed exploration of a new kind, in which we reveal progressively the structure of G n and the zero-average GFF on it. We also need to refine some properties of the component C h in T d shown in [START_REF] Abächerli | Level-set percolation of the Gaussian free field on regular graphs I: regular trees[END_REF]. In a work in progress [START_REF] Conchon-Kerjan | Supercritical GFF percolation on a regular tree : anatomy of the infinite cluster[END_REF], we study closely C h and the random walk on it, proving its ballisticity.

It would be exciting to determine what happens when h = h : are we in (I) or (II), or does the GFF level-set percolation lead to a new universality class of critical random graphs?

The mixing time

Mixing time and cutoff. A natural way to explore a graph and record information on its structure is to study random walks on its vertices. A quantity of great interest is the mixing time, i.e. the time it takes for a random walk to be approximately distributed according to the invariant probability measure on the graph. Formally, let P n be the transition matrix of the discrete-time Simple Random Walk (SRW) on V n . Suppose that G n is connected, so that there is a unique invariant distribution π n . For ε > 0 and for a vertex x of G n , the ε-mixing time from x is t (n) x (ε) := inf{t ∈ N ∪ {0}, P t n (x, •) -π n T V ≤ ε}.

One often examines the worst-case mixing time t

(n) max (ε) := sup{t (n)
x (ε), x ∈ V n }. Other distances than the total variation distance might be considered.

The mixing time delivers essential information on the geometry of the graph. For instance, it provides a lower bound for the typical distance between vertices. If the latter is much smaller than the mixing time, then the graph may have a weak expansion, or traps that delay the random walk. If the 1/10-mixing time is much larger than the 1/2-mixing time, then there might be a bottleneck in the graph that is hard to cross for the random walk, and so on.

The mixing time also plays a crucial role in Monte Carlo simulations, as it gives a good estimate for their running time. The books of Levin, Peres and Wilmer [START_REF] Levin | For The Reader[END_REF] and of Montenegro and Tetali [START_REF] Montenegro | Mathematical aspects of mixing times in Markov chains[END_REF] give a broad panorama of both historical and modern techniques on mixing times.

We say that there is cutoff when for all ε, ε ∈ (0, 1),

t (n) max (ε)/t (n) max (ε ) → 1 as n → +∞.
In other words, the cutoff means that the mixing of the random walk happens abruptly. First examples of this phenomenon were given in the 1980s for random walks on finite groups (see [START_REF] Aldous | Shuffling cards and stopping times[END_REF] or [START_REF] Diaconis | Generating a random permutation with random transpositions[END_REF] on the symmetric group) or on spaces that can be factored into a n-product of a base space (such as Z n 2 in [START_REF] Aldous | Random walks on finite groups and rapidly mixing markov chains[END_REF]). This "algebraic" direction is still investigated nowadays. See for instance [START_REF] Hermon | Cutoff for almost all random walks on abelian groups[END_REF][START_REF] Hermon | Cutoff for Mixing Times on Random Abelian Cayley Graphs[END_REF] on random Cayley graphs of abelian groups.

Expanders. A wide class of graphs where random walks mix fast and cutoff might happen are the expander graphs. We say that the sequence (G n ) n≥1 is expanding if its isoperimetric constant is bounded away from 0, i.e. there exists c > 0 such that for large enough n,

c n := inf S⊆Vn,|S|≤n/2 |∂S| |S| ≥ c,
where ∂S ⊆ V n \S is the set of vertices having one neighbour in S. The very accessible survey of Hoory, Linial and Widgerson [START_REF] Hoory | Expander graphs and their applications[END_REF] provides a good overview of the study of these graphs. Some explicit constructions of expanders have been achieved since the 1980s, using number theory and combinatorial arguments. The book of Lubotzky [START_REF] Lubotzky | Discrete groups, expanding graphs and invariant measures[END_REF] gives a good account on it. Finding such constructions is a difficult task. It is much easier to prove the existence of expanders via a probabilistic argument. If G n is a d-regular graph on n vertices (i.e. all vertices have degree d) chosen uniformly at random, then a combinatorial computation shows that with high probability, G n is an expander.

This expansion property is equivalent to the existence of a spectral gap: let λ n be the second largest eigenvalue of P n (the largest is 1, P n being a stochastic matrix). Then lim inf n→+∞ c n > 0 if and only if lim inf n→+∞ λ n > 0. The implication (expansion ⇒ spectral gap) was first formulated by Cheeger [START_REF] Cheeger | A lower bound for the smallest eigenvalue of the laplacian[END_REF] in 1969, for Riemannian manifolds. To our knowledge, the earliest adaptation to random walks on discrete graphs is due to Alon and Milman [START_REF] Alon | Eigenvalues, expanders and superconcentrators[END_REF] in 1984. It is classical that a spectral gap bounded away from 0 implies in turn that the SRW on G n mixes in O(log n) steps, provided we make it lazy, i.e. at each step, the random walker flips a balanced coin and stays on its position if and only if the result is heads. Conversely, if the mean degree of a vertex of G n is bounded, and the degree distribution is sufficiently regular, one might expect that the SRW does not mix in less than Θ(log n) steps. Therefore, the mixing should happen after Θ(log n) steps.

Cutoff on sparse expanders. A major breakthrough came in 2010, when Lubetzky and Sly proved that the SRW on the random d-regular graph has a cutoff [START_REF] Lubetzky | Cutoff phenomena for random walks on random regular graphs[END_REF].

Theorem [START_REF] Lubetzky | Cutoff phenomena for random walks on random regular graphs[END_REF]] If G n is a uniform d-regular random graph, then for every ε ∈ (0, 1),

t (n) max (ε) -d d-2 log d-1 n log d-1 n P -→ 2 d(d -1) (d -2) 3/2 Φ -1 (ε),
where Φ(x) := +∞ x (2π) -1/2 exp(-u 2 /2)du for x ∈ R is the tail distribution of the standard normal.

Rather than a simple asymptotic on the mixing time, this result is very similar to a central limit theorem, with a precise cutoff window Θ( √ log n). In fact, it reflects the behaviour of the entropy of the SRW on T d . If P (d) is the transition matrix of the SRW on T d , then

t -1/2 (log d-1 P t (d) (X 0 , X t ) -d-2 d t)
converges in distribution to a centred Gaussian variable as t → +∞. Then, by counting arguments limiting w.h.p. the occurrence of cycles around the trajectory of the SRW on G n , one can show that log d-1 P t (d) (X 0 , X t ) and log d-1 P t n (X 0 , X t ) are close. The mixing happens when P t n (X 0 , X t ) 1/n, and this occurs with positive probability for t = d d-2 log d-1 n + Θ( log d-1 n). Since then, several results were proved on similar models, following roughly this pattern: cutoff happens at an entropic time, the constant being given by a law of large numbers for the entropy of the random walk on the local limit of (G n ) n≥1 . In the same article, Lubetzky and Sly also showed a cutoff for the Non-Backtracking Random Walk (NBRW, i.e. a SRW conditioned at each step on not going back along the edge it has just crossed). See also [START_REF] Berestycki | Random walks on the random graph[END_REF] for the SRW on the largest component of a supercritical Erdős-Rényi random graph, or [START_REF] Ben | Cutoff for nonbacktracking random walks on sparse random graphs[END_REF] for the NBRW on a configuration model. Very recently, Hermon, Sly and Sousi [START_REF] Hermon | Universality of cutoff for graphs with an added random matching[END_REF] found a generic way to produce cutoff by adding new edges to a deterministic graph G n (on which one makes mild assumptions), via a perfect matching of its vertices.

Over the last few years, there has been increasing interest in mixing times on dynamical graphs (typically, edges are re-sampled at random at a given rate), when the mixing time profile is already well-known on a static version of the graph, see [START_REF] Avena | Random walks on dynamic configuration models:a trichotomy[END_REF][START_REF] Caputo | Mixing time of PageRank surfers on sparse random digraphs[END_REF][START_REF] Sousi | Cutoff for random walk on dynamical Erdős-Rényi graph[END_REF].

In Chapter 4, we study the random walk on random lifts of a fixed graph G, which is a natural way to combine the "product of a base space" and the "expanding sparse graph" perspectives for cutoff. To obtain a uniform random n-lift of G, take n copies of the vertices of G, and for every pair of vertices u, v sharing an edge e in G, draw an edge from the i-th copy of u to the σ e (i)-th copy of v, the σ e 's being independent uniform random permutations of {1, . . . , n}.

Our main result is Theorem 4.1.1: we prove the cutoff for the random walk at entropic time, with a cutoff window Θ( √ log n) (with an explicit Gaussian profile only for the lower bound).

We make very few assumptions on the base graph G. In particular, we allow weights on the edges that make the random walk non-reversible. The proof relies on a fine understanding of the random walk on the universal cover of G, a periodic tree T G , which is the local limit of G n . Then, a careful exploration, where the structure of G n is revealed along the trajectory of a random walk, allows to transpose properties from T G to G n . A byproduct of our method is the computation of the diameter of random lifts (Theorem 4.3.1).

In addition, we prove that the NBRW on non-weighted random lifts has a cutoff, with a Gaussian profile for the cutoff window (Theorem 4.2.3).

Chapter 2

Scaling limit of a critical configuration model with power-law degrees This chapter stems from the preprint [START_REF] Conchon-Kerjan | The stable graph: the metric space scaling limit of a critical random graph with i.i.d. power-law degrees[END_REF], that has been submitted to Annals of Probability and accepted with major revisions.

Abstract. We prove a metric space scaling limit for a critical random graph with independent and identically distributed degrees having power-law tail behaviour with exponent α + 1, where α ∈ (1, 2). The limiting components are constructed from random R-trees encoded by the excursions above its running infimum of a process whose law is locally absolutely continuous with respect to that of a spectrally positive α-stable Lévy process. These spanning R-trees are measure-changed α-stable trees. In each such R-tree, we make a random number of vertexidentifications, whose locations are determined by an auxiliary Poisson process. This generalises results which were already known in the case where the degree distribution has a finite third moment (a model which lies in the same universality class as the Erdős-Rényi random graph)

and where the role of the α-stable Lévy process is played by a Brownian motion.

Contents 2.1 Introduction

Overview

In recent years, a wide variety of random graph models have been introduced and studied. Many of these models undergo a phase transition of the following type: below some threshold, the connected components are microscopic in size (in the sense that they each contain a negligible proportion of the vertices) and possess few cycles, whereas above the threshold, there is a component which occupies a positive fraction of the vertices and contains many cycles, and all other components are again microscopic. We are particularly interested in the behaviour exactly at the point of the phase transition, and in a precise description of the sizes and geometric properties of the components, which is typically much more delicate than in the sub-and supercritical cases. We will first give a brief overview of the setting in which we are interested, and of our main results, deferring a more detailed account with proper definitions, as well as a summary of the pre-existing literature, to the next section.

We consider a uniform random graph on n vertices with a given degree sequence, where the degrees themselves are independent and identically distributed random variables, D 1 , . . . , D n .

(If n i=1 D i is odd, we replace D n by D n + 1.) For simplicity, we impose the condition that P (D 1 ≥ 1) = 1, so that there are no isolated vertices. We also assume that P (D 1 = 2) < 1 (otherwise we have a random 2-regular graph, which contains many cycles of macroscropic size [START_REF] Richard Arratia | Logarithmic combinatorial structures: a probabilistic approach[END_REF]) and that var (D 1 ) < ∞ (otherwise the graph behaves very differently; see [START_REF] Van Leeuwaarden Souvik Dhara | Critical percolation on scale-free random graphs: New universality class for the configuration model[END_REF]). The phase transition then occurs when the parameter θ :

= E [D 1 (D 1 -1)] /E [D 1 ] passes through 1: if θ < 1 then
the proportion of vertices in the largest component tends to 0 in probability as n → ∞, whereas if θ > 1, this proportion instead converges to a strictly positive constant, again in probability as n → ∞.

At the critical point θ = 1, there is a sequence of components whose sizes are comparable (rather than a single giant component, as in the supercritical case) and which, even after rescaling, retain some randomness in the limit. The sizes and geometric properties of these components depend on the tail of the distribution of D 1 . In particular,

• if E D 3 1 < ∞ then the largest components have sizes on the order of n 2/3 and diameters on the order of n 1/3 ; α+2) for some constant c > 0 and α ∈ (1, 2) then the largest components have sizes on the order of n α/(α+1) and diameters on the order of n (α-1)/(α+1) .

• if P (D 1 = k) ∼ ck -(
(It will be convenient to refer to the first of these as the "α = 2 case".)

These scaling properties are either proved or conjectured to be universal, that is to hold for whole families of random graph models with similar asymptotic degree distributions. We will discuss the issue of universality in some detail below.

Our results

Let us now state our scaling limit theorem. Let G n 1 , G n 2 , . . . be the (vertex-sets of the) components of the critical random graph, listed in decreasing order of size, with ties broken arbitrarily.

We think of these as measured metric spaces, by endowing G n i with the graph distance, d n i , and the counting measure on its vertices, µ n i . Formally, each is an element of the Polish space of isometry-equivalence classes of measured metric spaces endowed with the Gromov-Hausdorff-Prokhorov distance, which we will define properly below.

G n i , n -(α-1)/(α+1) d n i , n -α/(α+1) µ n i , i ≥ 1 d -→ ((G i , d i , µ i ) , i ≥ 1)
in the sense of the product Gromov-Hausdorff-Prokhorov topology.

The same result also holds for a multigraph with the same degree sequence generated according to the configuration model (see Section 2.2.1 for more details).

In the terminology of [START_REF] Addario-Berry | The scaling limit of the minimum spanning tree of the complete graph[END_REF], (G i , d i ) is a random R-graph for each i ≥ 1. For reasons which will shortly become clear, we refer to the whole limiting object as the α-stable graph if α ∈ (1, 2) or the Brownian graph (instances of which have already occurred several times in the literature)

if α = 2.
This theorem, in particular, implies the scaling properties mentioned above. The α = 2 case may be deduced from a more general theorem due to Bhamidi and Sen [START_REF] Bhamidi | Geometry of the vacant set left by random walk on random graphs, wright's constants, and critical random graphs with prescribed degrees[END_REF], proved by different methods. For α ∈ (1, 2), Bhamidi, Dhara, van der Hofstad and Sen [START_REF] Bhamidi | Universality for critical heavy-tailed network models: Metric structure of maximal components[END_REF] considered the setting of critical percolation on a supercritical uniform random graph with given degree sequence, having similar tail behaviour to ours, and proved a scaling limit theorem in the sense of the product Gromov-weak topology. (We understand that this will be improved to a convergence in the product Gromov-Hausdorff-Prokhorov topology for critical degree sequences satisfying certain conditions in forthcoming work [START_REF] Bhamidi | Universality for critical heavy-tailed network models: Gromov-Hausdorff-Prohorov convergence[END_REF].) We will describe the results of [START_REF] Bhamidi | Universality for critical heavy-tailed network models: Metric structure of maximal components[END_REF] in more detail below and will, for the moment, simply observe that there are situations which are covered by both theorems, and where the limit objects must therefore be the same, but where this is certainly not obvious from their respective constructions.

One of the most striking aspects of our results is the characterisation of the limit spaces which we are able to give, which is completely new for α ∈ (1, 2), and generalises one which was already known for α = 2. In order to give this characterisation, we must first introduce some stochastic processes which play a key role.

Let µ = E [D 1 ]. For α ∈ (1, 2
), let L be a spectrally positive α-stable Lévy process with

Laplace transform

E [exp(-λL t )] = exp cΓ(2 -α) µα(α -1) λ α t , λ ≥ 0, t ≥ 0,
where c > 0 is the constant such that α+2) . Such a process can be thought of as encoding a forest of continuum trees; the standard way to do this goes via a (somewhat complicated) functional of L called the height process H (we will define this properly below).

P (D 1 = k) ∼ ck -(
Let

C α = cΓ(2 -α) α(α -1
) .

We will create a new pair ( L, H) of processes via change of measure as follows: for suitable

test-functions f : D([0, t], R) 2 → R, let E f ( L u , H u , 0 ≤ u ≤ t) = E exp - 1 µ t 0 sdL s - C α t α+1 (α + 1)µ α+1 f (L u , H u , 0 ≤ u ≤ t) .
(2.1)

For α = 2, letting µ = E [D 1 ] and β = E [D 1 (D 1 -1)(D 1 -2)],
we instead take

L t = β µ B t and H t = 2 µ β B t -inf 0≤s≤t B s ,
where B is a standard Brownian motion (in the Brownian setting, the associated height process has the same distribution as a reflected Brownian motion, up to a scaling constant). In this case, define

E f ( L u , H u , 0 ≤ u ≤ t) = E exp - 1 µ t 0 sdL s - βt 3 6µ 3 f (L u , H u , 0 ≤ u ≤ t) . (2.2)
In either case, let

R t = L t -inf 0≤s≤t L s , t ≥ 0. Now write (ζ i , i ≥ 1)
for the ordered sequence of lengths of excursions of R above 0. These excursions give rise to spanning R-trees for the limiting components. For i ≥ 1, let (Equivalently, we can think of this as a Poisson point process of intensity 1/µ in the area under the graph of R.) The points tell us how to identify vertices in the R-trees in order to create cycles. For i ≥ 1, suppose that a number M i ≥ 0 of points fall within the ith longest excursion

ε i : [0, ζ i ] → R + be
ε i . Given ε i , we then have M i ∼ Poisson 1 µ ∞ 0 ε i (u)du . If M i ≥ 1, write (s i,1 , x i,1 ) , (s i,2 , x i,2 ) , . . . , (s i,M i , x i,M i )
for the points themselves (with their first co-ordinates translated to the interval [0, ζ i ]). For

i ≥ 1 and 1 ≤ k ≤ M i , let t i,k = inf{t ≥ s i,k : ε i (t) ≤ x i,k }. Now for i ≥ 1, let (G i , d i , µ i ) be the measured metric space obtained from ( T i , d i , µ i ) by making no change if M i = 0 or, if M i ≥ 1, by identifying the M i pairs of points (p i (s i,1 ), p i (t i,1 )) , . . . , (p i (s i,M i ), p i (t i,M i )) .
(Formally, this is done by taking the quotient metric space in a standard way which is described in detail, for example, just before Lemma 21 of First consider α ∈ (1, 2), and let e be a normalised (i.e. length 1) excursion of the stable process L, and let h be the associated normalised excursion of the height process, which would encode an α-stable tree. Now for m ∈ Z + , define tilted excursions e (m) and h (m) via

E g( e (m) , h (m) ) = E g(e, h) 1 0 e(u)du m E 1 0 e(u)du m , (2.3)
for suitable test-functions g :

D([0, 1], R + ) × C([0, 1], R + ) → R. Let ( T (m) , d (m) , µ (m) ) be the R-tree ( T (m) , d (m)
) encoded by h (m) , along with its natural mass measure µ (m) . Write p (m) for the projection [0, 1] → T (m) . If m ≥ 1, now sample m pairs of points in the tree as follows. First pick (s 1 , x 1 ), . . . , (s m , x m ) independently and uniformly from the area below the excursion e (m) and above the x-axis according to the normalised Lebesgue measure. Define

t i = inf{t ≥ s i : e (m) (t) ≤ x i }. Finally, identify p (m) (s i ) and p (m) (t i ) for 1 ≤ i ≤ m in order to obtain (G (m) , d (m) , µ (m) ). Set (G (0) , d (0) , µ (0) ) = ( T (0) , d (0) , µ (0) ).
Something very similar works in the Brownian case. Here, we take e to be a normalised Brownian excursion (which is, in particular, continuous); in this context, h = 2e, so there is no need to consider two different excursions. The function 2e encodes the Brownian continuum random tree (in the normalisation adopted by Aldous [START_REF] Aldous | The continuum random tree[END_REF]). Again define e (m) as at (2.3) and let ( T (m) , d (m) , µ (m) ) be the measured R-tree encoded by 2 e (m) , and write p (m) for the projection [0, 1] → T (m) . If m ≥ 1, now sample m pairs of points in the tree as follows. First pick (s 1 , x 1 ), . . . , (s m , x m ) independently and uniformly from the area below the excursion e (m) and above the x-axis according to the normalised Lebesgue measure. Define

t i = inf{t ≥ s i : e (m) (t) ≤ x i }. Finally, identify p (m) (s i ) and p (m) (t i ) for 1 ≤ i ≤ m in order to obtain (G (m) , d (m) , µ (m) ). Set (G (0) , d (0) , µ (0) ) = ( T (0) , d (0) , µ (0) ).
M 1 , M 2 , . . . of points, the measured R-graphs (G 1 , d 1 , µ 1 ), (G 2 , d 2 , µ 2 ), . . . are independent with (G i , d i , µ i ) d = G (M i ) , ζ (α-1)/α i d (M i ) , ζ i µ (M i ) for each i ≥ 1.
This shows that, in order to understand further the geometric properties of our limit object, a key role will be played by the family of random R-graphs ((G (m) , d (m) , µ (m) ), m ≥ 0). These are studied in depth for α ∈ (1, 2) in the companion paper [START_REF] Goldschmidt | Stable graphs: distributions and line-breaking construction[END_REF]; the Brownian case was the subject of the earlier paper [START_REF] Addario-Berry | Critical random graphs: limiting constructions and distributional properties[END_REF]. From the absolute continuity relation (2.3), for any α ∈ [START_REF] Abächerli | Local picture and level-set percolation of the gaussian free field on a large discrete torus[END_REF][START_REF] Abächerli | Level-set percolation for the gaussian free field on a transient tree[END_REF] it is straightforward to see that (G (m) , d (m) , µ (m) ) has Hausdorff dimension α/(α -1) almost surely, since this is true of the appropriate Brownian/α-stable tree and one cannot change the fractal dimension by making finitely many vertex-identifications.

The branch-points of the α-stable tree are almost surely all infinitary (i.e. removing any of them breaks the tree into infinitely many connected components), and this property is inherited, via absolute continuity, by ( T (m) , d (m) ) for α ∈ (1, 2). It follows from the properties of the excursion e (m) (see [START_REF] Goldschmidt | Stable graphs: distributions and line-breaking construction[END_REF] for an in-depth discussion) that the vertex-identifications in ( T (m) , d (m) ) are almost surely all from a leaf to a branch-point of infinite degree. In contrast, in the α = 2 case, the vertex-identifications are almost surely all from a leaf to a point of degree 2 (see [START_REF] Addario-Berry | The continuum limit of critical random graphs[END_REF][START_REF] Addario-Berry | Critical random graphs: limiting constructions and distributional properties[END_REF]).

In [START_REF] Addario-Berry | Critical random graphs: limiting constructions and distributional properties[END_REF][START_REF] Goldschmidt | Stable graphs: distributions and line-breaking construction[END_REF], it is further shown that one may explicitly determine the law of the kernel of G (m) (that is, the multigraph with edge-lengths which encodes its cycle structure), and that G (m) may be constructed by gluing together randomly rescaled Brownian/stable trees. Finally, it is shown in [START_REF] Addario-Berry | Critical random graphs: limiting constructions and distributional properties[END_REF][START_REF] Goldschmidt | Stable graphs: distributions and line-breaking construction[END_REF] that G (m) possesses a line-breaking construction, that is, a recursive construction which starts from the kernel and successively glues on line-segments of random lengths to random points, obtaining a convergent sequence of approximations to the final R-graph.

Background

In this section, we give some background material on our random graph model, and discuss the previously known results on its critical behaviour. We also give a brief account of the scaling limit theory for Galton-Watson trees. We then give an overview of the proof of Theorem 2.1.1. This is followed by a brief summary of some related literature, and some open problems. Finally, at the end of this section, we give a plan of the rest of the paper.

For a sequence of random variables (A n ) n≥0 and a sequence (a n ) n≥0 of real numbers, we write A n = O P (a n ) to mean that (A n /a n ) n≥0 is tight. We write A n = Θ P (a n ) to mean that

A n = O P (a n ) and A -1 n = O P (a -1 n ). We write A n = o P (a n ) to mean A n /a n p → 0 as n → ∞.

The configuration model

We wish to sample a graph uniformly at random from among the graphs with the given degrees

D 1 , D 2 , . . . , D n .
There is a standard method for doing this, which originated (in varying degrees of generality) in the work of Bender and Canfield [START_REF] Bender | The asymptotic number of labeled graphs with given degree sequences[END_REF], Bollobás [START_REF] Bollobás | A probabilistic proof of an asymptotic formula for the number of labelled regular graphs[END_REF] and Wormald [START_REF] Wormald | Some problems in the enumeration of labelled graphs[END_REF], called the configuration model. (We refer the reader to the recent book of van der Hofstad [START_REF] Van Der Hofstad | Random graphs and complex networks: volume I[END_REF] for a full account of the configuration model and for proofs of the results quoted below.) We begin by first describing the setting where the vertex degrees are deterministic. More precisely, suppose that we have vertices labelled 1, 2, . . . , n where vertex i has degree d i , for 1 ≤ i ≤ n. Suppose that d i ≥ 1 for all 1 ≤ i ≤ n and that n i=1 d i is even. To vertex i, attach d i stubs or halfedges. Label the n i=1 d i half-edges in some arbitrary way, and then choose a pairing of them, uniformly at random. Join the paired half-edges together to make full edges, and then forget the labelling of the half-edges. In general, this procedure yields a multigraph (i.e. with self-loops, or multiple edges). However, if there exist one or more simple graphs with the given degree sequence (that is, if the degree sequence is graphical ) then, conditionally on the event that the configuration model yields a simple graph, that graph is uniform among the possibilities.

We are concerned with the setting where the degrees themselves are independent and identically distributed random variables D 1 , D 2 , . . . , D n . An immediate issue is that we cannot guarantee that n i=1 D i is even. We get around this problem by always assuming that if n i=1 D i is odd, then we in fact give vertex n degree D n + 1. For the regime and properties in which we are interested, this makes only a negligible difference, and we will ignore it in the sequel. Write ν = (ν k ) k≥1 for the probability mass function of D 1 , that is

ν k = P (D 1 = k), k ≥ 1. Let
M n (ν) be the multigraph resulting from the configuration model with these degrees. It remains to resolve the issue that the degree sequence may, in principle, be non-graphical. However, it is possible to show that if D 1 has finite variance and

θ = θ(ν) = E [D 1 (D 1 -1)] /E [D 1 ] then lim n→∞ P (M n (ν) is simple) = exp(-θ/2 -θ 2 /4),
and the right-hand side is strictly positive (see for instance Proposition 7.13 of [START_REF] Van Der Hofstad | Random graphs and complex networks: volume I[END_REF]). Let G n (ν) be a graph with the distribution of M n (ν) conditioned to be simple; this is our uniform random graph with i.i.d. ν-distributed degrees, and is the main object of study in this paper.

If ν k ∼ ck -(α+2) for some α ∈ (1, 2) as k → ∞, we will have that max 1≤i≤n D i = Θ P (n 1/(α+1) ). We will see in the sequel that vertices of degree Θ(n 1/(α+1) ) play an important role in the structure of the graph, and "show up" in the scaling limit as vertices of infinite degree (often known as hubs). However, since α > 1, with high probability we will not observe edges directly joining two vertices of degree Θ(n 1/(α+1) ) and, indeed, the vertices of highest degree will be typically well-separated. If E D 3 1 < ∞, on the other hand, then max 1≤i≤n D i = o P (n 1/3 ) and there are no hubs in the limit.

An important property of the configuration model is that the pairing of the edges may be generated in a progressive manner. This makes possible the use of an exploration process in order to capture properties of the (multi-)graph. We do this in a depth-first manner, conditionally on the vertex-degrees, and making use of the arbitrary labelling we gave the half-edges, as follows.

Start from a vertex v chosen with probability proportional to its degree D v . We will maintain a stack, namely an ordered list of half-edges which we have seen but not yet explored. Put the D v half-edges attached to v onto this stack, in increasing order of label, so that the lowest labelled half-edge is on top of the stack. At every subsequent step, if the stack is non-empty, take the top half-edge and sample its pair uniformly at random from those available (i.e. the others on the stack and those which we have not yet observed in our exploration). If the pair half-edge belongs to a vertex w which has not yet been observed (i.e. if the pair half-edge does not lie in the stack), remove the paired half-edges from the system, and add the remaining D w -1 half-edges attached to w to the top of the stack, again in increasing order of label. If ever the stack becomes empty, select a new vertex with probability proportional to its degree, and put all of its half-edges onto the stack. Repeat until the whole graph has been exhausted.

Notice that the stack is empty at the end of a step if and only if we have reached the end of a component, and that in each step except the one at the start of a component, we pair two half-edges. Let R n (k) be the size of the stack at step k. Then, for example, we may read off the numbers of edges in the successive components as the lengths minus 1 of the excursions above 0 of the process (R n (k), k ≥ 0). It turns out that this process, as we shall explain below, in fact encodes much more information about the multigraph.

Write |G| for the size of the vertex set of a graph G. For a connected graph G, write s(G) for its surplus, that is how many more edges it has than any of its spanning trees (which necessarily have |G| -1 edges). Write G n 1 , G n 2 , . . . for the connected components of G n (ν), in decreasing order of size, with ties broken arbitrarily. Similarly, write M n 1 , M n 2 , . . . for the ordered connected components of M n (ν).

The phase transition and critical behaviour of the component sizes

As we have already mentioned, G n (ν) undergoes a phase transition in its component sizes depending on its parameters [START_REF] Molloy | A critical point for random graphs with a given degree sequence[END_REF][START_REF] Molloy | The size of the giant component of a random graph with a given degree sequence[END_REF][START_REF] Janson | A new approach to the giant component problem[END_REF]

. Indeed, if θ(ν) ≤ 1, then the largest connected component G n 1 of G n (ν) is such that |G n 1 |/n p → 0. On the other hand, if θ(ν) > 1 then |G n 1 |/n p → ρ(ν)
, where ρ(ν) is some strictly positive constant. These results also hold for M n (ν). To give an idea of why the quantity θ(ν) is important, imagine performing the depth-first exploration outlined above, but ignoring any edges which create cycles. Then it is not hard to see that, at each step which is not the start of a component, the degree of the vertex to which the half-edge on the top of the stack connects (as long as it does not connect to something on the stack and thus create a cycle) is a size-biased pick from among the remaining possibilities.

So, at least close to the beginning, the exploration process should look approximately like a branching process with offspring distribution given by D * -1, where

P (D * = k) = kν k /E [D 1 ]. But then θ(ν) = E [D * -1]
, and so the critical point for the approximating branching process is indeed θ(ν) = 1. Our interest is in this precisely critical case, and a significant part of this paper is devoted to making the heuristic argument just outlined precise.

The following theorem, due to Joseph [START_REF] Joseph | The component sizes of a critical random graph with given degree sequence[END_REF], summarises some of the possible behaviours for the component sizes in the case θ(ν) = 1. A version of part (i) was proved independently by

Riordan [START_REF] Riordan | The phase transition in the configuration model[END_REF] (see below for further discussion). Let

2 ↓ =    (x 1 , x 2 , . . .) ∈ R N : x 1 ≥ x 2 ≥ . . . ≥ 0, i≥1 x 2 i < ∞    . Theorem 2.2.1. (i) Suppose that P (D 1 = 2) < 1, E [D 1 ] = µ and E [D 1 (D 1 -1)(D 1 -2)] = β.
Let B be a standard Brownian motion, and let

L t = β µ B t - β 2µ 2 t 2 , t ≥ 0 and R t = L t -inf 0≤s≤t L s , t ≥ 0. (2.4)
Then

n -2/3 (|G n 1 |, |G n 2 |, . . .) d -→ (ζ 1 , ζ 2 , . . .)
as n → ∞ in (ii) Suppose that lim k→∞ k α+2 P (D 1 = k) = c for some constant c > 0 and some α ∈ (1, 2),

and that E [D 1 ] = µ. Let X be the process with independent increments whose law is specified by its Laplace transform

E [exp(-λX t )] = exp t 0 ds ∞ 0 dx(e -λx -1 + λx) c µ 1 x α+1 e -xs/µ , λ ≥ 0, t ≥ 0,
and let

L t = X t - cΓ(2 -α) α(α -1)µ α t α , t ≥ 0 and R t = L t -inf 0≤s≤t L s t ≥ 0. (2.5) Then n -α/(α+1) (|M n 1 |, |M n 2 |, . . .) d -→ (ζ 1 , ζ 2 , . . .)
as n → ∞ in 2 , where (ζ 1 , ζ 2 , . . .) are the lengths of the excursions above 0 of the process

(R t ) t≥0 .
The sequences (ζ 1 , ζ 2 , . . .) appearing in Theorem 2.2.1 must, of course, have the same distributions as the lengths of the excursions above 0 of the processes (R t , t ≥ 0) from the Introduction. Indeed, the processes L defined in (2.4) and (2.5) have the same distributions as those defined at (2.2) and (2.1), respectively. We prove this in Proposition 2.3.2 below.

Joseph [START_REF] Joseph | The component sizes of a critical random graph with given degree sequence[END_REF] conjectures that Theorem 2.2.1 (ii) should also hold with M n (ν) replaced by G n (ν). We show in the sequel (Section 2.5.3) that this is indeed true (this has been proved independently by Dhara, van der Hofstad, van Leeuwaarden and Sen [START_REF] Dhara | Heavy-tailed configuration models at criticality[END_REF]). In consequence, all of our scaling limit results hold interchangeably for G n (ν) and M n (ν).

The common structure exhibited by the two parts of Theorem 2.2.1 is no coincidence. In both cases, the proof proceeds via an exploration of the graph similar to the one described earlier.

As outlined above, locally, the components resemble critical branching processes. Since the components have small surplus, the excursions of the stack-size process above 0 approximately encode the component sizes. Moreover, the stack-size process behaves approximately like a reflected random walk. A weak convergence result for the stack-size process then yields the convergence of the component sizes.

Riordan [START_REF] Riordan | The phase transition in the configuration model[END_REF], in fact, proves a more refined version of Theorem 2.2.1 (i), but under the (non-optimal) assumption that the degrees are bounded. Firstly, his results are stated for a uniform random graph with a given n-dependent deterministic degree sequence (d

(n) i ) i≥1
, where the moment conditions on D 1 are replaced by appropriate convergence results for the moments of the degree of a uniformly chosen vertex. In particular, he is able to consider the components anywhere in the critical window, rather than precisely at θ = 1. (We refer the reader to [START_REF] Riordan | The phase transition in the configuration model[END_REF] for the details.) Secondly, he takes account also of the surplus of each component. Jointly with the convergence of the rescaled component sizes, he shows that

(s(G n 1 ), s(G n 2 ), . . .) d -→ (M 1 , M 2 , . . .)
for a non-trivial random sequence (M 1 , M 2 , . . .) ∈ Z N + . The sequence (M 1 , M 2 , . . .) is again obtained using the process R in (2.4): on top of the graph of the random function R, superpose a Poisson point process of intensity 1/µ in the plane. Then M i is the number of points falling in the area beneath the excursion ε i and above the x-axis, for i ≥ 1.

The first result of this kind was proved by Aldous [START_REF] Aldous | Brownian excursions, critical random graphs and the multiplicative coalescent[END_REF] Theorem 2.2.2 (Aldous [START_REF] Aldous | Brownian excursions, critical random graphs and the multiplicative coalescent[END_REF]). As n → ∞,

n -2/3 (|G ER,n 1 |, |G ER,n 2 |, . . .), (s(G ER,n 1 ), s(G ER,n 1 ), . . .) d -→ ((ζ 1 , ζ 2 , . . .), (M 1 , M 2 , . . .))
where the convergence is in 2 for the component sizes and in the sense of the product topology for the surpluses.

The limit is the same as in Theorem 2.2.1 (i) in the case β = µ = 1. This should be intuitively unsurprising, since the vertex degrees in the Erdős-Rényi random graph approximately behave like i.i.d. Poisson(1) random variables, for which Theorem 2.2.1 would apply with β = µ = 1.

(Aldous' theorem in fact treats the whole critical window, i.e. G(n, 1/n + λn -4/3 ) for λ ∈ R.

The effect is to introduce an extra drift of λt into the process L; we omit the very similar details for the sake of brevity.)

Branching processes and their metric space scaling limits

As alluded to above, the components of our critical random graphs behave approximately like critical branching process trees. It will be useful to spend a little time now exploring what happens in the true branching process setting, since what we do later will be analogous. Suppose that we take a sequence of i.i.d. Galton-Watson trees, with offspring distribution represented by some non-negative random variable Y with E [Y ] = 1 and P (Y = 1) < 1. (This entails that each of the trees has finite size almost surely.) We use the standard encoding of this forest in terms of its Lukasiewicz path or depth-first walk, given by S(0) = 0 and Le Gall [95] or Duquesne and Le Gall [START_REF] Duquesne | Random trees, Lévy processes and spatial branching processes[END_REF] for more details). Here, as usual, we explore the vertices of the forest in depth-first order, and Y i is the number of children of the ith vertex that we visit; these get added to the stack to await processing. The stack-size process is essentially a reflected version of S, given by (1 + S(k) -min 0≤j≤k S(j)) k≥0 . It is straightforward to see that the individual trees correspond to excursions above the running minimum of (S(k)) k≥0 ; it is technically easier to work with the depth-first walk than with the stack-size process, since S it is an unreflected random walk. An even more convenient encoding of the forest is given by the height process, which tracks the generation of the successive vertices listed in depth-first order.

S(k) = k i=1 (Y i -1) for k ≥ 1 (see
(It is, however, considerably harder to understand its distribution.) In terms of the depth-first walk, the height process (G(n)) n≥0 is defined by G(0) = 0 and

G(n) := # j ∈ {0, 1, . . . , n -1} : S(j) = min j≤k≤n S(k) .
(2.6)

The different trees now correspond to excursions above 0 of G.

The following generalised functional central limit theorem indicates some of the possible scaling limits for S in this setting (see, for example, Theorem 3.7.2 of Durrett [START_REF] Durrett | Probability: theory and examples[END_REF]).

Theorem 2.2.3. (i) Suppose that E [Y ] = σ 2 < ∞. Then n -1/2 (S( nt ), t ≥ 0) d -→ σ(B t , t ≥ 0) as n → ∞, in D(R + , R)
, where B is a standard Brownian motion.

(ii) Suppose that lim k→∞ k α+1 P (Y = k) = c for some constant c > 0 and some α ∈ (1, 2).

Then

n -1/α (S( nt ), t ≥ 0) d -→ (L t , t ≥ 0) as n → ∞, in D(R + , R),
where L is a spectrally positive α-stable Lévy process, with Laplace transform

E [exp(-λL t )] = exp cΓ(2 -α) α(α -1) λ α t , λ ≥ 0, t ≥ 0.
We now turn to the behaviour of the height process. In the Brownian case, this turns out to be asymptotically the same as that of the reflected depth-first walk, up to a scaling constant.

In the stable case, however, matters are a little more complicated. Consider the α-stable Lévy process L. Chapter 1 of Duquesne & Le Gall [START_REF] Duquesne | Random trees, Lévy processes and spatial branching processes[END_REF] shows that it is possible to make sense of a corresponding continuous height process, defined as follows. First, for 0

≤ s ≤ t, let L(t) s = L t -L (t-s)-and let M (t) s = sup 0≤r≤s L(t) r .
Then define H t to be the local time at level 0 of the process L(t) -M (t) . We may choose the normalization in such a way that

H t = lim ↓0 1 t 0 1 { M (t) s - L(t) s ≤ } ds (2.7)
in probability. Theorem 1.4.3 of [START_REF] Duquesne | Random trees, Lévy processes and spatial branching processes[END_REF] shows that H has continuous sample paths with probability 1, and so we may (and will) work with a continuous version in the sequel. Corollary 2.5.1 of [START_REF] Duquesne | Random trees, Lévy processes and spatial branching processes[END_REF] entails the following joint convergences.

Theorem 2.2.4 (Duquesne & Le Gall [70]). (i) Suppose that E [Y ] = σ 2 < ∞. Then (n -1/2 S( nt ), n -1/2 G( nt ), t ≥ 0) d -→ σB t , 2 σ B t -inf 0≤s≤t B s , t ≥ 0 . as n → ∞ in D(R + , R 2 ).
(ii) Suppose that lim k→∞ k α+1 P (Y = k) = c for some constant c > 0 and some α ∈ (1, 2).

Then we have

n -1/α S( nt ), n -(α-1)/α G( nt ), t ≥ 0 d -→ (L t , H t , t ≥ 0) as n → ∞ in D(R + , R 2 ).
There is also a conditional version of Theorem 2.2.4 for the depth-first walk S n and height process G n of a single Galton-Watson tree, conditioned to have total progeny n. (Let us assume that P (Y = k) > 0 for all k ≥ 0, so that the event of having total progeny n has positive probability for all n; this is not really necessary, but will facilitate the statement of the theorem.)

Theorem 2.2.5. (i) (Marckert & Mokkadem [107]). Suppose that E [Y ] = σ 2 < ∞. Then (n -1/2 S n ( nt ), n -1/2 G n ( nt ), 0 ≤ t ≤ 1) d -→ σe(t), 2 σ e(t), 0 ≤ t ≤ 1 , as n → ∞ in D([0, 1], R 2 )
, where e is a standard Brownian excursion.

(ii) (Duquesne [START_REF] Duquesne | A limit theorem for the contour process of conditioned Galton-Watson trees[END_REF]). Suppose that lim k→∞ k α+1 P (Y = k) = c for some constant c > 0 and some α ∈ (1, 2). Then we have

n -1/α S n ( nt ), n -(α-1)/α G n ( nt ), t ≥ 0 d -→ (e(t), h(t), 0 ≤ t ≤ 1) as n → ∞ in D([0, 1], R 2 )
, where e is a normalised excursion of L and h is the corresponding normalised excursion of H.

We now describe briefly how a limiting height process excursion may be used to define a limit R-tree (and the reader to the survey paper of Le Gall [95] for more details). Suppose

first that h : [0, ζ h ] → R + is any continuous function such that h(0) = h(ζ h ) = 0. Define a pseudo-metric on [0, 1] via d h (x, y) = h(x) + h(y) -2 min x∧y≤z≤x∨y h(z), x, y ∈ [0, ζ h ]. Define an equivalence relation ∼ on [0, ζ h ] by declaring x ∼ y if d h (x, y) = 0. Now let T h = [0, ζ h ]
/∼ and endow it with the distance d h in order to obtain a metric space (which is compact:

one checks easily that it is a Hausdorff space, and that it is sequentially compact). Then

(T h , d h ) is the R-tree encoded by h. Write p h : [0, ζ h ] → T h
for the canonical projection. We may additionally endow (T h , d h ) with a natural "uniform" measure µ h having total mass ζ h , obtained as the push-forward of the Lebesgue measure on [0, ζ h ] onto the tree (and concentrated on the leaves of T h ). Write M for the space of compact metric spaces each endowed with a finite (non-negative) Borel measure, up to measure-preserving isometry. We equip M with the Gromov-Hausdorff-Prokhorov distance d GHP , defined as follows. (See Section 2.1 of [START_REF] Addario-Berry | The scaling limit of the minimum spanning tree of the complete graph[END_REF] for more details and proofs of the results claimed below, as well as further references to the literature.)

Let (X, d, µ) and (X , d , µ ) be elements of M. We say that C is a correspondence between X and X if C ⊆ X × X and, whenever x ∈ X, there exists x ∈ X such that (x, x ) ∈ C and vice versa. The distortion of the correspondence

C is dist(C) := sup{|d(x 1 , x 2 ) -d (x 1 , x 2 )| : (x 1 , x 1 ), (x 2 , x 2 ) ∈ C}.
Write C(X, X ) for the set of correspondences between X and X . Write M (X, X ) for the set of non-negative Borel measures on X × X . Write p and p for the canonical projections from X × X to X and X respectively. We define the discrepancy of π ∈ M (X, X ) with respect to µ and µ to be

disc(π; µ, µ ) = µ -p * π + µ -p * π ,
where ν is the total variation of the signed measure ν. We define the Gromov-Hausdorff-

Prokhorov distance by d GHP ((X, d, µ), (X , d , µ )) := inf C∈C(X,X ), π∈M (X,X ) 1 2 dist(C) ∨ disc(π; µ, µ ) ∨ π(C c ) .
Then (M, d GHP ) is a Polish space. We observe a very useful upper bound for the Gromov-Hausdorff-Prokhorov distance between R-trees encoded by continuous excursions:

d GHP ((T h , d h , µ h ), (T g , d g , µ g )) ≤ 2 max sup 0≤x≤ζ h ∧ζg |h(x) -g(x)|, sup ζ h ∧ζg<x≤ζ h h(x) + sup ζ h ∧ζg<x≤ζg g(x), 1 2 |ζ h -ζ g | , (2.8)
The random R-trees encoded by 2e for α = 2 and h for α ∈ (1, 2) are known as the Brownian continuum random tree, for which we will write (T (2) , d (2) ) (with mass measure µ (2) ), and the α-stable tree, for which we will write (T (α) , d (α) ) (with mass measure µ (α) ), respectively. (Note that because of our choice of Laplace exponent, this is a constant multiple of the usual α-stable tree.)

Let T n be our Galton-Watson tree conditioned to have size n. The natural way to take a scaling limit of the tree itself is to consider it as a metric space using the graph distance d n .

Create a (probability) measure µ n by assigning mass 1/n to each vertex of T n . An important consequence of Theorem 2.2.5 and the bound (2.8) is the following. 2) , d (2) , µ (2) .

Theorem 2.2.6. (i) Suppose that E [Y ] = σ 2 < ∞. Then T n , σ √ n d n , µ n d -→ T (
as n → ∞ for the Gromov-Hausdorff-Prokhorov topology.

(ii) Suppose that lim k→∞ k α+1 P (Y = k) = c for some constant c > 0 and some α ∈ (1, 2).

Then

T n , n -(α-1)/α d n , µ n d -→ T (α) , d (α) , µ (α) ,
as n → ∞ for the Gromov-Hausdorff-Prokhorov topology.

Returning now to the setting of Theorem 2.2.4, the excursions of the limiting height process can heuristically be thought of as defining a forest of random R-trees. (Since there is neither a shortest nor a longest excursion, there is no sensible way to list these trees. For definiteness, let us instead think of restricting to an interval [0, t] in time, for which there is a longest excursion, and then list the trees in decreasing order of size.) Using the scaling properties of the underlying Lévy processes, these consist of randomly rescaled copies of the Brownian continuum random tree in case (i) or α-stable trees in case (ii), respectively. We refer to these as the Brownian and stable forests.

Our method

Our approach to proving Theorem 2.1.1 is as follows. Firstly, we show that the law of the depth-first walk of the graph is (up to a small error) absolutely continuous with respect to that of a centred random walk which is in the domain of attraction of the spectrally positive α-stable Lévy process L. This enables us to give an alternative (and perhaps more "conceptual") proof of Theorem 2.2.1. We also show that the convergence of the depth-first walk can be boosted to a joint convergence with the corresponding height process. The joint convergence of this pair of coding functions in the setting of a sequence of i.i.d. Galton-Watson trees, Theorem 2.2.4, is a highly non-trivial result. The corresponding result in our setting, however, follows relatively straightforwardly from Theorem 2.2.4 via absolute continuity and some integrability lemmas.

The height process is the key ingredient in proving a metric space convergence for these graphs, and allows us to show the convergence of a spanning forest of our graph. In order to obtain the full metric space convergence, we must also control the edges which form cycles. We call these back-edges. We prove that the number of back-edges edges in the "large components" is a tight quantity. This firstly allows us to resolve Conjecture 8.5 of Joseph [START_REF] Joseph | The component sizes of a critical random graph with given degree sequence[END_REF], by showing that all of the above results extend to the case where the multigraph is conditioned to be simple.

Secondly, we are able to capture the full graph structure by tracking also the locations of these back-edges in the spanning forest. We finally show that all of these quantities can be passed through to the limit in such a way that we get convergence to the stable graph.

Related work on scaling limits of critical random graphs, universality, and open problems

This paper is a contribution to a now extensive literature on scaling limits of critical random graphs. In this section, we will place our work in context by giving a summary of related results.

As mentioned above, the first critical random graph to be studied from the perspective of scaling limits was the Erdős-Rényi random graph, in the work of Aldous [START_REF] Aldous | Brownian excursions, critical random graphs and the multiplicative coalescent[END_REF], who considered both component sizes and surpluses. Addario-Berry, Broutin and Goldschmidt [START_REF] Addario-Berry | The continuum limit of critical random graphs[END_REF][START_REF] Addario-Berry | Critical random graphs: limiting constructions and distributional properties[END_REF] built on Several models have been proved to lie in the same universality class as the Erdős-Rényi random graph, which is roughly characterised by the property that the degree of a uniformly chosen vertex converges to a limit with finite third moment. Already in [START_REF] Aldous | Brownian excursions, critical random graphs and the multiplicative coalescent[END_REF], Aldous had, in fact, also considered another model: a rank-one inhomogeneous random graph in which, for each n ≥ 1, we are given a sequence of weights w (n) = (w

(n) 1 , w (n) 2 , . . . , w (n)
n ) and each pair of vertices {i, j} is connected independently with probability 1-exp(-q (n) w

(n) i w (n) j )
, for 1 ≤ i, j ≤ n. Such graphs may be constructed dynamically by assigning an exponential clock to each potential edge and including the edge when the clock rings. It is straightforward to see that, in consequence, the component sizes then evolve according to the multiplicative coalescent. In his Proposition 4, Aldous gave conditions on sequences (w (n) , q (n) ) n≥1 for which one gets convergence of the rescaled component weights to the same limit as for the component sizes in the Erdős-Rényi case. These results were generalised by Bhamidi, van der Hofstad and van Leeuwaarden [START_REF] Bhamidi | Scaling limits for critical inhomogeneous random graphs with finite third moments[END_REF] to give convergence of the rescaled component sizes in the Norros-Reittu model [START_REF] Norros | On a conditionally Poissonian graph process[END_REF] (for which q (n) above is replaced by 1/ n i=1 w

(n) i ) to the sequence (ζ 1 , ζ 2 , . . .) appearing in Theorem 2.2.1 (i), with a general β and µ. The convergence of the component sizes was also treated in a similar setting but with i.i.d. vertex weights by Turova [START_REF] Turova | Diffusion approximation for the components in critical inhomogeneous random graphs of rank 1[END_REF].

Nachmias and Peres [START_REF] Nachmias | Critical percolation on random regular graphs[END_REF] proved the convergence of the rescaled component sizes for critical percolation on a random d-regular graph, for d ≥ 3, to the excursion lengths of the reflected Brownian motion with parabolic drift for appropriate β and µ. As we have already detailed above, Riordan [START_REF] Riordan | The phase transition in the configuration model[END_REF] and Joseph [START_REF] Joseph | The component sizes of a critical random graph with given degree sequence[END_REF] proved analogous results for the critical configuration model with asymptotic degree distribution possessing finite third moment, with Riordan treating the surpluses as well as the component sizes. Dhara, van der Hofstad, van Leeuwaarden and Sen [START_REF] Dhara | Critical window for the configuration model: finite third moment degrees[END_REF] improved these results to give the scaling limit of the sizes and the surpluses under a minimal set of conditions on the (deterministic) vertex degrees, which essentially amount to the convergence in distribution of the degree of a uniform vertex, along with the convergence of its third moment. In a somewhat different direction, Bhamidi, Budhiraja and Wang [START_REF] Bhamidi | The augmented multiplicative coalescent, bounded size rules and critical dynamics of random graphs[END_REF] considered critical random graphs generated by Achlioptas processes [START_REF] Bohman | Avoiding a giant component[END_REF] with bounded size rules. They again proved convergence of the rescaled component sizes, along with the surpluses, as a process evolving in the critical window, building on results for the barely subcritical regime proved in [START_REF] Bhamidi | Bounded-size rules: the barely subcritical regime[END_REF]. Federico [START_REF] Federico | Critical scaling limits of the random intersection graph[END_REF] has recently proved a scaling limit for the component sizes of the random intersection graph which is related to that of the Erdős-Rényi model.

Turning now to the metric structure, very general results concerning the domain of attraction of the Brownian graph have been proved by Bhamidi, Broutin, Sen and Wang [START_REF] Bhamidi | Scaling limits of random graph models at criticality: Universality and the basin of attraction of the Erdős-Rényi random graph[END_REF], building on earlier work for the Norros-Reittu model by Bhamidi, Sen and Wang [START_REF] Bhamidi | Continuum limit of inhomogenous random graphs[END_REF]. In particular, [START_REF] Bhamidi | Scaling limits of random graph models at criticality: Universality and the basin of attraction of the Erdős-Rényi random graph[END_REF] gives a set of sufficient conditions under which one obtains convergence in the Gromov-Hausdorff-Prokhorov sense to the Brownian graph. It is also demonstrated in that paper that these conditions are fulfilled for certain critical inhomogeneous random graphs (of the stochastic block model variety), and for critical percolation on a supercritical configuration model with finite third moment degree distribution. A crucial role is played by dynamical constructions of the graphs in question, and by the idea that some pertinent statistic of the evolving graph may be well approximated by the multiplicative coalescent. Bhamidi and Sen [START_REF] Bhamidi | Geometry of the vacant set left by random walk on random graphs, wright's constants, and critical random graphs with prescribed degrees[END_REF] later proved convergence to the Brownian graph for the critical configuration model (rather than for percolation on the supercritical case) in the Gromov-Hausdorff-Prokhorov sense, under the same set of minimal conditions as in [START_REF] Dhara | Critical window for the configuration model: finite third moment degrees[END_REF], and used it to deduce geometric properties of the vacant set left by a random walk on various models of graph.

We have mentioned a few examples of critical percolation on graphs for which the resulting cluster sizes lie in the universality class of the Brownian graph. This is expected to be true in much greater generality: for a wide variety of finite base graphs which are sufficiently "high dimensional", although the percolation critical point will be model-dependent, the behaviour in the vicinity of that critical point should essentially be the same as in the mean-field case of percolation on the complete graph, i.e. the Erdős-Rényi model. We refer the reader to the book of Heydenreich and van der Hofstad [START_REF] Heydenreich | Progress in high-dimensional percolation and random graphs[END_REF] for an in-depth discussion of this universality conjecture.

The results of the present paper primarily concern cases where the degree of a uniformly chosen vertex has infinite third moment and a power-law tail with exponent α + 1 ∈ (2, 3), and in this context the picture is more complicated. As in the Brownian case, it is to be expected that, as long as the degree of a uniformly chosen vertex has the right properties, we should get the same scaling limit irrespective of precisely which model we consider. It is technically more straightforward to consider rank-one inhomogeneous random graphs than the configuration model. In the context of component sizes, this was first done by Aldous and Limic [START_REF] Aldous | The entrance boundary of the multiplicative coalescent[END_REF] for the rank-one model treated by Aldous in [START_REF] Aldous | Brownian excursions, critical random graphs and the multiplicative coalescent[END_REF] but with appropriately altered conditions on the weight sequence. These different conditions correspond to different extremal entrance laws for the multiplicative coalescent. Aldous and Limic obtained the analogue of Theorem 2.2.1 for the component weights, where the limit is now given by the ordered lengths of the excursions above the running infimum of the thinned Lévy process,

  κB t + λt + i≥1 ϑ i (1 {E i ≤t} -ϑ i t)   t≥0 , (2.9)
where E i ∼ Exp(ϑ i ) for each i ≥ 1, κ ≥ 0, λ ∈ R, and ϑ 1 ≥ ϑ 2 ≥ . . . ≥ 0 is a sequence such that i≥1 ϑ 3 i < ∞ and, if κ = 0, also i≥1 ϑ 2 i = ∞. This was extended in the κ = 0 case by Bhamidi, van der Hofstad and van Leeuwaarden [START_REF] Bhamidi | Novel scaling limits for critical inhomogeneous random graphs[END_REF] to give the convergence of the component sizes for the Norros-Reittu model with a specific weight sequence. Heuristically, the choice of entrance law for the multiplicative coalescent is determined by the properties of the barely subcritical graph. For the configuration model, the first work in the power-law setting was that of Joseph [START_REF] Joseph | The component sizes of a critical random graph with given degree sequence[END_REF] detailed above for the case of i.i.d. degrees. The convergence of the sizes and surpluses for much more general (deterministic) degree sequences were treated by Dhara, van der Hofstad, van Leeuwaarden and Sen [START_REF] Dhara | Heavy-tailed configuration models at criticality[END_REF], with the possible scaling limits being driven by the same κ = 0 thinned Lévy processes as in the Norros-Reittu model. A significant challenge in obtaining a metric space convergence in the power-law setting is that one often does not have direct access to a scaling limit result for the height process of the spanning forest discovered by a depth-first exploration. (That we have such a result in the case of i.i.d. degrees is of considerable help to us.) The first metric space scaling limit in the power-law setting was obtained by Bhamidi, van der Hofstad and Sen [START_REF] Bhamidi | The multiplicative coalescent, inhomogeneous continuum random trees, and new universality classes for critical random graphs[END_REF] for the Norros-Reittu model with the specific weight sequence used in [START_REF] Bhamidi | Novel scaling limits for critical inhomogeneous random graphs[END_REF]. Here, the convergence is in the product Gromov-Hausdorff-Prokhorov sense, and the limit object is constructed by making vertex identifications in tilted inhomogeneous continuum random trees (of the sort introduced by Aldous and Pitman in [START_REF] Aldous | Inhomogeneous continuum random trees and the entrance boundary of the additive coalescent[END_REF]).

Broutin, Duquesne and Wang [START_REF] Broutin | Limits of multiplicative inhomogeneous random graphs and lévy trees: The continuum graphs[END_REF][START_REF] Broutin | Limits of multiplicative inhomogeneous random graphs and lévy trees: Limit theorems[END_REF] use a very different approach in order to prove a unified metric space scaling limit for the Norros-Reittu model with very general weight sequences.

They are able to treat situations where the scaling limit of the depth-first walk is a thinned Lévy process for any κ ≥ 0, λ ∈ R and sequence (ϑ 1 , ϑ 2 , . . .), recovering the generality of Aldous and Limic's paper [START_REF] Aldous | The entrance boundary of the multiplicative coalescent[END_REF]. They embed spanning subtrees of the components of the graph inside a forest of Galton-Watson trees, and exploit the convergence of this (bigger) forest on rescaling to the sequence of R-trees encoded by a Lévy process (as in Duquesne and Le Gall [START_REF] Duquesne | Random trees, Lévy processes and spatial branching processes[END_REF]), whose height process also converges. This enables them to obtain the convergence of the height process of the true spanning forest in the Gromov-Hausdorff-Prokhorov sense; the surplus edges can also be tracked, in order to obtain a product Gromov-Hausdorff-Prokhorov convergence of the whole ordered sequence of graph components.

Let us finally turn to the work of Bhamidi, Dhara, van der Hofstad and Sen [START_REF] Bhamidi | Universality for critical heavy-tailed network models: Metric structure of maximal components[END_REF], who proved a metric space scaling limit analogous to that of [START_REF] Bhamidi | The multiplicative coalescent, inhomogeneous continuum random trees, and new universality classes for critical random graphs[END_REF] for critical percolation on a supercritical configuration model. Among the settings studied so far, theirs is the closest to ours, although the technical content is very different (their work relies on the analysis of some susceptibility functions in the barely subcritical regime, while we do not study the latter). We will describe it precisely, in order to provide a comparison with Theorem 2.1.1. They take a (deterministic)

degree sequence d n 1 , d n 2 , . . . , d n n such that n i=1 d n i is even and, if D n is the degree of a typical vertex, then (i) n -1/(α+1) d n i → ϑ i as n → ∞ for each i ≥ 1, where ϑ 1 ≥ ϑ 2 ≥ . . . ≥ 0 is such that i≥1 ϑ 3 i < ∞ but i≥1 ϑ 2 i = ∞; (
(d n i ) 3 = 0. Let θ n = E [D n (D n -1)] /E [D n ]
(which, by (ii), converges to θ > 1). They then perform percolation at parameter

p n (λ) = 1 θ n + λn -(α-1)/(α+1) ,
for some λ ∈ R, which yields a graph in the critical window. In this setting, their Theorem 2.2 is the precise analogue of our Theorem 2.1.1 but with the convergence in the product Gromov-weak topology and with the limit object ((G i , d i , µ i ), i ≥ 1) constructed by making vertex identifications in the tilted inhomogeneous continuum random trees mentioned above. (We understand that this convergence will be improved to a product Gromov-Hausdorff-Prokhorov convergence under an extra technical condition in work in preparation [START_REF] Bhamidi | Universality for critical heavy-tailed network models: Gromov-Hausdorff-Prohorov convergence[END_REF].) A precise description of the limit object would be too lengthy to undertake here, but it is instructive to compare the scaling limit of the depth-first walk in the two settings. For us, this is the measure-changed stable Lévy process L; for Bhamidi, Dhara, van der Hofstad and Sen it is the thinned Lévy process in (2.9)

with κ = 0. To make the connection between the results, suppose now we take D such that similarly to the order statistics of a i.i.d. sample which satisfy the conditions of our theorem. (In particular, using results of Janson [START_REF] Janson | On percolation in random graphs with given vertex degrees[END_REF], such mild dependence can be shown to have a negligible effect on the properties of the graph.) Then it should be the case that Bhamidi, Dhara, van der Hofstad and Sen's limit object is the same as the stable graph. In particular, the process defined at (2.9) with κ = 0 and λ = 0 should, for this particular random sequence (ϑ 1 , ϑ 2 , . . .), have the same law as L. Similarly, if it is the case that the scaling limit is the same as for the analogous inhomogeneous random graph setting, then our limit object should also coincide with a particular annealed version of that of Broutin, Duquesne and Wang [START_REF] Broutin | Limits of multiplicative inhomogeneous random graphs and lévy trees: The continuum graphs[END_REF][START_REF] Broutin | Limits of multiplicative inhomogeneous random graphs and lévy trees: Limit theorems[END_REF].

P (D = 1) > 0, E [D] = µ, E [D(D -1)] /E [D] = θ > 1 and P (D = k) ∼ ck -α-2 . Let
It is perhaps worth emphasising that, in contrast to the bulk of the other papers cited here, the multiplicative coalescent (and its relationship to percolation) appears nowhere in our proofs, and is conceptually absent from our approach.

Let us now give a list of open problems and conjectures arising from our work.

(i) Prove that the stable graph is, indeed, an annealed version of the limit object from [START_REF] Bhamidi | Universality for critical heavy-tailed network models: Metric structure of maximal components[END_REF] or [START_REF] Broutin | Limits of multiplicative inhomogeneous random graphs and lévy trees: The continuum graphs[END_REF][START_REF] Broutin | Limits of multiplicative inhomogeneous random graphs and lévy trees: Limit theorems[END_REF].

(ii) The convergence in our main theorem occurs with respect to the product Gromov-Hausdorff-Prokhorov topology. For sequences A = (A 1 , A 2 , . . .) and B = (B 1 , B 2 , . . .) of compact measured metric spaces, we may obtain stronger topologies using the distances

dist p (A, B) =   i≥1 d GHP (A i , B i ) p   1/p (2.10)
for p ≥ 1. For the Erdős-Rényi random graph, the analogous convergence to the Brownian graph holds in the sense of dist 4 . We conjecture that it should be possible to improve our main result for α ∈ (1, 2] to a convergence in the sense of dist 2α/(α-1) .

(iii) One reason for wanting to prove such a result is that it would imply the convergence in distribution of the diameter of the whole graph (i.e. the largest distance between any two vertices in the same component). In order to prove convergence in dist 2α/(α-1) , we would need bounds on the component diameters in terms of powers of their sizes for the whole graph (we can do this for the parts explored up to time O(n α/(α+1) ) using the methods of this paper, but that is not sufficient). A finer understanding of the barely subcritical regime for the configuration model would presumably help to resolve this issue.

(iv) As shown in Proposition 2.6.2, the measure change used in this paper makes sense for a large family of spectrally positive Lévy processes (see Section 2.6.1 for the precise conditions). Any such Lévy process may be intuitively thought of as encoding a forest of continuum trees, although the analogue of Theorem 2.2.4 holds only with the imposition of extra regularity conditions (see Theorem 2.3.1 of [START_REF] Duquesne | Random trees, Lévy processes and spatial branching processes[END_REF]). Is it possible to find a sequence of degree distributions (ν n ) n≥1 , depending now on n and such that the regularity conditions are satisfied, so that if we take

D (n) 1 , . . . , D (n)
n to be i.i.d. random variables with distribution ν n then we get convergence of our discrete measure change to its continuum analogue? Or does the self-similarity inherent in the Brownian and stable settings play a key role? If a generalisation to the Lévy case is possible, what is the connection to thinned Lévy processes, or to the approach of Broutin, Duquesne and Wang [START_REF] Broutin | Limits of multiplicative inhomogeneous random graphs and lévy trees: The continuum graphs[END_REF][START_REF] Broutin | Limits of multiplicative inhomogeneous random graphs and lévy trees: Limit theorems[END_REF]?

For simplicity we have restricted our attention in this paper to the case where θ(ν) = 1.

The critical window is obtained by considering the situation where the degrees D n 1 , . . . , D n n are i.i.d. but now with some n-dependent degree distribution ν n , such that E [D n 1 ] → µ for some µ as n → ∞, θ(ν n ) = 1 + λn -(α-1)/(α+1) and P (D n 1 = k) ∼ ck -α-2 as k → ∞, for some fixed λ ∈ R. This regime is the subject of work in progress by Serte Donderwinkel.

Plan of the rest of the paper

In Section 2.3, we study the process L which gives rise to the stable graph. In particular, we establish the local absolute continuity relation between L and L, and present some results in excursion theory. The section concludes with the proof of Theorem 2.1.2. In Section 2.4, we study a forest which is closely related to M n (ν). We show that the absolute continuity relation (2.1), (2.2) may be seen as the limit of a discrete measure change between the degrees in the order we observe them when we explore this forest in a depth-first manner and an i.i.d. sequence of random variables whose law is the size-biased version of ν. The main result of this section is Theorem 2.4.1, which gives the joint convergence of the depth-first walk and height process of the discrete forest to their continuum counterparts. In Section 2.5, we explore the multigraph M n (ν) in a depth-first manner, and record its structure via coding functions close to those of the forest in Section 2.4, and show their convergence in law. We also deal with the occurrence of the back-edges, and prove that M n (ν) and G n (ν) cannot have different scaling limits. We must then extract the individual components of the graph in decreasing order of size, and prove that their individual coding functions converge. We adapt an approach of Aldous [START_REF] Aldous | Brownian excursions, critical random graphs and the multiplicative coalescent[END_REF] using size-biased point processes; this is perhaps the most technical part of the paper. Section 2.5 culminates in the proof of Theorem 2.1.1. The Appendix contains various technical results. In particular, in Section 2.6.1 we give a formulation of the measure change in (2.1) and (2.2) for a general class of Lévy processes, which may be of independent interest. In Section 2.6.4, we show the natural result that a single component of G n (ν) or M n (ν) conditioned to have size xn α/(α+1) has a component of the α-stable graph of size x as its scaling limit.

2.3

The limit object: the stable graph 2.3.1 An absolute continuity relation for spectrally positive α-stable Lévy processes We begin by discussing the coding function R discovered by Joseph [START_REF] Joseph | The component sizes of a critical random graph with given degree sequence[END_REF], which was defined in (2.5). Fix α ∈ (1, 2), µ ∈ (1, 2) and c > 0. Recall that L is the spectrally positive α-stable Lévy process having Lévy measure π(dx) = c µ x -(α+1) dx. This process has Laplace transform

E [exp(-λL t )] = exp(tΨ(λ)), λ ≥ 0, t ≥ 0,
where

Ψ(λ) = ∞ 0 c µ x -(α+1) dx(e -λx -1 + λx) = C α µ λ α , with C α = cΓ(2 -α) α(α -1) .
Recall also that X is the unique process with independent increments such that

E [exp(-λX t )] = exp t 0 ds ∞ 0 dx(e -λx -1 + λx) c µ 1 x α+1 e -xs/µ , λ ≥ 0, t ≥ 0. Let A t = -C α t α µ α
and define

L t = X t + A t .
We observe that X is a martingale and A is a finite-variation process, so this is, in fact, the Doob-Meyer decomposition of the process L.

Proposition 2.3.1. We have

L t → -∞ a.s.
as t → ∞.

Proof. Lemma B.3 of Joseph [START_REF] Joseph | The component sizes of a critical random graph with given degree sequence[END_REF] gives the convergence in probability; we adapt his argument.

Since A t is deterministic and tends to -∞, it will be sufficient to prove that lim sup t→∞ t -α X t = 0 a.s.

Consider a Poisson point process on R + × R + of intensity c µ x -(α+1) e -xs/µ ds dx, with points {(s, ∆ s )}. Let X (1) be the martingale arising from compensating the jumps of magnitude at most 1, formally defined as the → 0 limit of the family {X (1, ) , > 0} of processes given by

X (1, ) t = s≤t ∆ s 1 { <∆s<1)} - t 0 ds 1 dx c µ x -α e -xs/µ ,
and let

X (2) t = s≤t ∆ s 1 {∆s≥1} - t 0 ds ∞ 1 dx c µ x -α e -xs/µ . Then X t = X (1) t + X (2)
t . By Doob's L 2 -inequality, we have

E sup 0≤s≤t X (1) s 2 ≤ 4E (X (1)
t ) 2 = 4 t 0 ds 1 0 dx c µ x -(α-1) e -xs/µ .
The integral on the right-hand side is bounded above by Ct α-1 for some constant C > 0.

Indeed, for every s > 0,

1 0 dx c µ x -(α-1) e -xs/µ = s α-2 s/µ 0 c µ α-1 u -(α-1) e -u du ≤ cΓ(2 -α) µ α-1 s α-2 .
Hence, applying Markov's inequality, we get

P sup n-1<s≤n X (1) s > n (α+1)/2 ≤ C n 2 .
As this is summable in n, the Borel-Cantelli lemma gives that P sup n-1<s≤n

X (1) s > n (α+1)/2 i.o. = 0.

Since α > 1, it follows that lim sup t→∞ t -α X

(1) t = 0 a.s.

Turning now to X (2) , for all t ≥ 0 we have the straightforward bound sup t≥0

X (2) s ≤ s≥0 ∆ s 1 {∆s≥1} .
The right-hand side has expectation

∞ 0 ds ∞ 1 dx c µ x -α e -xs/µ = c ∞ 1 dx x -(α+1) ∞ 0 ds x µ e -xs/µ = c ∞ 1 x -(α+1) dx = c α .
This computation also entails that

inf t≥0 X (2) t ≥ - c α .
Hence by Markov's inequality, we have for all n ≥ 1:

P sup n-1<s≤n X (2) s > n (α+1)/2 ≤ c αn (α+1)/2 .
As for X (1) , the Borel-Cantelli lemma gives that lim sup t→∞ t -α X (2) t = 0 a.s.

The result follows.

The main purpose of this section is to expand considerably our understanding of the processes L and R. Our first new result says that the law of the process L is absolutely continuous with respect to the law of the Lévy process L on compact time-intervals. Proposition 2.3.2. For every t ≥ 0, we have the following absolute continuity relation: for every non-negative integrable functional f :

D([0, t], R) → R + , E f ( L s , 0 ≤ s ≤ t) = E exp - 1 µ t 0 sdL s -C α t α+1 (α + 1)µ α+1 f (L s , 0 ≤ s ≤ t) .
This proposition is a consequence of a more general change of measure for spectrally positive Lévy processes, Proposition 2.6.2, which is proved in the Appendix below.

In the Brownian case, we instead have L t = β µ B t , where B is a standard Brownian motion,

L t = β µ B t - β 2µ 2 t 2 , (2.11)
and Proposition 2.6.2 gives

E f ( L s , 0 ≤ s ≤ t) = E exp - 1 µ t 0 sdL s - β 6µ 3 t 3 f (L s , 0 ≤ s ≤ t) .
In order to harmonise notation, let us define (α+1)µ α+1 , t ≥ 0 is a martingale which is not uniformly integrable.)

C 2 := β/2,
Recall that H is the height process which corresponds to L. Then for any α ∈ (1, 2], we may define a pair ( L, H) of processes via change of measure as follows: for suitable test-functions

f : D([0, t], R) 2 → R, E f ( L u , H u , 0 ≤ u ≤ t) = E exp - 1 µ t 0 sdL s - C α t α+1 (α + 1)µ α+1 f (L u , H u , 0 ≤ u ≤ t) .

Excursion theory

We begin with some notation. Write D + (R + , R + ) for the space of càdlàg functions f : R + → R + with only positive jumps. We write E for the space of excursions, that is

E = {ε ∈ D + (R + , R + ) : ∃t > 0 s.t. ε(s) > 0 for s ∈ (0, t) and ε(s) = 0 for s ≥ t}.
For ε ∈ E, let ζ(ε) be the lifetime of ε, that is the smallest t such that ε(s) = 0 for s ≥ t. Let Let I t = inf 0≤s≤t L s . It is standard that the process -I acts as a local time at 0 for the reflected Lévy process (L t -inf 0≤s≤t L s , t ≥ 0) (see Chapter VII of Bertoin [START_REF] Bertoin | Lévy processes[END_REF] or Section 1.1.2 of Duquesne and Le Gall [START_REF] Duquesne | Random trees, Lévy processes and spatial branching processes[END_REF]). Indeed, we may decompose the path of the reflected process into excursions above 0. Now write σ = inf {t ≥ 0 : I t < -} , so that (σ , ≥ 0) is the inverse local time. We observe that σ is a stopping time for the (usual augmentation of the) natural filtration of (L t ) t≥0 . For ≥ 0, write

ε ( ) =    ( + L σ -+u , 0 ≤ u ≤ σ -σ -) if σ -σ -> 0 ∂ otherwise.
Then the following theorem is standard (see Theorem VII.1.1 of Bertoin [START_REF] Bertoin | Lévy processes[END_REF], converting from the spectrally negative case, or Miermont [START_REF] Miermont | Self-similar fragmentations derived from the stable tree. I. Splitting at heights[END_REF] for a convenient reference).

Theorem 2.3.4. The inverse local time process (σ , ≥ 0) is a stable subordinator of index 1/α and, more specifically, with Lévy measure

µ 1/α C 1/α α αΓ(1 -1/α) x -1-1/α dx.
Moreover, the point measure on R + × E given by s≥0:σs-σ s->0

δ (s,ε (s) ) (2.12)
is a Poisson random measure of intensity d ⊗ N(de), where the excursion measure N satisfies

N(ζ(e) ∈ dx) = µ 1/α C 1/α α αΓ(1 -1/α)
x -1-1/α dx.

Consider the excursions occurring before time σ . With probability 1, only finitely many of these are longer than η in duration for any η > 0. So, in particular, they may be listed in decreasing order of length as (ε

( ) i , i ≥ 1). Since L is self-similar, it is possible to make sense of normalised versions of N i.e. N (x) (•) = N(•|ζ(e) = x
), which are probability measures. (Again see Miermont [START_REF] Miermont | Self-similar fragmentations derived from the stable tree. I. Splitting at heights[END_REF] for more details.) For example, the law of e under N (x) is the same as the law of (x/ζ(e)) 1/α e(ζ(e)s/x), 0 ≤ s ≤ x under N(•|ζ(e) > η) for any fixed η > 0. In particular, we have that under N (x) , the rescaled excursion (x -1/α e(xu), 0 ≤ u ≤ 1) has the same law as e under N (1) . It follows that the excursions ε (s) appearing in (2.12) may be thought of in two parts: as their lengths ζ(ε (s) ) = σ s -σ s-and their normalised "shapes" e (s) := ζ(ε (s) ) -1/α ε (s) (ζ(ε (s) )u), 0 ≤ u ≤ 1 where, crucially, the collection of shapes (e (s) , s ≥ 0) is independent of the collection of excursion lengths (ζ(ε (s) ), s ≥ 0). We observe that the excursions of the Lévy process L above its running infimum and the excursions of the height process H are in one-to-one correspondence and have the same lengths.

In particular, we can make sense of an excursion of the height process h derived from e, under N or its conditioned versions. The scaling relation for the height process is that under N (x) the rescaled excursion (x -(α-1)/α h(xu), 0 ≤ u ≤ 1) has the same law as h under N (1) . The usual stable tree is encoded by (a scalar multiple of) a height process with the distribution of h under N (1) .

Much of this structure can be transferred into our setting, by absolute continuity. Recall that

R t = L t -inf 0≤s≤t L s , t ≥ 0.
We will make use of the following properties.

Lemma 2.3.5. The following statements hold almost surely.

(i) For each > 0, R has only finitely many excursions of length greater than or equal to .

(ii) The set {t : R t = 0} has Lebesgue measure 0.

(iii) If (l 1 , r 1 ) and (l 2 , r 2 ) are excursion-intervals of R and l 1 < l 2 , then L l 1 > L l 2 .

(iv) For a ≥ 0, let B a = {b > a : L b-= inf a≤s≤b L s }. Then B a does not intersect the set of jump times of L.

Proof. Part (i) is a consequence of Lemma B.3 of Joseph [START_REF] Joseph | The component sizes of a critical random graph with given degree sequence[END_REF]. For parts (ii), (iii) and (iv), we first argue that the claimed properties are almost surely true for the Lévy process L and then use absolute continuity to deduce them for L.

The analogues of both (ii) and (iii) are standard for L (see, for example, Chapter VII of Bertoin [START_REF] Bertoin | Lévy processes[END_REF]; indeed, these properties are necessary for Theorem 2.3.4 to work). It follows by absolute continuity that P (Leb({s ≤ t : R s = 0}) = 0) = 1 and

P L l 1 > L l 2 for all (l 1 , r 1 ), (l 2 , r 2 ) excursion-intervals of R with l 1 < l 2 ≤ t = 1,
for fixed t > 0. But then (ii) and (iii) follow by monotone convergence.

By the stationarity and independence of increments of L, it is sufficient to prove (iv) for a = 0. But this then follows from Corollary 1 of Rogers [START_REF] Rogers | A new identity for real levy processes[END_REF]. In particular, if we let J be the set of jump-times of L, by absolute continuity we get P (B a ∩ [0, t] ∩ J = ∅) = 0 for any t > 0.

By monotone convergence again, we obtain P (B a ∩ J = ∅) = 0.

Let I t = inf 0≤s≤t L s . As for the reflected stable process, we have that -I acts as a local time at 0 for R. We write ( σ , ≥ 0) with σ = inf{t > 0 : I t < -} for the inverse local time, ( ε ( ) , ≥ 0) for the collection of excursions above 0, indexed by local time (with ε ( ) = ∂ if σ -σ -= 0), and ( e ( ) , ≥ 0) for their shapes. In order to understand the laws of these quantities, we first need to prove two preliminary results, Lemma 2.3.6 and Proposition 2.3.7.

Lemma 2.3.6. Let α ∈ [START_REF] Abächerli | Local picture and level-set percolation of the gaussian free field on a large discrete torus[END_REF][START_REF] Abächerli | Level-set percolation for the gaussian free field on a transient tree[END_REF]. Then for any θ > 0,

E N (1) exp θ 1 0 e(t)dt < ∞.
Proof. This is well known in the α = 2 case; see, for example, Section 13 of Janson [START_REF] Janson | Brownian excursion area, Wright's constants in graph enumeration, and other Brownian areas[END_REF]. For α ∈ (1, 2), observe that

1 0 e(t)dt ≤ sup t∈[0,1]
e(t).

By Theorem 9 of Kortchemski [START_REF] Kortchemski | Sub-exponential tail bounds for conditioned stable Bienaymé-Galton-Watson trees[END_REF] (see also the discussion at the top of the 12th page), for any δ ∈ (0, α α-1 ), there exist constants C 1 , C 2 > 0 such that

E N (1) sup t∈[0,1] e(t) ≥ u ≤ C 1 exp(-C 2 u δ ),
for every u ≥ 0. (Note that since Kortchemski works with the Lévy process having Laplace exponent λ α , his normalised excursions are a constant scaling factor different from ours. But this changes the bound only by a constant.) Since we may take δ > 1, the result follows.

For t ≥ 0 write

Φ(t) := exp - 1 µ t 0 sdL s - C α t α+1 (α + 1)µ α+1 .
Proposition 2.3.7. Let α ∈ [START_REF] Abächerli | Local picture and level-set percolation of the gaussian free field on a large discrete torus[END_REF][START_REF] Abächerli | Level-set percolation for the gaussian free field on a transient tree[END_REF]. For any ≥ 0 we have that (Φ(t ∧ σ ), t ≥ 0) is a uniformly integrable martingale and, thus, E [Φ(σ )] = 1.

Proof. By Lemma 2.6.1 with θ = 1 µ , γ = δ = 0 and π(dx) = c µ x -(α+1) dx, (Φ(t), t ≥ 0) is a non-negative martingale of mean 1. Since for any ≥ 0, σ is a stopping time for L and since Φ has right-continuous trajectories, (Φ(t ∧ σ ), t ≥ 0) is a martingale (w.r.t to the natural filtration of L). So by the almost sure martingale convergence theorem, we must have

Φ(t ∧ σ ) → Φ(σ ) almost surely as t → ∞. Then (Φ(t ∧ σ ), t ≥ 0) is uniformly integrable if
and only if this convergence also holds in L 1 . By Fatou's lemma, we get E [Φ(σ )] ≤ 1, so that Φ(σ ) is integrable. Now, for any t > 0,

E [|Φ(σ ) -Φ(t ∧ σ )|] = E |Φ(σ ) -Φ(t)|1 {σ >t} ≤ E Φ(σ )1 {σ >t} + E Φ(t)1 {σ >t} .
Observe that by the definition of the measure-changed process, we have E Φ(t)

1 {σ >t} = P ( σ > t). So E [|Φ(σ ) -Φ(t ∧ σ )|] ≤ P ( σ > t) + E Φ(σ )1 {σ >t} .
Since Φ(σ ) is integrable (and hence uniformly integrable) and since σ < ∞ almost surely, we have lim t→∞ E Φ(σ )1 {σ >t} = 0. By Proposition 2.3.1, we have that L t → -∞ almost surely as t → ∞, and so σ < ∞ almost surely. So lim t→∞ P ( σ > t) = 0 and we get

E [|Φ(σ ) -Φ(t ∧ σ )|] → 0 as t → ∞.
Hence, (Φ(t ∧ σ ), t ≥ 0) is uniformly integrable and, in particular, we may deduce that E [Φ(σ )] = 1.

We are now in a position to characterise the joint law of ( σ s , 0 ≤ s ≤ ) and ( ε (s) , s ≤ ). We will find it convenient to list the excursions occurring before local time has been accumulated in decreasing order of length, as ( ε

( ) i , i ≥ 1)
E f ( σ s , 0 ≤ s ≤ ) n i=1 g i ε ( ) i = E exp 1 µ 0 σ r dr - C α σ α+1 (α + 1)µ α+1 f (σ s , 0 ≤ s ≤ ) ×E   exp   1 µ j>n a(ε ( ) j )   ζ(ε ( ) k ), k > n   n i=1 E exp 1 µ a(ε ( ) i ) g i ε ( ) i ζ(ε ( ) i )   .
In particular, the excursions ( ε

( )
i , i ≥ 1) are conditionally independent given their lengths. Moreover, for any i ≥ 1 and any suitable test function g,

E g ε ( ) i ζ( ε ( ) i ) = x = E N (x) exp 1 µ x 0 e(t)dt g(e) E N (x) exp 1 µ x 0 e(t)dt = E N (1) exp x 1+1/α µ 1 0 e(t)dt g(x 1/α e(•/x)) E N (1) exp x 1+1/α µ 1 0 e(t)dt .
Proof. By integration by parts and writing L s = I s + (L s -I s ), noting that L σ = -, we get

- 1 µ σ 0 sdL s = σ µ + 1 µ σ 0 L s ds = σ µ + 1 µ σ 0 I s ds + 1 µ σ 0 (L s -I s )ds.
Changing variable in the middle term, and using the fact that I σs = -s, we obtain

σ µ + 1 µ 0 I σs dσ s + 1 µ σ 0 (L s -I s )ds = σ µ - 1 µ 0 sdσ s + 1 µ σ 0 (L s -I s )ds.
Another integration by parts yields that this is equal to

1 µ 0 σ s ds + 1 µ σ 0 (L s -I s )ds.
Finally, we can integrate the excursions of L -I separately to obtain that this is equal to

1 µ 0 σ s ds + 1 µ s≤ a(ε (s) ).
Hence,

Φ(σ ) = exp   1 µ 0 σ r dr + 1 µ s≤ a(ε (s) ) -C α σ α+1 (α + 1)µ α+1   .
(2.13)

Now,

E f ( σ s , 0 ≤ s ≤ ) n i=1 g i ε ( ) i = E   exp   1 µ 0 σ r dr + 1 µ s≤ a(ε (s) ) -C α σ α+1 (α + 1)µ α+1   f (σ s , 0 ≤ s ≤ ) n i=1 g i ε ( ) i   = E exp 1 µ 0 σ r dr -C α σ α+1 (α + 1)µ α+1 f (σ s , 0 ≤ s ≤ ) ×E   exp   1 µ s≤ a(ε (s) )   n i=1 f i ε ( ) i (σ s , 0 ≤ s ≤ )     ,
As discussed below Theorem 2.3.4, the excursions of the stable Lévy process are conditionally independent given their lengths, which yields the first expression in the statement of the proposition. The final statement is an immediate consequence of the scaling property for stable excursions; we observe that this change of measure for the excursions is well-defined by Lemma 2.3.6.

Lemma 2.3.5 (i) implies that we can list all the excursions of R in decreasing order of length: write ( ε i , i ≥ 1) for this list. Write ( h i , i ≥ 1) for the corresponding height process excursions.

Proposition 2.3.9. The pairs of excursions ( ε i , h i , i ≥ 1) are conditionally independent given their lengths (ζ( ε i ), i ≥ 1), with law specified by

E g ε i , h i ζ( ε i ) = x = E N (x) exp 1 µ x 0 e(t)dt g(e, h) E N (x) exp 1 µ x 0 e(t)dt = E N (1) exp x 1+1/α µ 1 0 e(t)dt g(x 1/α e(•/x), x (α-1)/α h(•/x)) E N (1) exp x 1+1/α µ 1 0 e(t)
E g ε i , h i 1 {M i =m} ζ( ε i ) = x = E g ε i , h i exp - 1 µ ∞ 0 ε i (u)du 1 m! 1 µ ∞ 0 ε i (u)du m ζ( ε i ) = x = E N (x) exp 1 µ x 0 e(t)dt g(e, h) exp -1 µ x 0 e(t)dt 1 m! 1 µ x 0 e(t)dt m E N (x) exp 1 µ x 0 e(t)dt
and so

E g ε i , h i ζ( ε i ) = x, M i = m = E N (x) 1 µ x 0 e(t)dt m g(e, h) E N (x) 1 µ x 0 e(t)dt m = E N (1) 1 0 e(t)dt m g(x 1/α e(•/x), x (α-1)/α h(•/x)) E N (1) 1 0 e(t)dt m .
The claimed result follows.

Convergence of a discrete forest

The multigraph M n (ν) contains cycles with probability tending to 1 as n → ∞. However, its components will turn out to be tree-like, in that they each have a finite surplus, with probability 1. In this section, we study an idealised version of the depth-first walk of the multigraph, ignoring cycles.

Let ( Dn 1 , Dn 2 , . . . , Dn k ) be D 1 , D 2 , . . . , D n arranged in size-biased random order. More precisely, let Σ be a random permutation of {1, 2, . . . , n} such that

P (Σ = σ|D 1 , . . . , D n ) = D σ(1) n j=1 D σ(j) D σ(2) n j=2 D σ(j) • • • D σ(n) D σ(n)
and define

( Dn 1 , Dn 2 , . . . , Dn n ) = (D Σ(1) , D Σ(2) , . . . , D Σ(n) ). Now let S n (0) = 0 and, for k ≥ 1, S n (k) = k i=1 ( Dn i -2).
Then S n is the depth-first walk of a forest of trees in which the ith vertex visited in depth-first order has Dn i -1 ≥ 0 children. Define the corresponding height process,

G n (k) = # j ∈ {0, 1, . . . , k -1} : S n (j) = inf j≤ ≤k S n ( ) .
The purpose of this section is to recover Theorem 8.1 of Joseph [START_REF] Joseph | The component sizes of a critical random graph with given degree sequence[END_REF] and, indeed, to strengthen it by adding the convergence of the height process to that of the depth-first walk. We will prove the following.

Theorem 2.4.1. We have

n -1 α+1 S n ( n α α+1 t ), n -α-1 α+1 G n ( n α α+1 t ), t ≥ 0 d -→ ( L t , H t , t ≥ 0) as n → ∞ in D(R + , R) 2 .
In order to prove this theorem, we will begin by showing that there is an analogue in the discrete setting of the change of measure used to define L.

Write Z 1 , Z 2 , . . . , Z n for i.i.d. random variables with the size-biased degree distribution, i.e.

P (Z 1 = k) = kν k µ , k ≥ 1.
Observe that µ ∈ (1, 2) since, firstly, D 1 ≥ 1 and, secondly, E D 2 1 = 2µ and we must have var

(D 1 ) = µ(2 -µ) > 0. Then we have E [Z 1 ] = 2, P (Z 1 ≥ 1) = 1 and P (Z 1 = k) ∼ c µ k -(α+1) as k → ∞ if α ∈ (1, 2), or var (Z 1 ) = β/µ if α = 2. Proposition 2.4.2. For any k 1 , k 2 , . . . , k n ≥ 1, we have P Dn 1 = k 1 , Dn 2 = k 2 , . . . , Dn n = k n = k 1 ν k 1 k 2 ν k 2 . . . k n ν kn n i=1 (n -i + 1) n j=i k j . Moreover, for 0 ≤ m ≤ n and k 1 , k 2 , . . . , k m ≥ 1, let φ n m (k 1 , k 2 , . . . , k m ) := E m i=1 (n -i + 1)µ m j=i k j + Ξ n-m
, where Ξ n-m has the same law as

D m+1 + D m+2 + • • • + D n .
Then for any suitable test-function

g : Z m + → R + , E g( Dn 1 , Dn 2 , . . . , Dn m ) = E [φ n m (Z 1 , Z 2 , . . . , Z m )g(Z 1 , Z 2 , . . . , Z m )] . (2.14)
We have not found a precise reference for the contents of Proposition 2.4.2. The analogue of (2.14) for continuous random variables is equation ( 1) of Barouch & Kaufman [START_REF] Barouch | Probabilistic modelling of oil and gas discovery[END_REF]; see Proposition 1 of Pitman & Tran [START_REF] Pitman | Size-biased permutation of a finite sequence with independent and identically distributed terms[END_REF] for a proof. The proof of Proposition 2.4.2 is elementary and may be found in the Appendix.

We now show that the Radon-Nikodym derivative in the change-of-measure formula converges in distribution under appropriate conditions. Until the end of this section, we restrict our attention to the case α ∈ (1, 2); the proof for the Brownian case is similar but a little more involved, so we defer it to Section 2.6.3 in the Appendix.

Proposition 2.4.3. Let Φ(n, m) := φ n m (Z 1 , Z 2 , . . . , Z m )
and recall that

Φ(t) = exp - 1 µ t 0 sdL s - C α t α+1 (α + 1)µ α+1 .
Then for fixed t > 0, Φ(n, tn In order to prove this, we will need some technical lemmas.

First, we consider the asymptotics of S(k) = k i=1 (Z i -2). The generalised functional central limit theorem, Theorem 2.2.3 (ii), entails that

n -1/(α+1) S tn α/(α+1) , t ≥ 0 d -→ (L t , t ≥ 0) (2.15) as n → ∞ in D(R + , R),
where L is the spectrally positive α-stable Lévy process introduced in the previous section. We will need to deal with functionals of S converging, which we will do via the continuous mapping theorem (Theorem 3.2.4 of Durrett [START_REF] Durrett | Probability: theory and examples[END_REF]). We give here the details for the functional which will arise most frequently in the sequel.

Lemma 2.4.4. For any t ≥ 0,

1 n tn α α+1 -1 k=0 S(k) d -→ t 0 L s ds as n → ∞.
Proof. We have

1 n tn α α+1 -1 k=0 S(k) = 1 n tn α α+1 0 S( v )dv = n -α α+1 tn α α+1 0 n -1 α+1 S( un α α+1
)du, by changing variable. Then the convergence in law follows from the fact that n -α α+1 tn α α+1

→ t and the continuous mapping theorem.

L(λ) = exp -µλ + µ(2 -µ) 2 λ 2 - C α λ α+1 (α + 1) + o(λ α+1 ) . (2.16)
Proof. First observe that α+2) , the right-hand side is finite and, by the Euler-Maclaurin formula, asymp-

L (λ) = -E D 3 1 exp(-λD 1 ) = - ∞ k=1 k 3 e -λk ν k . Since ν k ∼ ck -(
totically equivalent to ∞ 0 cx 1-α e -λx dx = cλ α-2 Γ(2 -α),
as λ → 0. In other words,

L (λ) = -cλ α-2 Γ(2 -α) + o(λ α-2 ),
where o is for λ → 0. We also have E [D 1 ] = µ and E D 2 1 = 2µ. So integrating three times, we obtain

L(λ) = 1 -µλ + µλ 2 - cΓ(2 -α) (α -1)α(α + 1) λ α+1 + o(λ α+1 ),
and it is straightforward to see that this implies

L(λ) = exp -µλ + µ(2 -µ) 2 λ 2 - C α λ α+1 (α + 1) + o(λ α+1 ) . Lemma 2.4.6. For m = O(n α/(α+1) ), we have exp m - (2 + µ) 2µ m 2 n L m nµ n-m = (1 + o(1)) exp - C α m α+1 (α + 1)µ α+1 n α . Proof. By (2.16), it is sufficient to show that m - (2 + µ) 2µ m 2 n + (n -m) - µm nµ + µ(2 -µ) 2 m 2 n 2 µ 2 - C α (α + 1) m α+1 n α+1 µ α+1 = - C α m α+1 (α + 1)µ α+1 n α + o(1),
as n → ∞. But this is now easily seen to be true on cancellation and using m = O(n α/(α+1) ).

The result follows.

Lemma 2.4.7. Let s(0) = 0 and s(i

) = i j=1 (k j -2) for i ≥ 1. Then if m = O(n α/(α+1) ), we have φ m n (k 1 , k 2 , . . . , k m ) ≥ exp 1 nµ m i=0 (s(i) -s(m)) - C α m α+1 (α + 1)µ α+1 n α (1 + o(1)), where the o(1) term is independent of k 1 , . . . , k m ≥ 1. Proof. First rewrite m i=1 (n -i + 1) = n m m-1 i=1 1 - i n .
Then

φ m n (k 1 , k 2 , . . . , k m ) = m-1 i=1 1 - i n E m i=1 nµ m j=i k j + Ξ n-m = E   exp   m-1 i=1 log 1 - i n - m i=1 log   Ξ n-m nµ + 1 nµ m j=i k j       .
Now note that for any x ∈ (-1, ∞), we have log(1 + x) ≤ x. We also have log(1

-i/n) ≥ -i/n -m 2 /n 2 for 1 ≤ i ≤ m -1. So φ m n (k 1 , k 2 , . . . , k m ) ≥ E   exp   - m-1 i=1 i n - m 3 n 2 -m Ξ n-m nµ -1 - 1 nµ m i=1 m j=i k j     = exp - m(m -1) 2n - m 3 n 2 + m + 1 nµ m i=1 s(i) -s(m) -2(m -i + 1) × E exp - m nµ Ξ n-m = exp - m(m -1) 2n - m 3 n 2 + m + 1 nµ m i=0 (s(i) -s(m)) - m(m + 1) nµ × E exp - m nµ Ξ n-m = exp 1 nµ m i=0 (s(i) -s(m)) exp m - (2 + µ) 2µ m 2 n L m nµ n-m × exp (µ -2)m 2µn - m 3 n 2 .
We have

m 3 /n 2 = O(n α-2 α+1 ) = o(1)
and so the final exponential tends to 1 as n → ∞. The desired result then follows from Lemma 2.4.6.

Lemma 2.4.8. Let (X n ) n≥1 , (Y n ) n≥1 be two sequences of non-negative random variables such that X n ≥ Y n and E [X n ] = 1 for all n. Suppose that X is another non-negative random variable such that E [X] = 1 and Y n d -→ X. Then X n d -→ X and (X n ) n≥1 is uniformly integrable.
Proof. We first prove that X n d -→ X, using Portmanteau's theorem. Let f be a bounded and uniformly continuous function, we have

E [f (Y n )] → E [f (X)]. It is enough to show that E [f (Y n + Z n )] -E [f (Y n )] → 0, where Z n := X n -Y n . Z n is a non-negative random variable such that E [Z n ] → 0, hence Z n converges to 0 in probability. Fix ε > 0, and let δ > 0 be such that |f (x 1 ) -f (x 2 )| ≤ ε for all x 1 , x 2 ∈ R satisfying |x 1 -x 2 | ≤ δ. For n large enough, P (|Z n | ≥ δ) ≤ δ. Thus, |E [f (Y n + Z n )] -E [f (Y n )] | ≤ ε + δ f ∞ .
Since ε and δ can be chosen arbitrarily small, this yields X n d -→ X. Now, we prove that X n → X in L 1 , which implies the uniform integrability. By Skorokhod's representation theorem, we may work on a probability space where X n → X almost surely.

Consider (X -X n ) + . For any given > 0, we may choose

K such that E X1 {X≥K} < (since E [X] < ∞). So E (X -X n ) + 1 {X≥K} < . But also E (X -X n ) + 1 {X<K} → 0 as n → ∞ because it is less than δ + KP ((X -X n ) + > δ) for any δ > 0. So E [(X -X n ) + ] → 0. Since E [X] = E [X n ], we also have E [(X n -X) + ] = E [(X -X n ) + ] → 0. So finally E [|X n -X|] → 0. Proof of Proposition 2.4.3. Recall that S(k) = k i=1 (Z i -2). By Lemma 2.4.7, for m = tn α α+1 , Φ(n, m) ≥ Φ(n, m) := exp 1 nµ m i=0 (S(i) -S(m)) - C α m α+1 (α + 1)µ α+1 n α (1 + o(1)).
By (2.15) and Lemma 2.4.4, we have

1 nµ m i=0 (S(i) -S(m)) d -→ 1 µ t 0 (L s -L t )ds.
Hence, by the continuous mapping theorem,

Φ(n, tn α α+1 ) d -→ exp 1 µ t 0 (L s -L t )ds - C α t α+1 (α + 1)µ α+1 = exp - 1 µ t 0 sdL s - C α t α+1 (α + 1)µ α+1 ,
and the right-hand side is, of course, Φ(t), which has mean 1.

We also have

E Φ(n, tn α α+1 ) = 1 for all n. So by Lemma 2.4.8, we must have Φ(n, tn α α+1 ) d -→ Φ(t)
as n → ∞, as well as the claimed uniform integrability.

We are now ready to prove Theorem 2.4.1.

Proof of Theorem 2.4.1. It is sufficient to show that for any t ≥ 0 and any bounded continuous

test-function f : D([0, t], R) 2 → R, E f n -1 α+1 S n ( n α α+1 u ), n -α-1 α+1 G n ( n α α+1 u ), 0 ≤ u ≤ t → E f ( L u , H u , 0 ≤ u ≤ t) , as n → ∞. Let us write S n (u) = n -1 α+1 S( n α α+1 u ) and, similarly, G n (u) = n -α-1 α+1 G( n α α+1 u ).
Then, by changing measure, we wish to show that for any t ≥ 0 and any bounded continuous test-function

f : D([0, t], R) 2 → R, E Φ(n, tn α α+1 )f (S n (u), G n (u), 0 ≤ u ≤ t) → E [Φ(t)f (L u , H u , 0 ≤ u ≤ t)] ,
as n → ∞. From the proof of Proposition 2.4.3, we have that

E Φ(n, tn α α+1 ) -Φ(n, tn α α+1 ) → 0
as n → ∞, and so it will suffice to show that

E Φ(n, tn α α+1 )f (S n (u), G n (u), 0 ≤ u ≤ t) → E [Φ(t)f (L u , H u , 0 ≤ u ≤ t)] . But Φ(n, tn α α+1 ) = exp 1 µ t 0 (S n (u) -S n (t))du - C α tn α α+1 α+1 (α + 1)µ α+1 n α .
In particular, for a path

x ∈ D([0, t], R), let Θ(x, t) = exp 1 µ t 0 (x(u) -x(t))du - C α t α+1 (α + 1)µ α+1
and observe that Θ is a continuous functional of its first argument. Then we have

E Φ(n, tn α α+1 ) -Θ(S n , t) → 0.
So it suffices to show that

E Θ(S n , t)f (S n (u), G n (u), 0 ≤ u ≤ t) → E [Θ(L, t)f (L u , H u , 0 ≤ u ≤ t)] .
But this now follows from Theorem 2.2.4 and uniform integrability.

The configuration multigraph

The processes S n and G n encode a forest of trees where the numbers of children of the vertices, visited in depth-first order, are Dn i -1, i ≥ 1. Let us write F n (ν) for this forest. In this section, we wish to encode similarly the multigraph M n (ν). Let us first describe the organisation of this section.

In Section 2.5.1, we simultaneously generate and explore M n (ν) using the depth-first approach outlined in the Introduction: we view each connected component of the graph as a spanning tree explored in a depth-first manner plus some additional edges, creating cycles, that we call back-edges. In Section 2.5.2, we prove that the exploration process is close enough to S n in order to have the same scaling limit, and add the joint convergence of the locations of the back-edges in Section 2.5.3 (Theorem 2.5.5). In Section 2.5.4, we split the multigraph into its components by showing that the (rescaled) ordered sequence of component sizes converges to the sequence of ordered excursion lengths of the continuous process R (Proposition 2.5.6).

We improve this result in Section 2.5.5 by adding in the locations of the back-edges under each excursion (Proposition 2.5.12). In Section 2.5.6, we study the height process and show in Proposition 2.5.16 the joint convergence of the height process excursions and the locations of the back-edges. This finally allows us to prove Theorem 2.1.1 in Section 2.5.7.

Exploration of the multigraph

We work conditionally on the sequence ( Dn 1 , Dn 2 , . . . , Dn n ). Let us declare that the vertex of degree Dn i is called v i . This means that we have already determined the (size-biased by degree) order in which we will observe new vertices.

We will couple F n (ν) and M n (ν) by using the same ordering on the new vertices we explore.

Recall that we start from vertex v 1 with degree Dn 1 . We maintain a stack, namely an ordered list of half-edges which we have seen but not yet explored (remember that the half-edges come with an arbitrary labelling for this purpose). We put the Dn 1 half-edges of v 1 onto this stack, in increasing order of label, so that the lowest labelled half-edge is on top of the stack. At a subsequent step, suppose we have already seen the vertices v 1 , v 2 , . . . , v k . If the stack is non-empty, take the top half-edge and sample its pair. This lies on the stack with probability proportional to the height of the stack minus 1 or belongs to v k+1 with probability proportional to n i=k+1

Dn

i . In the first case we simply remove both half-edges from the stack. In the second, we remove the half-edge at the top of the stack (which has just been paired) and replace it by the remaining half-edges (if any) of v k+1 . If the stack is empty, we start a new component at

v k+1 .
Let us now describe the forest F n (ν) from which we will recover M n (ν). Whenever there is a back-edge in M n (ν), say from vertex v k to vertex v i with i ≤ k, remove the back-edge and replace it by two edges, one from v k to a new leaf and the other from v i to a new leaf. To recover M n (ν) it is then sufficient to remove the edges to the new leaves and put in a new edge

from v i to v k .
Our aim is to encode the forest F n (ν), firstly via its depth-first walk and then by its height process. We will simultaneously keep track of marks which tell us which vertices we should identify in order to recover the multigraph.

Our first observation is that the vertex-sets of pairs of components in F n (ν) correspond precisely to the vertex-sets of subcollections of components in M n (ν). (This is illustrated in Figure 2.1 to which the reader is referred in the following argument.) More precisely, without loss of generality, let the pair of components of F n (ν) be the first two, on vertices v 1 , v 2 , . . . , v m 1 and v m 1 +1 , v m 1 +2 , . . . , v m 1 +m 2 . Suppose that the same vertices in M n (ν) are adjacent to b back-edges. Now vertices v 1 and v m 1 +1 each possess one more half-edge in M n (ν) than they do in F n (ν) (since in F n (ν), if v i is the first vertex of a component, it has Di -1 edges and not Di ). In particular, adding an edge between v 1 and v m 1 +1 clearly produces a tree with

m 1 + m 2 vertices and 1 2 m 1 +m 2 i=1
Dn i = m 1 + m 2 -1 edges. We now "rewire" this tree to obtain the relevant components of M n (ν). The effect of adding a back-edge is to shunt all of the subsequent subtrees along in the depth-first order. (See Figure 2.1.) The overall effect is that each back-edge causes a new component to come into existence. Each time we observe a back-edge, it occupies two half-edges, so there are two subtrees which get pushed out of the component. The earlier of these subtrees in the depth-first order becomes the basis for the next component of M n (ν). The root of this component has one more child than it had in the original tree. This allows the absorption of the second subtree, whose root gets attached by its free half-edge to the root of the component. Subsequent back-edges similarly each generate one new component. Following this through, we see that we end up with b+1 components of M n (ν).

(For the purposes of intuition, note that because the vast majority of vertices lie in components of size o(n α/(α+1) ), with high probability at most one of them will be of size Θ(n α/(α+1) ) and thus show up in the limit. So, at least heuristically, this rewiring process cannot affect what we see in the limit.)

It is clear that the effects of adding back-edges are relatively local and so it is at least intuitively clear that the depth-first walk of the forest F n (ν) should be similar to that of F n (ν), as long as there are not too many back-edges. Let X n denote the depth-first walk of F n (ν). We will now describe how to construct X n from S n , and also how to keep track of the back-edges.

We will write R n (k) for the number of half-edges on the stack at step k. We will let N n count the occurrences of back-edges, and U n the positions of their targets on the stack. We will write M n (k) for a set of marks (in N) at step k, indicating back-edges which have not yet been closed, and τ n (k) for the number of vertices already seen at step k (note that we see a new vertex if and only if the current step does not involve a back-edge). Finally, let C n (k) be the number of components of F n (ν) we have fully explored by time k. So we will have that for all k ≥ 1,

C n (k) = -min 0≤ ≤k X n ( ), and that R n (k) = X n (k) + C n (k).
We start from X n (0) = N n (0) = 0, M n (0) = ∅ and τ n (0) = 0. For k ≥ 0, we might encounter the following three situations.

• New component.

If X n (k) = min 0≤i≤k-1 X n (i) -1 or k = 0, let τ n (k + 1) = τ n (k) + 1, N n (k + 1) = N n (k), M n (k + 1) = M n (k) and X n (k + 1) = X n (k) + S n (τ n (k) + 1) -S n (τ n (k)) + 1.
• Start a back-edge or not. Top: the first two components of the forest F n (ν). Middle: the first four components of M n (ν), on the same vertices. Three back-edges are marked in red. The subtree surgery required to get from F n (ν) to M n (ν) is indicated. The back-edge from 5 to 8 moves the subtree from 9 to the next available half-edge, also belonging to 5. This shifts further the subtrees from 13 and 16:

13 is connected to 1, which has one more edge in

M n (ν) than in F n (ν) ( D1 instead of D1 -1),
as any first vertex of a connected component of M n (ν). 16 starts a new component. Bottom:

If X n (k) > min 0≤i≤k-1 X n (i) -1 and X n (k) / ∈ M n (k),
-With probability

X n (k) -min 0≤i≤k X n (k) -|M n (k)| X n (k) -min 0≤i≤k X n (k) -|M n (k)| + n j=τn(k)+1 Dn j , let τ n (k + 1) = τ n (k) and X n (k + 1) = X n (k) -1. Let N n (k + 1) = N n (k) + 1, sample U n (k + 1) uniformly from min 0≤i≤k X n (i), . . . , X n (k) -1 \ M n (k),
and let

M n (k + 1) = M n (k) ∪ {U n (k + 1)}.
-With probability

n j=τn(k)+1 Dn j X n (k) -min 0≤i≤k X n (k) -|M n (k)| + n j=τn(k)+1 Dn j let τ n (k + 1) = τ n (k) + 1, X n (k + 1) = X n (k) + S n (τ n (k) + 1) -S n (τ n (k)), N n (k + 1) = N n (k) and M n (k + 1) = M n (k).
• Close a back-edge?

If X n (k) > min 0≤i≤k-1 X n (i) -1 and X n (k) ∈ M n (k) then let τ n (k + 1) = τ n (k), X n (k + 1) = X n (k) -1, N n (k + 1) = N n (k), and M n (k + 1) = M n (k) \ {X n (k)}.
It is straightforward to check that this is the depth-first walk of the forest F n (ν). We observe that for k ≥ 1 we have

X n (k) + min 0≤i≤k X n (i) = S n (τ n (k)) + 1 -N n (k) -#{i ≤ k : |M n (i)| < |M n (i -1)|}.
Hence, X n (k) + min 0≤i≤k X n (i) is the number of half-edges seen but not yet paired or reserved for back-edges, in the currently explored connected component (the (j + 1)-th component if

-min 0≤i≤k X n (i) = j). Indeed, S n (τ n (k)) + 1 = ( D1 -1) -1 + D2 -2 + . . . + Dτn(k)-1 -2 + Dτn(k) -1
is the number of half-edges seen and not paired in the spanning forest of the components explored so far.

N n (k) + #{i ≤ k : |M n (i)| < |M n (i -1)
|} is the number of half-edges that have been paired in or reserved for back-edges. Note that |M n (i)| increases (resp. decreases) by 1 exactly when a back-edge is initiated (resp. closed). N n (k) counts the number of back-edges started.

In particular,

S n (τ n (k)) -2N n (k) ≤ X n (k) + min 0≤i≤k X n (i) -1 ≤ S n (τ n (k)).
(2.17)

Convergence of the depth-first walk and marks

Let us first prove a bound on the number N n (k) of back-edges which have occurred by step k.

Lemma 2.5.1. For every t > 0, the sequences of random variables N n ( tn α/(α+1) ) n≥1 and

sup 0≤k≤ tn α/(α+1) |τ n (k) -k| n≥1 are tight.
Proof. Fix t > 0. We observe that k -2N n (k) ≤ τ n (k) ≤ k so that it is enough to prove that N n ( tn α/(α+1) ) n≥1 is tight. At time i, the number of half-edges on the stack is R n (i), and the total number of unpaired half-edges is n j=τn(i)+1

Dn j , so that the probability to start a back-edge is

R n (i) n j=τn(i)+1 Dn j
Note that if the component we are exploring at step i began at step j, then R n (i) ≤ 2 +

τn(i) k=τn(j) ( Dn k -2
), since at most Dn τn(j) + . . . + Dn τn(i) half-edges have been seen, and at least 2(τ n (i) -τ n (j)) of them have been used to connect the first τ n (i) -τ n (j) + 1 vertices of the component. Hence,

R n (i) ≤ 2 + max 0≤j≤i S n (τ n (j)) -min 0≤j≤i S n (τ n (j)) ≤ 2 + max 0≤j≤i S n (j) -min 0≤j≤i S n (j), since τ n ({1, . . . , i}) ⊆ {1, . . . , i}.
As underlined in the beginning of Section 2.5.1, we can realize M n (ν) by first drawing the sequence ( Dn i ) 1≤i≤n , and then proceeding to the pairings of the half-edges through the exploration. Conditionally on ( Dn i ) 1≤i≤n , for every i ≥ 1 and whatever happened in the first i -1 steps of the exploration, the probability to create a back-edge at time i is at most 2 + max 0≤j≤ tn α/(α+1) S n (j) -min 0≤j≤ tn α/(α+1) S n (j)

n j=τn(i)+1 Dn j .
Note also that τ n (i) + 1 ≤ tn α/(α+1) + 1 for every i ≤ tn α/(α+1) . Therefore, conditionally on ( Dn i ) 1≤i≤n , the random variable N n ( tn α/(α+1) ) is stochastically dominated by a Binomial random variable with parameters tn α/(α+1) and 2 + max 0≤j≤ tn α/(α+1) S n (j) -min 0≤j≤ tn α/(α+1) S n (j)

n j= tn α/(α+1) +1 Dn j .
For K > 0, define

E 1 = 2 + max 1≤i≤ tn α/(α+1) S n (i) - min 0≤i≤ tn α/(α+1) S n (i) ≤ Kn 1/(α+1)
and

E 2 =    n j= tn α/(α+1) +1 Dn j ≥ n    .
Fix > 0. Theorem 2.4.1, Lemma 2.6.5 and the fact that µ > 1 imply that there exists K > 0 such that for n large enough,

P (E 1 ∩ E 2 ) ≥ 1 -.
On the event E 1 ∩ E 2 , we have 2 + max 0≤j≤ tn α/(α+1) S n (j) -min 0≤j≤ tn α/(α+1) S n (j)

n j= tn α/(α+1) +1 Dn j ≤ K n α/(α+1) .
Let Y ∼ Bin( tn α/(α+1) , 2K/n α/(α+1) ). Then there exists K > 0 such that for n large enough,

P N n ( tn α/(α+1) ) ≥ K ≤ P Y ≥ K + P ((E 1 ∩ E 2 ) c ) ≤ 2 .
Since > 0 was arbitrary, the result follows.

In particular, the steps on which back-edges occur are negligible on the timescale in which we are interested. Write I t = inf 0≤s≤t L s for t ≥ 0 and recall that

R t = L t -I t . Proposition 2.5.2. As n → ∞, n -1/(α+1) S n ( tn α/(α+1) ), n -1/(α+1) R n ( tn α/(α+1) ), n -1/(α+1) C n ( tn α/(α+1) ), t ≥ 0 d -→ L t , R t , -1 2 I t , t ≥ 0 , in D(R + , R) 3 .
Proof. Let k ≥ 1, and let j be such that X n (j) = min 0≤ ≤k X n ( ). Then X n (j)+min 0≤ ≤j X n ( ) =

2X n (j), and by (2.17), we have

2X n (j) ≥1 + min 0≤ ≤k ( S n (τ n ( )) -2N n ( )) ≥1 + min 0≤ ≤k S n (τ n ( )) -2N n (k), since (N n (i)) i≥0 is non-increasing.
Thus, by (2.17), we have

1 + min 0≤ ≤k S n (τ n ( )) -2N n (k) ≤ 2 min 0≤ ≤k X n ( ) ≤ 1 + min 0≤ ≤k S n (τ n ( )).
By Theorem 2.4.1 and the continuous mapping theorem,

n -1/(α+1) min 0≤ ≤ sn α/(α+1) S n ( ), s ≥ 0 d -→ ( I s , s ≥ 0)
and combining this with Lemma 2.5.1 and recalling that

C n (k) = -min 0≤ ≤k X n ( ) yields n -1/(α+1) C n ( sn α/(α+1) ), s ≥ 0 d -→ (-1 2 I s , s ≥ 0)
Another application of (2.17) gives

1 + S n (τ n (k)) -2N n (k) -2 min 0≤i≤k X n (i) ≤ R n (k) ≤ 1 + S n (τ n (k)) -2 min 0≤i≤k X n (i)
and since

(n -1/(α+1) S n (τ n ( tn α/(α+1) )), t ≥ 0) d -→ ( L t , t ≥ 0)
we have

n -1/(α+1) R n ( tn α/(α+1) ), t ≥ 0 d -→ (R t , t ≥ 0),
jointly with the convergence of the minimum.

Thus the exploration of F n (ν) sees approximately twice as many components as that of F n (ν) but the limiting reflected process is the same for both. In particular, asymptotically the two processes have the same longest excursions. This fact will play an important role in the sequel.

Back-edges

We will now show that the parts of the multigraph we observe up until well beyond the timescale in which we are interested are, with high probability, simple. To this end, let A n (k) be the number of loops and edges created parallel to an existing edge, up until step k of the depth-first exploration of F n (ν). Call these anomalous edges.

Proposition 2.5.3. Suppose α α+1 < β < α 2 . Then we have

P A n ( n β ) > 0 → 0 as n → ∞.
Proof. We adapt the proof of Lemma 7.1 of Joseph [START_REF] Joseph | The component sizes of a critical random graph with given degree sequence[END_REF] (which applies in the finite third moment setting). Self-loops are obviously associated with a unique vertex. We associate extra edges created parallel to an existing edge with their vertex which is discovered first in the depth-first exploration. Consider a particular vertex of degree d in the exploration before time n β . Its kth half-edge (in the order that we process them) creates a self-loop with probability bounded above by

d -k n i= n β +1 Dn i .
It creates a multiple edge with probability at most

k -1 n i= n β +1 Dn i .
This vertex therefore possesses an anomalous edge with probability bounded above by

d(d -1) n i= n β +1 Dn i .
Hence, by the conditional version of Markov's inequality,

P A n ( n β ) > 0 Dn 1 , Dn 2 , . . . , Dn n ≤ n β i=1 ( Dn i ) 2 n i= n β +1 Dn i ∧ 1. But n i= n β +1 Dn i = n i=1 Dn i - n β i=1 Dn i ≥ n i=1 Dn i - n β i=1 ( Dn i ) 2
. By Lemmas 2.6.4 and 2.6.5 and the bounded convergence theorem, we obtain that

P A n ( n β ) > 0 → 0 as n → ∞. Let ρ(n) = inf{k ≥ 0 : A n (k) > 0} and note that the event that M n (ν) is simple is equal to {ρ(n) = ∞}.
The last proposition shows that we observe any anomalous edges long after the timescale in which we explore the largest components of the graphs. This allows us to conclude that all of the results we prove using only the timescale n α/(α+1) for the multigraph are also true conditionally on {ρ(n) = ∞}. In this way, we may give a proof of Conjecture 8.6 of Joseph [START_REF] Joseph | The component sizes of a critical random graph with given degree sequence[END_REF].

(See also Theorem 3 of [START_REF] Dhara | Heavy-tailed configuration models at criticality[END_REF].)

Theorem 2.5.4. Conditional on {ρ(n) = ∞}, as n → ∞, n -1/(α+1) R n ( sn α/(α+1) , n -1/(α+1) C n ( sn α/(α+1) , s ≥ 0 d -→ R s , -1 2 I s , s ≥ 0 , in D(R + , R) 2 .
Proof. Given Propositions 2.5.2 and 2.5.3, this follows in exactly the manner as Joseph's Theorem 3.2 follows from his Theorem 3.1.

Henceforth, using exactly the same argument, statements about our processes should be understood to hold either unconditionally or conditionally on the event {ρ(n) = ∞}.

We now turn to the locations of the back-edges that do occur. Recall that if a back-edge

occurs at step k, then U n (k) is its index in the stack. For steps k such that N n (k)-N n (k-1) = 0, declare U n (k) = ∂,
where ∂ denotes that no mark occurs.

Theorem 2.5.5. We have

n -1/(α+1) R n ( sn α α+1 ), n -1/(α+1) C n ( sn α/(α+1) , N n ( sn α α+1 ), n -1/(α+1) U n ( sn α α+1 ), s ≥ 0 d -→ (R s , -1 2 I s , N s , U s , s ≥ 0),
where ((N s , U s ), s ≥ 0) is a marked Cox process of intensity R s /µ at time s ≥ 0, and the marks are uniform on [0, R s ] i.e.

U s    = ∂ if N s -N s-= 0 ∼ U[0, R s ] if N s -N s-= 1.
Equivalently, conditionally on (R s , s ≥ 0),

0≤s≤t: Ns-N s-=1 δ (s,Us) is a Poisson point process on {(s, x) ∈ R + × R + : x ≤ R s } of constant intensity 1/µ.
Here, the convergence is in the Skorokhod topology for the first three co-ordinates and in the topology of vague convergence for counting measures on R 2 + for the fourth.

We observe that, in particular, for fixed t ≥ 0 we have sup 0≤s≤t R s < ∞ a.s. and so N t < ∞ a.s.

Proof. We refine the argument from the proof of Lemma 2.5.1. At step k, conditionally on the sequence ( Di ) 1≤i≤n and on the first k -1 steps of the exploration, the probability of seeing a back-edge is

R n (k) -|M n (k)| R n (k) -|M n (k)| + n j=τn(k)+1 Dn j , where |M n (k)| ≤ N n (k). Now n i=k+1 Dn i ≤ R n (k) -|M n (k)| + n i=τ n (k)+1 Dn i ≤ n i=1 Dn i + R n (k).
But then by Lemma 2.5.1, Proposition 2.5.2, Lemma 2.6.5 (in the Appendix) and Slutsky's lemma, we obtain

n α/(α+1) R n ( sn α/(α+1) ) -|M n ( sn α/(α+1) )| R n ( sn α/(α+1) ) -|M n ( sn α/(α+1) )| + n j=τn( sn α/(α+1) )+1 Dn j , 0 ≤ s ≤ t d -→ 1 µ (R s , 0 ≤ s ≤ t). (2.18) Let F n k = σ(( Dn 1 , Dn 2 , . . . , Dn n ), X n (i), N n (i), M n (i), 0 ≤ i ≤ k). Then (N n (k), k ≥ 0) is a counting process with compensator N n comp (k) = k-1 j=1 R n (j) -|M n (j)| R n (j) -|M n (j)| + n i=τn(j)+1 Dn i 1 {X n (j) / ∈M n (j)} . Since k-1 j=1 R n (j) -|M n (j)| R n (j) -|M n (j)| + n i=τn(j)+1 Dn i 1 {X n (j)∈M n (j)} ≤ N n (k -1) max 0≤j≤k-1 R n (j) -|M n (j)| R n (j) -|M n (j)| + n i=τn(j)+1
Dn i and n -α/(α+1) N n ( tn α/(α+1) ) p → 0 by Lemma 2.5.1, using (2.18) and the continuous mapping theorem we get that

E n ( tn α/(α+1) ) := tn α/(α+1) -1 j=0 R n (j) -|M n (j)| R n (j) -|M n (j)| + n i=τn(j)+1 Dn i 1 {X n (j)∈M n (j)} p → 0.
So by (2.18) and another application of the continuous mapping theorem, we obtain

N n comp ( tn α/(α+1) ) = k-1 j=1 R n (j) -|M n (j)| R n (j) -|M n (j)| + n i=τn(j)+1 Dn i -E n ( tn α/(α+1) ) d -→ 1 µ t 0 R s ds.
Finally, Theorem 14.2.VIII of Daley and Vere-Jones [START_REF] Daley | An introduction to the theory of point processes[END_REF] (with

F (n) 0 = σ( Dk i , k, i ≥ 1
) for all n ≥ 1, recall the construction of M n (ν) in the beginning of Section 2.5.1) yields that

n -1/(α+1) R n ( sn α/(α+1) ), N n ( sn α/(α+1) ), 0 ≤ s ≤ t d -→ (R s , N s , 0 ≤ s ≤ t).
The marks are uniform on the vertices of the stack which do not already carry marks, and so it is straightforward to see that they must be uniform in the limit.

Components of the finite graph

We now turn to the consideration of the individual components of the multigraph. Let σ n (0) = 0 and for k ≥ 1, write

σ n (k) = inf{j ≥ 0 : C n (j) ≥ k}.
This is the time at which we finish exploring the kth component of the forest F n (ν). Let

ζ n (k) = σ n (k) -σ n (k -1),
the corresponding length of the excursion, which is equal to the total number vertices within the component, since precisely one of these is killed at each step. But then ζ n (k) is also equal to the number of vertices in the corresponding component of M n (ν), plus twice the number of back-edges. Let

ε n k (t) = n -1/(α+1) X n (σ n (k -1) + tn α/(α+1) ) -X n (σ n (k -1)) for 0 ≤ t ≤ n -α/(α+1) ζ n (k) be the kth rescaled excursion of X n , with length ζ(ε n k ) = n -α/(α+1) ζ n (k) and rescaled left endpoint g n k = n -α/(α+1) σ n (k -1)
. Recall from Section 2.3 the notation ( ε i , i ≥ 1) for the ordered excursions of R above 0 and ζ( ε i ) for the lifetime of ε i . Denote by g i the left endpoint of ε i . Recall also that

2 ↓ = {(x 1 , x 2 , . . .) ∈ R N : x 1 ≥ x 2 ≥ . . . ≥ 0, i≥1 x 2 i < ∞}.
Let Γ be a countable index set and write 2 + (Γ) for the set of non-negative sequences (

x γ : γ ∈ Γ) such that γ∈Γ x 2 γ < ∞. Write ord : 2 + (Γ) → 2
↓ for the map which puts the elements of (x γ : γ ∈ Γ) into decreasing order. For a sequence (ε

k , A k ) k≥1 where ε k is an excursion of length ζ(ε k ) and A k is some other random variable, write ord (ζ(ε k ), A k , k ≥ 1)
for the same sequence put in decreasing order of ζ(ε k ). This section is devoted to proving the following proposition. Proposition 2.5.6. We have

(ζ( ε i ), i ≥ 1) ∈ 2 ↓ , ord (ζ(ε n k ), k ≥ 1) d -→ (ζ( ε i ), i ≥ 1)
as n → ∞ in 2 ↓ , and

ord (ζ(ε n k ), g n k , k ≥ 1) d -→ (ζ( ε i ), g i , i ≥ 1),
where the convergence is in 2 ↓ for the first coordinate and in the product topology for the second.

We apply a method outlined in Proposition 15 of Aldous [START_REF] Aldous | Brownian excursions, critical random graphs and the multiplicative coalescent[END_REF], which is most conveniently recounted in Section 2.6 of Aldous and Limic [START_REF] Aldous | The entrance boundary of the multiplicative coalescent[END_REF]. This is very similar to Theorem 8.3 of Joseph [START_REF] Joseph | The component sizes of a critical random graph with given degree sequence[END_REF], who omits many of the details of the proof. We feel that the argument is sufficiently subtle to merit a full account, which we now give.

Essentially, there are two steps to proving the desired convergence. First, we need to show that the longest excursions of R n and R occurring before some finite time match up for large enough n. Then we need to show that long excursions of R n cannot "wander off to time ∞".

Proposition 2.5.7 below is designed to deal with these issues.

Following Aldous, we introduce the concept of a size-biased point process. Suppose we have

random variables Y = (Y γ : γ ∈ Γ) in 2 + (Γ). Given Y, let E γ ∼ Exp(Y γ ) independently for different γ ∈ Γ. Set Σ(a) = γ∈Γ Y γ 1 {Eγ <a} (2.19) and note that Σ(a) < ∞ a.s. Let Σ γ = Σ(E γ ). Then Ξ = {(Σ γ , Y γ ) : γ ∈ Γ} is the size-biased
point process (SBPP) associated with Y. Write π for the projection onto the second co-ordinate, so that π({(s γ , y γ )}) = {y γ }. Proposition 2.5.7 (Proposition 15 of [START_REF] Aldous | Brownian excursions, critical random graphs and the multiplicative coalescent[END_REF] and Proposition 17 of [START_REF] Aldous | The entrance boundary of the multiplicative coalescent[END_REF]). Let Y n ∈ 2 + (Γ n ) for each n > 1, let Σ n be the analogue of (2.19) and let Ξ n be the associated SBPP. Suppose that

Ξ n d -→ Ξ ∞ ,
as n → ∞, for the topology of vague convergence of counting measures on [0, ∞) × (0, ∞), where

Ξ ∞ is some point process satisfying 1. sup{s : (s, y) ∈ Ξ ∞ for some y} = ∞ a.s. 2. if (s, y) ∈ Ξ ∞ then (s ,y )∈Ξ ∞ :s <s y = s a.s. 3. max{y : (s, y) ∈ Ξ ∞ for some s > t} p → 0 as t → ∞. Then Y ∞ := ord(π(Ξ ∞ )) ∈ 2 ↓ and ord(Y n ) d -→ ord(Y ∞ ) in 2 ↓ . In addition, ord(Y n γ , Σ n γ , γ ∈ Γ n ) d -→ ord(Y γ , Σ γ , γ ∈ Γ)
as n → ∞, where the convergence is in 2 ↓ for the first coordinate and in the product topology for the second.

The original statement does not mention the last convergence. It is, in fact, implicitly contained in the proof of Proposition 15 given in [START_REF] Aldous | Brownian excursions, critical random graphs and the multiplicative coalescent[END_REF], more precisely in the assertion that the tightness of the sequence (Σ n (a)) n≥1 for arbitrary a > 0 and the convergence Ξ

n d -→ Ξ ∞ together imply the convergence ord(Y n ) d -→ ord(Y ∞ ) in 2 .
For k ≥ 1, we let

Y n k = n -α/(α+1) [ζ n (k) -(N n (σ n (k)) -N n (σ n (k -1)) + 1)] .
Recall that at the end of the exploration of the (k -1)th component of M n (ν) in depthfirst order, we choose a new vertex from the unexplored parts of the graph with probability proportional to its degree. So we pick a component with probability proportional to the sum of its degrees, which is twice the number of its edges. Since the number of steps it takes to explore a component of M n (ν) is the number of its vertices (which is the number of its non-back-edges plus one) plus twice the number of back-edges (unlike the non-back-edges, a back-edge takes one step to create and another step to close), it follows that Y n k is the number of edges of the

kth component of M n (ν) times n -α/(α+1) . For k ≥ 1, let Σ n k = k-1 i=1 Y n i = n -α/(α+1) [σ n (k -1) -N n (σ n (k -1)) -k + 1] ,
and put

Ξ n = {(Σ n k , Y n k ) : k ≥ 1}.
It is easy to see that Ξ n then has the same distribution as the SBPP associated with

n -α/(α+1) (ζ n (k) -N n (σ n (k)) + N n (σ n (k -1)) -1, k ≥ 1).
Recall from Section 2.3 the notation ( ε ( ) , ≥ 0) for the excursions of the reflected process (R t , t ≥ 0) indexed by local time , and ( σ , ≥ 0) for the inverse local time process. Let Ξ ∞ be the point process given by

Ξ ∞ = {( σ -, ζ( ε ( ) )) : ≥ 0, σ -σ -> 0}.
By Proposition 2.3.1 and Lemma 2.3.5, properties (1), ( 2) and (3) from Proposition 2.5.7 above hold for Ξ ∞ . In order to apply Proposition 2.5.7, it thus remains to establish the convergence of Ξ n to Ξ ∞ . We do this by first proving a deterministic result for a suitable class of functions, extending Lemma 7 of Aldous [START_REF] Aldous | Brownian excursions, critical random graphs and the multiplicative coalescent[END_REF] from the setting of continuous functions to the setting of càdlàg functions satisfying certain conditions.

For a càdlàg function f : [0, ∞) → R with only positive jumps, let E(f ) be the set of nonempty intervals e = (l, r) such that f (l) = inf s≤l f (s) = f (r) and f (s) > f (l) for all s ∈ (l, r).

We say that such intervals are excursions of f . Let S denote the set of functions f : [0, ∞) → R satisfying the following conditions:

1. f is càdlàg and has only non-negative jumps.

2. f (x) → -∞ as x → ∞. 3. If 0 ≤ a < b and f (b-) = inf a≤s≤b f (s) then f (b) = f (b-).
4. For each > 0, E(f ) contains only finitely many excursions of length greater than or equal to .

5. The complement of ∪ (l,r)∈E(f ) (l, r) has Lebesgue measure 0.

6. If (l 1 , r 1 ), (l 2 , r 2 ) ∈ E(f ) and l 1 < l 2 , then f (l 1 ) > f (l 2 ).

Lemma 2.5.8. Let f ∈ S and let (f n ) n≥1 be a sequence of càdlàg functions such that f n → f as n → ∞ in the Skorokhod sense. For each n ∈ N, let (t n,i ) i≥1 be a strictly increasing sequence such that

(i) t n,1 = 0 and lim i→∞ t n,i = ∞, (ii) f n (t n,i ) = inf s≤t n,i f n (s), (iii) for each s < ∞, lim n→∞ max i:t n,i ≤s (f n (t n,i ) -f n (t n,i+1 )) = 0. Write Ξ = {(l, r -l) : (l, r) ∈ E(f )} and Ξ n = {(t n,i , t n,i+1 -t n,i ) : i ≥ 1} for n ≥ 1. Then Ξ n → Ξ
as n → ∞, where the convergence holds in the topology of vague convergence of counting measures on [0, ∞) × (0, ∞).

Proof. We adapt the proof of Lemma 4.8 of Martin & Ráth [START_REF] Martin | Rigid representations of the multiplicative coalescent with linear deletion[END_REF]. Suppose that (l, r) is an excursion of f . Fix ∈ (0, r -l). Since f ∈ S, there exists δ > 0 such that

f (x) ≥ f (l) + δ for all x ∈ [0, l -/2] f (x) ≥ f (l) + δ for all x ∈ [l + /2, r -/2] f (x) ≤ f (l) -δ for some x ∈ (r, r + /2].
The first line is a consequence of conditions ( 1) and ( 6) in the definition of the set S: if for every n > 0, there exists

x n ∈ [0, l-/2] such that f (x n ) < f (l)+1
/n, then by condition (1) there would be an accumulation point

x ∞ ∈ [0, l-/2] of the sequence (x n ) n≥1 such that f (x ∞ ) ≤ f (l),
and hence there would exist an interval (l , r ) with r < r such that f (r ) ≤ f (x ∞ ) ≤ f (r). But this contradicts [START_REF] Addario-Berry | The continuum limit of critical random graphs[END_REF].

The second line follows from a similar argument (there cannot be an accumulation point

x ∞ ∈ [l + /2, r -/2] such that f (x ∞ ) ≤ f (l) since (l, r
) is an excursion interval). The third line is again a consequence of condition [START_REF] Addario-Berry | The continuum limit of critical random graphs[END_REF] in the definition of the set S.

Since we have f n → f in the Skorokhod sense, there exist n 0 and a sequence of continuous strictly increasing functions

λ n : [0, ∞) → [0, ∞) such that λ n (0) = 0, lim t→∞ λ n (t) = ∞ and for all n ≥ n 0 , |f n (λ n (x)) -f (x)| < δ/2 for all x ∈ [0, r + /2]
and

|λ n (x) -x| < /2 for all x ∈ [0, r + /2].
Then for n ≥ n 0 ,

f n (λ n (x)) ≥ f (l) + δ/2 for all x ∈ [0, l -/2] f n (λ n (x)) > f (l) -δ/2 for all x ∈ [l -/2, l + /2] f n (λ n (l)) < f (l) + δ/2 f n (λ n (x)) ≥ f (l) + δ/2 for all x ∈ [l + /2, r -/2] f n (λ n (x)) ≤ f (l) -δ/2 for some x ∈ [r, r + /2].
For the second point, note that

f n (λ n (x)) > f (x) -δ/2 ≥ f (r) -δ/2 = f (l) -δ/2 for every x ≤ l + /2 < r. Therefore, f n (x) ≥ f (l) + δ/2 for all x ∈ [0, l -] f n (x) > f (l) -δ/2 for all x ∈ [l -, l + ] f n (x) < f (l) + δ/2 for some x ∈ [l -/2, l + /2]. f n (x) ≥ f (l) + δ/2 for all x ∈ [l + , r -] f n (x) ≤ f (l) -δ/2 for some x ∈ [r -/2, r + ].
From the first and third inequalities, the set

I := {x ∈ [l -, r + ], f n (x) ≤ inf y≤l-f n (y)} is not empty. Let l (n) := inf I and let (x k ) k≥1 be a decreasing sequence in I converging to l (n) . Since f n is right-continuous, f n (l (n) ) = lim k→∞ f n (x k ), and f n (l (n) ) ≤ inf x≤l-f n (x), so that l (n) = min I. Clearly, f n (l (n) ) = inf x≤l (n) f n (x)
Similarly, from the first, second, fourth and fifth inequalities, we obtain the existence of r

(n) ∈ [r -, r + ] such that f n (r (n) ) = inf x≤r (n) f n (x).
Now fix η > 0. We can find t > 0 such that there are no excursions of length exceeding η in E(f ) which intersect [t, ∞). Then there exists a finite collection {(l i , r i ) : 1 ≤ i ≤ m} of excursions in E(f ) with l i ≤ t + η for 1 ≤ i ≤ m and such that ∪ 1≤i≤m (l i , r i ) covers all of [0, t + η] except for a set of Lebesgue measure at most η/2. Set = η/4m and apply the above argument for each excursion, to see that for n sufficiently large, there exist disjoint intervals (l

(n) 1 , r (n) 1 ), . . . , (l (n) m , r (n) m ) such that |l (n) i -l i | < η/4m and |r (n) i -r i | < η/4m for 1 ≤ i ≤ m.
But then the remaining length in [0, t + η] is at most η and so, in particular, we must have captured all possible intervals (t n,i , t n,i+1 ) with t n,i ≤ t+η and t n,i+1 -t n,i ≥ η, up to an error of at most η/4m at each end-point. The required vague convergence follows straightforwardly.

Lemma 2.5.9. We have L ∈ S almost surely. Moreover,

Ξ n d -→ Ξ ∞
as n → ∞, where the convergence holds in the topology of vague convergence of counting mea-

sures on [0, ∞) × (0, ∞).
Proof. L is clearly càdlàg with only non-negative jumps almost surely. The other conditions required for a function to lie in S follow from Proposition 2.3.1 and Lemma 2.3.5.

Now let f = L and f n = n -1/(α+1) X n ( n α/(α+1) • ). It is clear that Ξ ∞ is Ξ of the previous lemma for this f . For n ≥ 1, i ≥ 0, let t n,i = n -α/(α+1) σ n (i). Then t n,0 = 0 and lim i→∞ t n,i = ∞.
By construction, the t n,i are times at which new infima of f n are reached. Moreover,

f n (t n,i ) -f n (t n,i+1 ) = n -1/(α+1) (X n (σ n (i)) -X n (σ n (i + 1))) = n -1/(α+1) .
Hence, the (t n,i ) i≥1 satisfy the conditions in Lemma 2.5.8. It follows that

Ξ n = {(t n,i , t n,i+1 -t n,i ), i ≥ 1} d -→ Ξ
as n → ∞. Now we have

Σ n i = t n,i -n -α/(α+1) [N n (σ n (i)) + i]
and

Y n i = t n,i -t n,i-1 -n -α/(α+1) [N n (σ n (i)) -N n (σ n (i -1)) + 1].
Since n -α/(α+1) N n (σ n (i)) → 0 for each i ≥ 0, it is straightforward to see that Ξ n and {(t n,i , t n,i+1 -t n,i ), i ≥ 1} can be made arbitrarily close in the vague topology by taking n large. Hence, Ξ n d -→ Ξ ∞ as desired.

Proposition 2.5.7 tells us that we may now extract the ordered excursion lengths, and that we can add the convergence of the starting points of the excursions. This completes the proof of Proposition 2.5.6. As an aside, we observe that Proposition 2.5.6 gives us an analogue of Corollary 16 of [START_REF] Aldous | Brownian excursions, critical random graphs and the multiplicative coalescent[END_REF], as follows.

Corollary 2.5.10. The point process

Ξ ∞ = {( σ ( ) , ζ( ε ( ) )) : ≥ 0, σ -σ -> 0} consisting of
the left endpoints and the lengths of the excursions of R is distributed as the SBPP associated with the set of excursion lengths of R.

Marked excursions

We now strengthen the convergence in Proposition 2.5.6 to a convergence of ordered marked excursions. Let us first prove a deterministic analytic result, similar in spirit to Lemma 2.5.8, which we shall use to handle the positions of the back-edges (recall that when a back-edge is discovered, its other endpoint is explored at the first time when the corresponding mark in the stack reaches the top of the stack).

Lemma 2.5.11. Let f ∈ S and let (f n ) n≥1 be a sequence of càdlàg functions such that f n → f as n → ∞ in the Skorokhod sense. For each n ∈ N, let (s n , y n ) be a pair of points such that

s n ≥ 0 and y n ≤ f n (s n ). Let (s, y) be such that s > 0, f (s-) = f (s) and 0 < y < f (s). Let t n = inf{u ≥ s n : f n (u) ≤ y n } and t = inf{u ≥ s : f (u) ≤ y}. Suppose that (s n , y n ) → (s, y), that t is not a local minimum of f and that f (t-) = f (t). Then t n → t as n → ∞. Proof. Fix 0 < < t -s. Since f ∈ S, y < f (s), f (s-) = f (s), f (t-) = f (t) and t is not a local minimum of f , there exists δ > 0 such that f (x) ≥ y + δ for all x ∈ [s -/2, t -/2] f (x) ≤ y -δ for some x ∈ (t, t + /2].
As f n → f , there exist n 0 and a sequence of continuous strictly increasing functions

λ n : [0, ∞) → [0, ∞) such that λ n (0) = 0, lim x→∞ λ n (x) = ∞ and, for all n ≥ n 0 , |f n (λ n (x)) -f (x)| < δ/2 for all x ∈ [0, t + /2]
and

|λ n (x) -x| < /4 for all x ∈ [0, t + ].
Then

f n (λ n (x)) ≥ y + δ/2 for all x ∈ [s -/2, t -/2] f n (λ n (x)) ≤ y -δ/2 for some x ∈ (t, t + /2].
By taking n 0 larger if necessary, we also have

|s -s n | < /4 and |y -y n | < δ/2
for all n ≥ n 0 . Then for all n ≥ n 0 , we have |λ -1 n (s n ) -s| < /2 and so

f n (x) ≥ y + δ/2 for all x ∈ [s n , t -].
It follows that

f n (x) > y n for all x ∈ [s n , t -].
Moreover, it must be the case that f n goes below y n in the time-interval [λ n (t-/2), λ n (t+ /2)],

i.e. we must have

t n ∈ [λ n (t -/2), λ n (t + /2)]. But then t n ∈ [t -, t + ] for all n ≥ n 0 .
As was arbitrary, the result follows.

Let

M n (k) = N n (σ n (k)) -N n (σ n (k -1)),
be the number of back-edges falling in the excursion ε n k . Suppose that M n (k) ≥ 1. Then for 1 ≤ r ≤ M n (k), let g n k + s n k,r be the rescaled time at which the rth back-edge is discovered in the kth component and let n 1/(α+1) x n k,r ≥ 0 be its position on the stack. Let g n k + t n k,r be the rescaled time at which the corresponding marked leaf in the stack is killed, thus closing the rth back-edge. It is the first time that the stack size goes below the height of that leaf after its discovery (the stack being a LIFO queue), so that we have

t n k,r = inf{t ≥ s n k,r : ε n k (t) ≤ x n k,r -n -1/(α+1) }. Finally, if M n (k) ≥ 1, let P n k = M n (k) r=1 δ (s n k,r ,t n k,r ) define a point measure on [0, ζ(ε n k )] 2 . If M n (k) = 0, let P n k be the null measure. Let Q n = k≥1 M n (k) r=1 δ (g n k +s n k,r ,g n k +t n k,r ) ,
the point measure encompassing all of the pairs of rescaled times at which a back-edge is opened and closed.

Turning now to the limiting process, recall that ( ε i , i ≥ 1) are the excursions of R listed in decreasing order of length, and that the sequence (ζ( ε i ), i ≥ 1) lies in 2 ↓ . Let M i be the number of marks falling in the excursion ε i , and if M i ≥ 1, write s i,1 , . . . , s i,M i for the times and x i,1 , . . . , x i,M i for the positions of the marks, respectively. For 1 ≤ r ≤ M i , let

t i,r = inf{t ≥ s i,r : ε i (t) ≤ x i,r }. If M i ≥ 1, write P i = M i r=1 δ (s i,r ,t i,r )
, and if M i = 0, let P i be the null measure. Finally, let

Q = i≥1 M i r=1 δ (g i +s i,r ,g i +t i,r ) .
Recall that

σ n (i) = min{k : S n (k) ≤ -i} and let ζ n (i) = σ n (i) -σ n (i -1
) be the length of the ith excursion of S n above its running minimum. Since the components of F n (ν) again appear in size-biased order, an argument completely analogous to that above gives that n -α/(α+1) ord(

ζ n (k), k ≥ 1) converges in distribution to (ζ( ε i ), i ≥ 1) in 2 ↓ . In particular, n -α/(α+1) max i≥1 ζ n (i) d -→ ζ( ε 1 ),
where we recall that ζ( ε 1 ) is the length of the longest excursion of R above 0; in particular, by Proposition 2.3.1 and Lemma 2.3.5 we have ζ( ε 1 ) < ∞ a.s.

For i ≥ 1, let M n (i) be the number of back-edges falling among the vertices corresponding to the (2i -1)th and 2ith components of F n (ν), and let N n max = max k≥1 M n (k). On the pair formed of the (2i -1)th and 2ith components of F n (ν), at corresponding vertices, we have that the size of the stack in F n (ν) is bounded above by the size of the stack in F n (ν) plus 1.

Proposition 2.5.12. We have

ord (ζ(ε n k ), ε n k , P n k , k ≥ 1) d -→ (ζ( ε i ), ε i , P i , i ≥ 1)
as n → ∞. Here, for each k ≥ 1, the convergence in the second co-ordinate is for the Skorokhod topology and in the third for the Hausdorff distance on R 2 + ; then we take the product topology over the different indices.

Proof. Because the points {(s i,r , x i,r ) : 1 ≤ r ≤ M i } are picked uniformly from the Lebesgue measure under the excursion ε i (recall that these points are the marks of a uniform Cox process under R, Theorem 2.5.5), for each i ≥ 1, and such an excursion has only countably many discontinuities (all of which are up-jumps), the conditions of Lemma 2.5.11 are fulfilled almost surely. Combining this with Theorem 2.5.5, we may deduce the joint convergence in distribution

as n → ∞ of (n -1/(α+1) R n ( tn α/(α+1) ), t ≥ 0) → (R t , t ≥ 0)
in the Skorokhod topology and Q n → Q in the topology of vague convergence on R 2 + . By Skorokhod's representation theorem, we may work on a probability space such that this joint convergence holds almost surely. Fix K ∈ N. We have already shown that, given δ > 0, there exist M > 0 and n 0 sufficiently large such that, with probability at least 1 -δ, the K longest excursions of both n -1/(α+1) R n ( • n α/(α+1) ) and R occur in the time-interval [0, M ] for all n ≥ n 0 . Since Q n → Q and Q has only finitely many points in [0, M ] 2 , we may deduce that the K smaller point processes obtained by restricting Q n to each of the K longest excursionintervals converge in the sense of the Hausdorff distance to P 1 , . . . , P K respectively. The result follows.

We conclude this section with some technical bounds on the number of back-edges in a given component, of which we will make use later.

Lemma 2.5.13. Almost surely, we have #{i ≥ 1 :

M i ≥ 2} < ∞ and N max := sup i≥1 M i < ∞.
Proof. We will bound

E #{i ≥ 1 : M i ≥ 2} (ζ( ε i )) i≥1 = i≥1 P M i ≥ 2 ζ( ε i ) ,
where we have used the independence of the excursions given their lengths. We use the crude bound P (Po(λ) ≥ 2) ≤ λ 2 . For any i ≥ 1, we have

P M i ≥ 2 ζ( ε i ) = x = P Po 1 µ x 0 ε i (u)du ≥ 2 ζ( ε i ) = x ≤ E N (x) 1 µ x 0 e(u)du 2 exp 1 µ x 0 e(u)du E N (x) exp 1 µ x 0 e(u)du ≤ E N (x)
1 µ

x 0 e(u)du

4 1/2 E N (x) exp 2 µ x 0 e(u)du 1/2
, by the Cauchy-Schwarz inequality and the fact that the denominator is bounded below by 1.

By the scaling property of the excursion e,

E N (x)
1 µ

x 0 e(u)du

4 1/2 ≤ Cx 2(1+1/α)
for some constant C > 0. Define

f (x) := E N (x) exp 2 µ
x 0 e(u)du

1/2 = E N (1) exp 2x 1+1/α µ 1 0 e(u)du 1/2
. This is clearly an increasing function of x, so that for

x ≤ ζ( ε 1 ) we have f (x) ≤ f (ζ( ε 1 )),
E #{i ≥ 1 : M i ≥ 2} (ζ( ε i )) i≥1 ≤ f (ζ( ε 1 )) i≥1 ζ( ε i ) 2(1+1/α) < ∞ a.s. It follows that #{i ≥ 1 : M i ≥ 2} < ∞ a.s.
Since the area of any individual excursion ε i is finite, it contains an almost surely finite number of points and this, together with the fact that only finitely many excursions contain more than 2 points, gives that N max < ∞ a.s.

Height process

In order to deal with the metric structure, we also need to know that the height process associated with X n converges. Let Our aim in this section is to prove that H n has H as its scaling limit, and that we can extract its marked excursions as for the exploration process. Recall that

H n (k) = # j ∈ {0, 1, . . . , k -1} : X n (j) = inf j≤ ≤k X n ( ) so that H n (k)
G n (k) = # j ∈ {0, 1, . . . , k -1} : S n (j) = inf j≤ ≤k S n ( ) ,
which is the height process corresponding to the forest F n (ν). We will compare H n and G n . It will be sufficient to do this for pairs of components of F n (ν). To this end, suppose that for a ≥ 0 and m ≥ 1, vertices v a+1 , v a+2 , . . . , v a+m form a pair of components of F n (ν), and that there are b back-edges on these vertices in M n (ν). Then the corresponding collection of components in F n (ν) together have m+2b vertices, which are visited at times c+a, c+a+1, . . . , c+a+m+2b-1 in the depth-first exploration, where c ≥ 0 is such that a = τ n (c + a). We compare H n with G n at the times τ n (c + a), τ n (c + a + 1), . . . , τ n (c + a + m + 2b -1).

Lemma 2.5.14. We have

max 0≤i≤m+2b-1 |H n (a + c + i) -G n (τ n (a + c + i))| ≤ 1 + b + 2b max 1≤i≤m | G n (a + i) -G n (a + i -1)|
Proof. Suppose first that b = 0. Until we come to the end of the first component of F n (ν), we have

H n (a + i) = G n (τ n (a + i)).
Thereafter, we have H n (a+i) = 1+ G n (τ n (a+i)), since the second component of F n (ν) becomes a subtree attached to the root of the first component of F n (ν).

For b > 0, we encourage the reader to refer back to Figure 2.1.

Write ∆ = max 1≤i≤m | G n (a + i) -G n (a + i -1)|.
The occurrence of a head or tail of a back-edge at time k implies τ n (k + 1) = τ n (k). To get from F n (ν) to F n (ν), we unplug a sequence of subtrees and plug them back in further along in the depth-first order. Within subtrees containing no back-edges, the increments of the height process remain the same as they are in the corresponding subtrees of F n (ν). Finally, every time we start a new subtree there is an extra difference of 1. It follows that the most by which the height process can be altered in going from F n (ν) to F n (ν) is an additive factor of 1 + b + 2b∆.

We are now ready to state and prove the main result of this section.

Proposition 2.5.15. Jointly with the convergence in Theorem 2.5.5, we have that as n → ∞,

n -α-1 α+1 H n ( tn α/(α+1) ), t ≥ 0 d -→ H t , t ≥ 0 in D(R + , R + ).
Proof. By Theorem 2.4.1, we have

n -α-1 α+1 G n ( n α α+1 u ), u ≥ 0 d -→ ( H u , u ≥ 0).
By Theorem 1.4.3 of Duquesne and Le Gall [START_REF] Duquesne | Random trees, Lévy processes and spatial branching processes[END_REF], (H u , u ≥ 0) is almost surely continuous. By the absolute continuity in Proposition 2.3.2, the same is true of ( H u , 0 ≤ u ≤ t). It follows that for any t > 0,

n -α-1 α+1 sup 1≤j≤ tn α/(α+1) | G n (j) -G n (j -1)| d -→ 0 (2.20)
as n → ∞.

By Lemma 2.5.13, if δ > 0, there exists K δ < ∞ such that

P (ζ( ε 1 ) > K δ ) < δ and P (N max > K δ ) < δ.
In particular, starting from any time k, the number of steps until we next reach the beginning of an odd-numbered component is bounded above by 2K δ n α/(α+1) with probability at least

1 -δ + o(1).
Now fix t > 0 and > 0. Then by Lemma 2.5.14, we have

P n -α-1 α+1 sup 0≤k≤ tn α/(α+1) |H n (k) -G n (τ n (k))| > ≤ P n -α/(α+1) max i≥1 ζ n (i) > K δ + P (N n max > K δ ) + P 1 + K δ + 2K δ sup 1≤j≤ (t+2K δ )n α/(α+1) | G n (j) -G n (j -1)| > n α-1 α+1
.

Using (2.20), we obtain that

P 1 + K δ + 2K δ sup 1≤j≤ (t+2K δ )n α/(α+1) | G n (j) -G n (j -1)| > n α-1 α+1 → 0 as n → ∞. It follows that lim sup n→∞ P n -α-1 α+1 sup 0≤k≤ tn α/(α+1) |H n (k) -G n (τ n (k))| > < 2δ.
But δ > 0 was arbitrary and so by Lemma 2.5.1 we may deduce that

n -α-1 α+1 H n ( un α/(α+1) ), 0 ≤ u ≤ t d -→ H u , 0 ≤ u ≤ t . Now let h n k (t) = n -(α-1)/(α+1) H n (σ n (k -1) + tn α/(α+1) ), 0 ≤ t ≤ n -α/(α+1) ζ n (k).
Proposition 2.5.16. We have

ord (ε n k , h n k , P n k , k ≥ 1) d -→ ε i , h i , P i , i ≥ 1 as n → ∞.
Here, for each k ≥ 1, the convergence in the first co-ordinate is for the Skorokhod topology and in the second for the topology of vague convergence of counting measures on [0, ∞) 2 ;

then we take the product topology over different k.

Proof. We derive this from Proposition 2.5.15 by applying the same reasoning as in the proof of Proposition 2.5.12, using the fact that R n and H n have the same excursion-intervals.

The convergence of the metric structure

We have now assembled all of the ingredients needed in order to prove Theorem 2.1.1. Recall that

M n 1 , M n 2 , .
. . are the components of the random multigraph M n (ν), listed in decreasing order of size. We will make the distance and measure explicit in each by writing (M n i , d n i , µ n i ) i≥1 . Recall also that F n (ν) is the forest encoded by H n . In order to recover M n (ν) from F n (ν) we remove pairs of marked leaves and replace them by back-edges as described above.

Proof of Theorem 2.1.1. Write (h n (i) , P n (i) ) for the ith element of the sequence ord (h n k , P n k , k ≥ 1). Write (T n i , d n i , µ n i )
for the tree with rescaled height process h n (i) and with mass n -α/(α+1) on each vertex. (For i ≥ 1, these are the trees of the forest F n (ν) listed in decreasing order of mass.) By Skorokhod's representation theorem, we may work on a probability space where the convergence

h n (i) , P n (i) , i ≥ 1 → ( h i , P i , i ≥ 1)
occurs almost surely. By (2.8), this entails that

d GHP ((T n i , d n i , µ n i ), ( T i , d i , µ i )) → 0 a.s.
as n → ∞. In order to obtain (

M n i , d n i , µ n i ) from (T n i , d n i , µ n i ) if P n (i) = m n (i) r=1 δ s n (i),r ,t n (i),r
we must remove the vertices encoded by s n (i),r and t n (i),r and replace them by a single edge between their parents, for each 1 ≤ r ≤ m n (i) . Since edges have rescaled length n -(α-1)/(α+1) → 0 it is straightforward to see (in the same manner as in the Proof of Theorem 22 in [START_REF] Addario-Berry | The continuum limit of critical random graphs[END_REF]) that we get

d GHP ((M n i , d n i , µ n i ), (G i , d i , µ i )) → 0 a.s. (2.21)
as n → ∞. The conclusion follows.

Appendix

A change of measure for spectrally positive Lévy processes

Let L be a spectrally positive Lévy process with Lévy measure π satisfying

∞ 0 (x ∧ x 2 )π(dx) < ∞. (2.22)
Then we may write the Laplace transform of L t as

E [exp(-λL t )] = exp(tΨ(λ)),
where

Ψ(λ) = γλ + δ 2 λ 2 2 + ∞ 0 π(dx)(e -λx -1 + λx).
We impose also that

γ ≥ 0, δ ≥ 0 (2.23)
and that at least one of the two following conditions holds: We note that t 0 Ψ(θs)ds < ∞ for all θ > 0 and all t > 0.

δ > 0 or ∞ 0 xπ(dx) = ∞. ( 2
Lemma 2.6.1. For any θ > 0, we have

E exp -θ t 0 sdL s = exp t 0 Ψ(θs)ds = E exp θ t 0 (L s -L t )ds .
In consequence, the process

exp -θ t 0 sdL s - t 0 Ψ(θs)ds , t ≥ 0 is a martingale.
Proof. Let M (ds, dx) be a Poisson random measure on R + of intensity ds ⊗ π(dx), and let M (ds, dx) be its compensated version. Then

E exp -θ t 0 ∞ 0 sx M (ds, dx) = exp t 0 ds ∞ 0 π(dx)(e -θsx -1 + θsx) .
If B is a standard Brownian motion, we obtain

E exp -θ t 0 sdB s = exp 1 2 θ 2 t 0 s 2 ds .
Since we may, in general, realise L as

L t = -γt + δB t + t 0 ∞ 0 x M (ds, dx),
where B and M are independent, we obtain

E exp -θ t 0 sdL s = E exp θ t 0 γsds -θ t 0 δsdB s -θ t 0 ∞ 0 xs M (ds, dx) = exp γ t 0 θsds + 1 2 δ 2 t 0 θ 2 s 2 ds + t 0 ds ∞ 0 π(dx)(e -θsx -1 + θsx) = exp t 0 Ψ(θs)ds .
The second equality in the statement of the lemma follows on integrating by parts, and the martingale property follows since L has independent increments. This martingale plays an important role as a Radon-Nikodym derivative. Fix θ > 0 and consider the process X with independent (but non-stationary) increments and Laplace transform

E [exp(-λX t )] = exp t 0 ds ∞ 0 π(dx)(e -λx -1 + λx)e -θxs .
(The process X may again be realised as a stochastic integral with respect to a compensated

Poisson random measure on R + × R + , but this time with intensity exp(-xs)dsπ(dx).) Let

A t = - 1 θ Ψ(θt) = -γt - 1 2 θδ 2 t 2 - 1 θ ∞ 0 π(dx)(e -θtx -1 + θtx)
and L t = δB t + X t + A t . (Note that this is expressed as the Doob-Meyer decomposition of L, with δB t + X t the martingale part.)

Proposition 2.6.2. For any θ > 0 and every t ≥ 0, we have the following absolute continuity relation: for every non-negative integrable functional F ,

E F ( L s , 0 ≤ s ≤ t) = E exp -θ t 0 sdL s - t 0 Ψ(θs)ds F (L s , 0 ≤ s ≤ t) .
Proof. Observe first that Lemma 2.6.1 entails that the change of measure is well-defined for each t ≥ 0.

Let us first deal with the case where γ = δ = 0. We use a decomposition of the Lévy measure similar to that in Bertoin [START_REF] Bertoin | Factorizing Laplace exponents in a spectrally positive Lévy process[END_REF] or the proof of Proposition 1 in Miermont [START_REF] Miermont | Self-similar fragmentations derived from the stable tree. II. Splitting at nodes[END_REF]:

t 0 ds ∞ 0 π(dx)(e -λx -1 + λx)e -θxs = t 0 ds ∞ 0 π(dx)(e -(λ+θs)x -1 + (λ + θs)x) - t 0 ds ∞ 0 π(dx)(e -θxs -1 + θxs) - t 0 ds ∞ 0 π(dx)λx(1 -e -θxs ) = t 0 Ψ(λ + θs)ds - t 0 Ψ(θs)ds - t 0 ds ∞ 0 π(dx)λx(1 -e -θxs ).
The last integral on the right-hand side makes sense because of the integrability condition (2.22).

Indeed, it may be calculated as follows:

t 0 ds ∞ 0 π(dx)λx(1 -e -θxs ) = λ ∞ 0 xπ(dx) t 0 (1 -e -θxs )ds = λ θ ∞ 0 π(dx)(e -θtx -1 + θtx) = λ θ Ψ(θt).
Hence,

E [exp(-λX t )] = exp t 0 ds ∞ 0 π(dx)(e -λx -1 + λx)e -θxs = exp - λ θ Ψ(θt) + t 0 Ψ(λ + θs)ds - t 0 Ψ(θs)ds
and we obtain

E [exp(-λ(X t + A t ))] = exp t 0 Ψ(λ + θs)ds - t 0 Ψ(θs)ds .
Consider the stochastic integral

t 0 (λ + θs)dL s = λL t + θ t 0 sdL s .
We have

E exp - t 0 (λ + θs)dL s = exp t 0 Ψ(λ + θs))ds
and so

E [exp(-λ(X t + A t ))] = E exp -λL t -θ t 0 sdL s - t 0 Ψ(θs)ds . Suppose now that 0 = t 0 < t 1 < • • • < t m = t. Let λ 1 , . . . , λ m ∈ R + .
Then, by the fact that

X has independent increments, E exp - m i=1 λ i ( L t i -L t i-1 ) = m i=1 E exp(-λ i ( L t i -L t i-1 )) .
By the same argument as above, we then have

E exp - m i=1 λ i ( L t i -L t i-1 ) = m i=1 E exp - t i t i-1 (λ i + θs)dL s - t i t i-1 Ψ(θs)ds = E exp - m i=1 t i t i-1 (λ i + θs)dL s - t 0 Ψ(θs)ds ,
since L also has independent increments. Again by integration by parts, we then get that the right-hand side is equal to

E exp - m i=1 λ i (L t i -L t i-1 ) -θ t 0 sdL s - t 0 Ψ(θs) .
This yields the claimed result for γ = δ = 0. Now let us instead suppose that γ ≥ 0, δ > 0 and there is no jump component i.e. L t = -γt + δB t and L t = -γt + δB t -δθt 2 /2. Then by the Cameron-Martin-Girsanov formula (see, for example, Section 5.6 of Le Gall [START_REF] Gall | Brownian motion, martingales, and stochastic calculus[END_REF]),

E f ( L s , 0 ≤ s ≤ t) = E exp -δθ t 0 sdB s -δ 2 θ 2 t 0 s 2 ds f (L s , 0 ≤ s ≤ t) = E exp -θ t 0 sdL s - t 0 Ψ(θs)ds f (L s , 0 ≤ s ≤ t) .
The result for general γ, δ and π now follows using the independence of X and B.

The α-stable case stated in Proposition 2.3.2 is obtained by setting γ = δ = 0, π(dx) = c µ x -(α+1) dx and θ = 1/µ. The Brownian case is obtained by taking γ = 0, δ = β/µ, θ = 1/µ and no Lévy measure π.

Size-biased reordering

In this section, we prove some elementary results about the size-biased reordering ( Dn

P Dn 1 = k 1 , Dn 2 = k 2 , . . . , Dn n = k n = P D Σ(1) = k 1 , D Σ(2) = k 2 , . . . , D Σ(n) = k n = σ∈Sn P D σ(1) = k 1 , D σ(2) = k 2 , . . . , D σ(n) = k n , Σ = σ = σ∈Sn P D σ(1) = k 1 , D σ(2) = k 2 , . . . , D σ(n) = k n k 1 n j=1 k j k 2 n j=2 k j . . . k n k n = n! ν k 1 ν k 2 . . . ν kn k 1 n j=1 k j k 2 n j=2 k j . . . k n k n ,
since D 1 , . . . , D n are i.i.d. with law ν. Rearrangement of this expression yields

P Dn 1 = k 1 , Dn 2 = k 2 , . . . , Dn n = k n = k 1 ν k 1 k 2 ν k 2 . . . k n ν kn n i=1 (n -i + 1) n j=i k j = k 1 ν k 1 µ k 2 ν k 2 µ • • • k n ν kn µ n i=1 (n -i + 1)µ n j=i k j . Now P Dn 1 = k 1 , Dn 2 = k 2 , . . . , Dn m = k m = k m+1 ,...,kn≥1 P Dn 1 = k 1 , Dn 2 = k 2 , . . . , Dn n = k n = k 1 ν k 1 µ k 2 ν k 2 µ • • • k m ν km µ µ m n! k m+1 ,...,kn≥1 k m+1 ν k m+1 • • • k n ν kn n i=1 1 n j=i k j = k 1 ν k 1 µ k 2 ν k 2 µ • • • k m ν km µ µ m n! × k m+1 ,...,kn≥1 m i=1 1 m j=i k j + n j=m+1 k j ν k m+1 . . . ν kn n =m+1 k n j= k j = k 1 ν k 1 µ k 2 ν k 2 µ • • • k m ν km µ µ m n! (n -m)! × k m+1 ,...,kn≥1 m i=1 1 m j=i k j + n j=m+1 k j P Dn-m 1 = k m+1 , . . . , Dn-m n-m = k n .
We have that n-m j=1 Dn-m

j d = n j=m+1 D j = Ξ n-m . It follows that the last expression is equal to k 1 ν k 1 µ k 2 ν k 2 µ • • • k m ν km µ n!µ m (n -m)! E m i=1 1 m j=i k j + Ξ n-m
, and the claimed result follows.

A simple consequence of Proposition 2.4.2 is the following stochastic domination.

Lemma 2.6.3. We have

( Dn 1 , Dn 2 , . . . , Dn n ) ≤ st (Z 1 , Z 2 , . . . , Z n ).
Proof. By Proposition 2.4.2 we have

P Dn 1 ≥ d 1 , Dn 2 ≥ d 2 , . . . , Dn n ≥ d n = E n i=1 (n -i + 1)µ n j=i Z j 1 {Z 1 ≥d 1 ,Z 2 ≥d 2 ,...,Zn≥dn} . Let f (k 1 , k 2 , . . . , k n ) = n i=1 (n -i + 1)µ n j=i k j and g(k 1 , k 2 , . . . , k n ) = 1 {k 1 ≥d 1 ,k 2 ≥d 2 ,...,kn≥dn} .
Then f is a decreasing function of its arguments and g is an increasing function of its arguments.

It follows from the FKG inequality that

E [f (Z 1 , Z 2 , . . . , Z n )g(Z 1 , Z 2 , . . . , Z n )] ≤ E [f (Z 1 , Z 2 , . . . , Z n )] E [g(Z 1 , Z 2 , . . . , Z n )] . But E [f (Z 1 , Z 2 , . . . , Z n )] = 1 and so P Dn 1 ≥ d 1 , Dn 2 ≥ d 2 , . . . , Dn n ≥ d n ≤ P (Z 1 ≥ d 1 , Z 2 ≥ d 2 , . . . , Z n ≥ d n ) ,
as required.

Lemma 2.6.4. Fix α ∈ (1, 2) and suppose m = O(n β ) for some β < α/2. Then as n → ∞,

1 n m i=1 ( Dn i ) 2 p → 0.
In particular, the above holds for m = tn α/(α+1) .

Proof. By Lemma 2.6.3, it is sufficient to prove that

1 n m i=1 Z 2 i p → 0.
By Theorem 2.5.9 of Durrett [START_REF] Durrett | Probability: theory and examples[END_REF], we have lim sup

m→∞ 1 m 1/β m i=1 Z 2 i = 0 a.s.
if and only if

∞ m=1 P Z 2 1 > m 1/β < ∞. But P Z 2 1 > m 1/β = P Z 1 > m 1/2β = O(m -α/2β
), which is summable since α > 2β.

Lemma 2.6.5. As n → ∞,

1 n n i= tn α/(α+1) +1 Dn i p → µ.
Proof. We have

n i= tn α/(α+1) +1 Dn i = n i=1 Dn i - tn α/(α+1) i=1 Dn i = n i=1 D i - tn α/(α+1) i=1 Dn i .
By the weak law of large numbers,

1 n n i=1 D i p → E [D 1 ] = µ
and, since Dn i ≥ 1 for 1 ≤ i ≤ n, by Lemma 2.6.4 we have that

1 n tn α/(α+1) i=1 Dn i p → 0.
The result follows.

Convergence of the measure-change in the Brownian case

Recall that we have µ :

= E [D 1 ], E D 2 1 = 2µ and β := E [D 1 (D 1 -1)(D 1 -2)], so that E D 3 1 = β + 4µ. Lemma 2.6.6. Let L(λ) := E [exp(-λD 1 )]. Then as λ → 0, L(λ) = exp -λµ + λ 2 µ(2 -µ) 2 - λ 3 6 (β + 4µ -6µ 2 + 2µ 3 ) + o(λ 3 ) . (2.25)
Proof. The first three cumulants of D 1 are

E [D 1 ] = µ, var (D 1 ) = µ(2 -µ), E (D 1 -µ) 3 = β + 4µ -6µ 2 + 2µ 3 ,
and the result follows immediately.

Recall from Proposition 2.4.2 that

φ n m (k 1 , k 2 , . . . , k m ) = E m i=1 (n -i + 1)µ m j=i k j + Ξ n-m .
We prove the following lemma.

Lemma 2.6.7. Let s(0) = 0 and s(i

) = i j=1 (k j -2) for i ≥ 1. Suppose that |s(i) -s(m)| ≤ n 1/3 log n for all 0 ≤ i ≤ m. Then if m = Θ(n 2/3 ), we have φ m n (k 1 , k 2 , . . . , k m ) ≥ exp 1 nµ m i=0 (s(i) -s(m)) - βm 3 6µ 3 n 2 (1 + o(1)),
where the o(1) term is independent of k 1 , . . . , k m ≥ 1 satisfying the conditions.

Proof. The method of proof is similar in spirit to, but somewhat more involved than, that of Lemma 2.4.7. Let us first introduce some useful notation. Let D i = D i -µ, the centred degree random variables, and let ∆ n-m := Ξ n-m -µ(n -m) be their sum. Let ψ be the log-Laplace

transform of D 1 , ψ(λ) = log E exp(-λD 1 )
, so that as λ → 0, we have

ψ(λ) = λ 2 µ(2 -µ) 2 - λ 3 6 (β + 4µ -6µ 2 + 2µ 3 ) + o(λ 3 ). (2.26) Now, φ m n (k 1 , . . . , k m ) = m i=1 1 - i -1 n × E exp - m i=1 log 1 + ∆ n-m + s(m) -s(i -1) + 2(m -i + 1) -µm µn .
We use Taylor expansion in order to approximate the exponent:

m i=1 log 1 - i -1 n -log 1 + ∆ n-m + s(m) -s(i -1) + (2 -µ)m -2(i -1) µn (2.27) = - m 2 2n - m 3 6n 2 + o(1) - m∆ n-m nµ + 1 nµ m i=0 (s(i) -s(m)) - (2 -µ)m 2 nµ + m 2 nµ + o(1) + m(∆ n-m ) 2 2n 2 µ 2 + 1 2µ 2 n 2 m i=0 (s(i) -s(m)) 2 + (2 -µ) 2 m 3 2µ 2 n 2 + 2m 3 3µ 2 n 2 - (2 -µ)m 3 µ 2 n 2 + o(1) + ∆ n-m µ 2 n 2 m i=0 (s(m) -s(i)) - (µ -1)m 2 ∆ n-m µ 2 n 2 + (2 -µ)m µ 2 n 2 m i=0 (s(m) -s(i)) - 2 µ 2 n 2 m i=0 i(s(m) -s(i)) + • • • As ∆ n-m
is a centred sum of i.i.d. random variables with finite variance, the central limit theorem applies and we have that

n -1/2 ∆ n-m d -→ N(0, µ(2 -µ))
as n → ∞. The desired lower bound will, however, be obtained by restricting to the moderate deviation event

E n = -(2 -µ)m -n 7/12 ≤ ∆ n-m ≤ -(2 -µ)m + n 7/12 .
On this event, for any 0 ≤ i ≤ m, we have

|∆ n-m | = O(n 2/3 ), |s(m) -s(i)| = O(n 1/3 log n) and |(2 -µ)m -2(i -1)| = O(n 2/3 ). So we have 1 2µ 2 n 2 m i=0 (s(i) -s(m)) 2 = o(1), (2 -µ)m µ 2 n 2 m i=0 (s(m) -s(i)) = o(1), ∆ n-m µ 2 n 2 m i=0 (s(m) -s(i)) = o(1), and - 2 µ 2 n 2 m i=0 i(s(m) -s(i)) = o(1),
and that the remainder term (hidden in the ellipsis) in the expansion of (2.27) is o(1). Using these facts we see that, on E n , the exponent (2.27) is equal to F n + o(1), where

F n := 1 nµ m i=0 (s(i) -s(m)) - (2 -µ)m 2 2µn - (2 -µ)(µ -1)m 3 3µ 2 n 2 + m(∆ n-m ) 2 2n 2 µ 2 - m µn + (µ -1)m 2 µ 2 n 2 ∆ n-m .
In order to find a lower bound on E [exp(F n )1 En ], we first consider the expectation of the stochastic part,

E exp m(∆ n-m ) 2 2n 2 µ 2 - m µn + (µ -1)m 2 µ 2 n 2 ∆ n-m 1 En .
Let θ > 0 (we shall choose a specific value for θ shortly) and define an equivalent measure Q

via dQ dP = exp (-θ∆ n-m -(n -m)ψ(θ)) .
Because the Radon-Nikodym derivative has a product form, under Q the random variables D 1 , D 2 , . . . , D n-m are still i.i.d. and each have mean

E Q D 1 = E [D 1 exp(-θD 1 )] E [exp(-θD 1 )] = -ψ (θ)
and variance var

Q (D 1 ) = ψ (θ). Now fix θ = m µn + m 2 µn 2 ,
so that

E Q [∆ n-m ] = -(n -m)ψ (θ) = -(n -m) (2 -µ)m n + (2 -µ)m 2 n 2 - (β + 4µ -6µ 2 + 2µ 3 )m 2 2µ 2 n 2 - (β + 4µ -6µ 2 + 2µ 3 )m 3 2µ 2 n 3 + o m 3 n 3 = -(2 -µ)m + O(n 1/3 ) and var Q (∆ n-m ) = (n -m)ψ (θ) = µ(2 -µ)(n -m) + O(n 1/3 ).
Using Chebyshev's inequality and the fact that n 1/3 n 7/12 , it follows that

Q(E c n ) ≤ Q(|∆ n-m + (2 -µ)m| > 1 2 n 7/12 ) ≤ 5var Q (∆ n-m ) n 7/6 = O(n -1/6 ). So E exp m(∆ n-m ) 2 2n 2 µ 2 - m µn + (µ -1)m 2 µ 2 n 2 ∆ n-m 1 En = exp (n -m)ψ m nµ + m 2 µn 2 E Q exp m(∆ n-m ) 2 2n 2 µ 2 + m 2 ∆ n-m µ 2 n 2 1 En ≥ exp (n -m)ψ m nµ + m 2 µn 2 Q(E n ) × exp m((2 -µ)m -n 7/12 ) 2 2n 2 µ 2 - m 2 ((2 -µ)m + n 7/12 ) µ 2 n 2 = (1 + o(1)) exp (n -m)ψ m nµ + m 2 µn 2 + (2 -µ) 2 m 3 2n 2 µ 2 - (2 -µ)m 3 µ 2 n 2 . Now, (n -m)ψ m nµ + m 2 µn 2 = (n -m) (2 -µ)m 2 2µn 2 + (2 -µ)m 3 µn 3 - (β + 4µ -6µ 2 + 2µ 3 )m 3 6µ 3 n 3 + o(1) = (2 -µ)m 2 2µn + (2 -µ)m 3 2µn 2 - (β + 4µ -6µ 2 + 2µ 3 )m 3 6µ 3 n 2 + o(1).
It follows that

φ m n (k 1 , . . . , k m ) ≥ (1 + o(1))E [exp(F n )1 En ] ≥ (1 + o(1)) exp (2 -µ)m 2 2µn + (2 -µ)m 3 2µn 2 - (β + 4µ -6µ 2 + 2µ 3 )m 3 6µ 3 n 2 + (2 -µ) 2 m 3 2n 2 µ 2 - (2 -µ)m 3 µ 2 n 2 - (2 -µ)m 2 2µn - (2 -µ)(µ -1)m 3 3µ 2 n 2 + 1 nµ m i=0 (s(i) -s(m)) = (1 + o(1)) exp 1 nµ m i=0 (s(i) -s(m)) - βm 3 6µ 3 n 2 ,
as claimed.

The event {|S(i)-S(m)| ≤ n 1/3 log n for 1 ≤ i ≤ m} has probability tending to 1 as n → ∞, and so the analogue of Proposition 2.4.3 now follows exactly as in the α ∈ (1, 2) case.

Convergence of a single large component for α ∈ (1, 2)

In this section, we consider a large component of the graph conditioned to have size xn α/(α+1) , for α ∈ (1, 2) only, and do the main technical work necessary to prove that it converges in distribution to a single component of the stable graph conditioned to have size x. By arguments analogous to those in Section 2.5, it is essentially sufficient to consider a single tree in the forest F n (ν) of size xn α/(α+1) , described by an excursion of the corresponding coding functions S n and G n . The main result of this section, Theorem 2.6.8, is a conditioned version of Theorem 2.4.1, which says that these excursions converge jointly in distribution to normalised excursions of L and H of length x. (This is precisely the analogue of Theorem 2.2.5 in the measure-changed setting.) At the end of the section, we sketch how to obtain the metric space scaling limit of a single large component of the graph.

For simplicity, we will make the assumption that the support of the law of D 1 is Z + so that excursions of any strictly positive length occur with positive probability. This assumption is not necessary, since the condition P (D 1 = k) ∼ ck -(α+2) implies that the greatest common divisor of {k ≥ 2 : P (D 1 -1 = k) > 0} is 1, so that the claimed results all hold for n sufficiently large.

Recall that (S(k), k ≥ 0) is a random walk which is skip-free to the left and in the domain of attraction of an α-stable Lévy process. Let

E m = S(k) ≥ 0 for 0 < k < m, S(m) = -1 ,
the event that the first m steps form an excursion above the running minimum. If m = n α/(α+1) x then, by Theorem 2.2.5,

E f n -1/(α+1) S( n α/(α+1) t ), n -(α-1)/(α+1) G( n α/(α+1) t ), 0 ≤ t ≤ x E m → N (x) [f (e, h)] .
More generally, write

E m 1 ,m 2 = S(m 1 ) = min 0≤k≤m 2 -1 S(k) = S(m 2 ) + 1
(so that E m = E 0,m ) and, similarly,

E n m 1 ,m 2 = S n (m 1 ) = min 0≤k≤m 2 -1 S n (k) = S n (m 2 ) + 1 ,
so that E n m 1 ,m 2 is the event that there is an excursion of S n above its running minimum between times m 1 and m 2 (recall that S n and G n have the same excursion intervals). This, of course, corresponds to a component of size m 2 -m 1 . Observe that the corresponding excursion of the height process starts and ends at 0. We will prove the following result.

Theorem 2.6.8. For any bounded continuous test function f , 0 ≤ t 1 < t 2 such that t 2 -t 1 = x,

and m 1 = t 1 n α/(α+1) , m 2 = t 2 n α/(α+1) , m = m 2 -m 1 then E f n -1/(α+1) [ S n ( (t 1 + t)n α/(α+1) ) -S n ( t 1 n α/(α+1) )], n -(α-1)/(α+1) G n ( (t 1 + t)n α/(α+1) ), 0 ≤ t ≤ n -α/(α+1) m E n m 1 ,m 2 → N (x) exp 1 µ x 0 e(t)dt f (e, h) N (x) exp 1 µ x 0 e(t)dt as n → ∞.
We need to prove a refinement of Lemma 2.4.7, to show that the change of measure is well-behaved at times when the process attains a new minimum. Proposition 2.6.9. Fix T > 0. For n ≥ 1 and m ≤ T n

α α+1 , let k (n) 1 , k (n) 2 , . . . , k (n) m ≥ 1 and let s (n) (i) = i j=1 (k (n) j -2) be such that s (n) (0) = 0 and s (n) (i) > s (n) (m) for 1 ≤ i ≤ m -1. Then φ m n (k (n) 1 , k (n) 2 , . . . , k (n) m ) = (1 + δ n ) exp 1 nµ m i=0 (s (n) (i) -s (n) (m)) - C α m α+1 (α + 1)µ α+1 n α ,
where δ n depends only on n and T , and δ n → 0 as n → ∞.

We start with a technical lemma.

Lemma 2.6.10. Suppose that m = O(n α/(α+1) ). Let E 1 , E 2 , . . . be i.i.d. standard exponential random variables. Suppose that for each n we have a sequence a

(n) 1 , a (n) 2 , . . . , a (n) m such that a (n) i ∈ (0, Km/n) for all 1 ≤ i ≤ m for some constant K. (a) We have m i=1 a (n) i (E i -1) → 0 in L 2 .
(b) For any p > 1, there exists a constant C > 0 such that

E exp p m i=1 a (n) i (E i -1) ≤ C exp 2pK 2 m 3 n 2 .
Both the convergence in (a) and the bound in (b) are uniform in sequences (a

(n)
i ) satisfying the above conditions.

Proof. (a) Since the sum is centred, we have

E   m i=1 a (n) i (E i -1) 2   = var m i=1 a (n) i (E i -1) = m i=1 (a (n) i ) 2 ≤ K 2 m 3 n 2 → 0 as n → ∞.
(b) For 0 < a < 1/2 we have

E [exp(aE 1 -a)] = e -a 1 -a = e -a 1 + a 1 -a ≤ exp -a + a 1 -a ≤ exp(2a 2 ).
So for sufficiently large n we have

E exp p m i=1 a (n) i (E i -1) ≤ exp 2p m i=1 (a (n) i ) 2 ≤ exp 2pK 2 m 3 n 2 .
Proof of Proposition 2.6.9. The lower bound does not rely on s (n) attaining a new minimum at time m, and has already been proved in Lemma 2.4.7; we need a matching upper bound. To ease readability, we will suppress the superscripts on k

(n) i and s (n) (i). Now, E m i=1 (n -i + 1)µ m j=i k j + Ξ n-m = m-1 i=1 1 - i n E m i=1 nµ s(m) -s(i -1) + 2(m -i + 1) + Ξ n-m = m-1 i=1 1 - i n E exp - m i=1 (s(m) -s(i -1) + 2(m -i + 1) + Ξ n-m nµ -1 E i ,
where E 1 , E 2 , . . . are i.i.d. standard exponential random variables, independent of Ξ n-m . We shall first consider the expectation conditionally on E 1 , E 2 , . . . , E m . Write A m for the quantity

m-1 i=1 1 - i n × E exp - m i=1 (s(m) -s(i -1) + 2(m -i + 1) + Ξ n-m nµ -1 E i E 1 , . . . , E m .
Let C > 0 be a constant to be chosen later. We will split E [A m ] into two parts, so that

E m i=1 (n -i + 1)µ m j=i k j + Ξ n-m = E A m 1 { m i=1 E i >Cn α/(α+1) } + E A m 1 { m i=1 E i ≤Cn α/(α+1) } .
We deal with the first term on the right-hand side first. Since k j ≥ 1 for all j and Ξ n-m ≥ n-m a.s., we have the crude bound

s(m) -s(i -1) + 2(m -i + 1) + Ξ n-m ≥ n -i + 1 > n -m > n/2
for 1 ≤ i ≤ m and all n sufficiently large that m/n < 1/2. Then

E A m 1 { m i=1 E i >Cn α/(α+1) } ≤ E exp 1 - 1 2µ m i=1 E i 1 { m i=1 E i >Cn α/(α+1) } = ∞ Cn α/(α+1) exp x - x 2µ e -x x m-1 Γ(m) dx = ∞ Cn α/(α+1) exp - x 2µ x m-1 Γ(m) dx = (2µ) m P 1 2µ m i=1 E i > Cn α/(α+1) .
By Markov's inequality, this last quantity is bounded above by

(2µ) m E exp 1 2µ m i=1 E i exp -Cn α/(α+1) = (2µ) 2 2µ -1 m exp(-Cn α/(α+1) ) → 0
as n → ∞, as long as we take C > T (2 log(2µ) -log(2µ -1)), which we henceforth assume.

Let us now turn to the expectation of A m on the event

{ m i=1 E i ≤ Cn α/(α+1) }. Since m-1 i=1 1 - i n ≤ exp - m(m -1) 2n ,
we have

A m 1 { m i=1 E i ≤Cn α/(α+1) } ≤ exp - m(m -1) 2n + 1 nµ m i=1 (s(i -1) -s(m))E i - m i=1 2(m -i + 1) nµ -1 E i × E exp - 1 nµ m i=1 E i Ξ n-m E 1 , . . . , E m 1 { m i=1 E i ≤Cn α/(α+1) } .
On the event

m i=1 E i ≤ Cn α/(α+1) , we have 1 nµ m i=1 E i = o(1)
. Hence, we may apply the asymptotic formula (2.16) for the Laplace transform of D 1 to obtain that on the event m i=1 E i ≤ Cn α/(α+1) we have

E exp - 1 nµ m i=1 E i Ξ n-m E 1 , . . . , E m = exp   - (n -m) n m i=1 E i + (2 -µ) 2µn m i=1 E i 2 - C α (α + 1)n α µ α+1 m i=1 E i α+1 + o(1)   .

It follows that

A m 1 { m i=1 E i ≤Cn α/(α+1) } ≤ exp   1 nµ m i=1 (s(i -1) -s(m))E i - C α (α + 1)n α µ α+1 m i=1 E i α+1 + o(1)   × exp   (2 -µ) 2µn m i=1 E i 2 - 2 nµ m i=1 (m -i + 1)E i + m n m i=1 E i - m 2 2n   × 1 { m i=1 E i ≤Cn α/(α+1) } .
Observe that

(2 -µ) 2µn m i=1 E i 2 - 2 nµ m i=1 (m -i + 1)E i + m n m i=1 E i - m 2 2n = (2 -µ) 2µn m + m i=1 E i m i=1 (E i -1) - 2 nµ m i=1 (m -i + 1)(E i -1) + m n m i=1 (E i -1) + m nµ . So A m 1 { m i=1 E i ≤Cn α/(α+1) } ≤ exp 1 nµ m i=0 (s(i) -s(m)) - C α m α+1 (α + 1)µ α+1 n α + o(1) exp(χ n )1 { m i=1 E i ≤Cn α/(α+1) } , (2.28)
where

χ n = 1 nµ m i=1 (s(i -1) -s(m))(E i -1) - 2 nµ m i=1 (m -i + 1)(E i -1) + m n m i=1 (E i -1) + (2 -µ) 2µn m + m i=1 E i m i=1 (E i -1) - C α m α+1 (α + 1)n α µ α+1   1 m m i=1 E i α+1 -1   .
We need to understand the asymptotics of the expectation of the right-hand side of (2.28).

Recall that s has steps down of magnitude at most 1, so that we have the crude bound s(i) -

s(m) ≤ m -i + 1 ≤ m for all 0 ≤ i ≤ m.
So by Lemma 2.6.10(a), we get

1 nµ m i=1 (s(i -1) -s(m))(E i -1) - 2 nµ m i=1 (m -i + 1)(E i -1) + m n m i=1 (E i -1) → 0 in L 2 , as n → ∞. We have E   (2 -µ) 2µn m + m i=1 E i m i=1 (E i -1) 2 1 { m i=1 E i ≤Cn α/(α+1) }   ≤ (2 -µ) 2 (T + C) 2 n 2α/(α+1) 4µ 2 n 2 E     m j=1 (E j -1)   2   ≤ (T + C) 2 T n 3α/(α+1) µ 2 n 2 → 0, since n α/(α+1) /n 2 = n (α-2)/(α-1) = o(1). If m → ∞ as n → ∞, it follows straightforwardly
from the weak law of large numbers that

1 m m i=1 E i α+1 p → 1
and so, for any m ≤ T n α/(α+1) , we have

C α m α+1 (α + 1)n α µ α+1   1 m m i=1 E i α+1 -1   p → 0.
These results imply that

χ n p → 0 on { m i=1 E i ≤ Cn α/(α+1) }, as n → ∞ and so exp(χ n )1 { m i=1 E i ≤Cn α/(α+1) } p → 1.
It remains to show that

E exp(χ n )1 { m i=1 E i ≤Cn α/(α+1)
} → 1 as n → ∞, for which we require uniform integrability. Now, we have

exp (2 -µ) 2µn m + m i=1 E i m i=1 (E i -1) 1 { m i=1 E i ≤Cn α/(α+1) } ≤ 1 + exp (2 -µ) 2µn (C + 1)m m i=1 (E i -1) . Hence, exp(χ n )1 { m i=1 E i ≤Cn α/(α+1) } ≤ exp C α m α+1 (α + 1)µ α+1 n α × exp m i=1 (s(i -1) -s(m)) nµ - 2(m -i + 1) nµ + m n (E i -1) + exp m i=1 (s(i -1) -s(m)) nµ - 2(m -i + 1) nµ + m n + (2 -µ)(C + 1)m 2µn (E i -1) .
Applying Lemma 2.6.10(b), we see that both terms are bounded in L p for p > 1. Hence, the sequence (exp(χ n )1 { m i=1 E i ≤Cn α/(α+1) } , n ≥ 1) is uniformly integrable and we may deduce that

E exp(χ n )1 { m i=1 E i ≤Cn α/(α+1) } → 1,
which concludes the proof.

We will also need the following lemma.

Lemma 2.6.11. Fix θ > 0 and let m = T n α/(α+1) . Then we have the following uniform integrability: for K > 0, lim sup

K→∞ sup n≥1 E exp θ n m i=0 (S(i) -S(m)) 1 { 1 n m i=0 (S(i)-S(m))>K} E m = 0.
Proof. The proof uses similar ingredients to the proof of Lemma 2.3.6. On the event E m we have

θ n m i=0 (S(i) -S(m)) ≤ θm n 1 + max 0≤i≤m S(i) ≤ θT (α+1)/α m -1/α 1 + max 0≤i≤m S(i) .
So it will be sufficient to show that we have lim sup

K→∞ sup m≥1 E exp θm -1/α max 0≤i≤m S(i) 1 {m -1/α max 0≤i≤m S(i)>K} E m = 0.
We have

E exp θm -1/α max 0≤i≤m S(i) 1 {m -1/α max 0≤i≤m S(i)>K} E m = ∞ k= Km 1/α +1 e θm -1/α k P max 0≤i≤m S(i) = k E m ≤ e (K+1)θ P m -1/α max 0≤i≤m S(i) > K E m + ∞ k= Km 1/α +2 θm -1/α e θm -1/α k P max 0≤i≤m S(i) ≥ k E m ,
by summation by parts and the fact that e θm -1/α k -e θm -1/α (k-1) ≤ m -1/α θe θm -1/α k . Theorem 9 of Kortchemski [START_REF] Kortchemski | Sub-exponential tail bounds for conditioned stable Bienaymé-Galton-Watson trees[END_REF] gives that for any δ ∈ (0, α/(α -1)), there exist universal constants

C 1 , C 2 > 0 such that P m -1/α max 0≤i≤m S(i) ≥ u E m ≤ C 1 exp(-C 2 u δ ).
We take δ ∈ (1, α/(α -1)). So then

E exp θm -1/α max 0≤i≤m S(i) 1 {m -1/α max 0≤i≤m S(i)>K} E m ≤ e (K+1)θ P m -1/α max 0≤i≤m S(i) > K E m + ∞ K θe θx P m -1/α max 0≤i≤m S(i) ≥ x -1 E m dx ≤ C 1 exp((K + 1)θ -C 2 K δ ) + ∞ K C 1 θ exp(θx -C 2 (x -1) δ )dx,
which clearly tends to 0 as K → ∞ since δ > 1. The result follows.

Proof of Theorem 2.6.8. Recall that S(k) = k i=1 (Z i -2), where Z 1 , Z 2 , . . . are i.i.d. with the size-biased degree distribution. Then we have

E f n -1 α+1 [ S n ( (t 1 + •)n α α+1 ) -S n ( t 1 n α α+1 )], G n ( (t 1 + •)n α α+1 ) E n m 1 ,m 2 = E f n -1 α+1 [ S n ( (t 1 + •)n α α+1 ) -S n ( t 1 n α α+1 )], n -α-1 α+1 G n ( (t 1 + •)n α α+1 ) 1 E n m 1 ,m 2 E 1 E n m 1 ,m 2
.

Using the change of measure, this is equal to

E Φ(n, t 2 n α α+1 )f n -1 α+1 [S( (t 1 +•)n α α+1 )-S( t 1 n α α+1 )],n -α-1 α+1 G( (t 1 +•)n α α+1 ) 1 Em 1 ,m 2 E Φ(n, t 2 n α α+1 )1 Em 1 ,m 2 = E Φ(n, t 2 n α α+1 )f n -1 α+1 [S( (t 1 +•)n α α+1 )-S( t 1 n α α+1 )],n -α-1 α+1 G( (t 1 +•)n α α+1 ) Em 1 ,m 2 E Φ(n, t 2 n α α+1 ) Em 1 ,m 2
.

(2.29) By Proposition 2.6.9, we have that on the event

E m 1 ,m 2 , Φ(n, t 2 n α α+1 ) = (1 + o(1)) exp 1 µn m 2 i=0 (S(i) -S(m 2 )) - C α t α+1 2 (α + 1)µ α+1 , (2.30)
where the o(1) is uniform on E m 1 ,m 2 . But using that S(m 2 ) = S(m 1 ) -1, we get

Φ(n, t 2 n α α+1 ) = (1 + o(1)) exp 1 µn m 1 -1 i=0 (S(i) -S(m 1 )) + m 1 µn - C α t α+1 2 (α + 1)µ α+1 × exp 1 µn m 2 i=m 1 ([S(i) -S(m 1 )] -[S(m 2 ) -S(m 1 )]) .
The increments of the random walk S are independent, and so the first and second terms in this product are independent. The first term is also independent of the argument of the function f . So in both the numerator and denominator of the fraction (2.29), we may cancel a factor of

E exp 1 µn m 1 -1 i=0 (S(i) -S(m 1 )) + m 1 µn - C α t α+1 2 (α + 1)µ α+1 E m 1 ,m 2 .
Using also the stationarity of the increments of S and the fact that m = m 2 -m 1 , we then obtain that (2. Chapter 3

Gaussian Free Field level-set percolation on regular random graphs This chapter stems from the preprint [START_REF] Conchon-Kerjan | Anatomy of a gaussian giant: supercritical level-sets of the free field on random regular graphs[END_REF], that has not yet been submitted.

Abstract. In this chapter, we study the level-set of the zero-average Gaussian Free Field on a uniform random d-regular graph above an arbitrary level h ∈ (-∞, h ), where h is the level-set percolation threshold of the GFF on the d-regular tree T d . We prove that w.h.p as the number n of vertices diverges, the GFF has a unique giant connected component C

(n)

1 of size η(h)n + o(n), where η(h) is the probability that the root percolates in the corresponding GFF level-set on T d . This gives a positive answer to the conjecture of [START_REF] Abächerli | Level-set percolation of the Gaussian free field on regular graphs II: finite expanders[END_REF] for most regular graphs. We also prove that the second largest component has size Θ(log n).

Moreover, we show that

C (n) 1
shares the following similarities with the giant component of the supercritical Erdős-Rényi random graph. First, the diameter and the typical distance between vertices are Θ(log n). Second, the 2-core and the kernel encompass a given positive proportion of the vertices. Third, the local structure is a branching process conditioned to survive, namely the level-set percolation cluster of the root in T d (in the Erdős-Rényi case, it is known to be a Galton-Watson tree with a Poisson distribution for the offspring).

Introduction

Overview

The Gaussian Free Field (GFF) on a transient graph G is a Gaussian process indexed by the vertices. Its covariance is given by the Green function, hence the GFF carries a lot of information on the structure of G and on the behaviour of random walks, giving a base motivation for its study.

Level-set percolation of the GFF has been investigated since the 1980s ( [START_REF] Bricmont | Percolation in strongly correlated systems: The massless gaussian field[END_REF][START_REF] Molchanov | Percolation in random fields[END_REF]). Lately, one important incentive has been to gain information on the vacant set of random interlacements ([104, 131]), via Dynkin-type isomorphism theorems ( [START_REF] Eisenbaum | A rayknight theorem for symmetric markov processes[END_REF][START_REF] Sabot | Inverting ray-knight identity[END_REF]). It was subject to much attention in the last decade on Z d ( [START_REF] Drewitz | The Sign Clusters of the Massless Gaussian Free Field Percolate on {Z d , d ≥slant 3} (and more)[END_REF]104,[START_REF] Popov | On decoupling inequalities and percolation of excursion sets of the gaussian free field[END_REF][START_REF] Sznitman | Random interlacements and the gaussian free field[END_REF]). On such a lattice where the Green function decays polynomially with the distance between vertices, it provides a percolation model with long-range interactions.

More recently, level-set percolation was studied on transient rooted trees ([3, 2, 132]). There is a phase transition at a critical threshold h ∈ R: if h < h , the connected component of the root in the level-set above h of the GFF has a positive probability to be infinite, and if h > h , this probability is zero.

One can define an analogous field on a finite connected graph, the zero-average Gaussian Free Field, whose covariance is given by the zero-average Green function (see Section 3.1.2). A natural question is whether some characteristics of the GFF on an infinite graph G can be transferred to a sequence of finite graphs (G n ) n≥0 whose local limit is G. For instance, one might ask whether a phase transition for the existence of an infinite connected component of the level-set in G corresponds to a phase transition for the emergence of a "macroscopic" [START_REF] Abächerli | Local picture and level-set percolation of the gaussian free field on a large discrete torus[END_REF] studied the zeroaverage GFF on the torus.

component of size Θ(|G n |) in the level set in G n . For G = Z d , Abächerli
Abächerli and Černý recently investigated the GFF on the d-regular tree T d [START_REF] Abächerli | Level-set percolation of the Gaussian free field on regular graphs I: regular trees[END_REF], and the zeroaverage GFF on some d-regular graphs (large girth expanders) in a companion paper [START_REF] Abächerli | Level-set percolation of the Gaussian free field on regular graphs II: finite expanders[END_REF]. In this setting, many essential questions (such as the value of h , or the sharpness of the phase transition at h for the zero-average GFF) remain open. In this paper, we answer some of them, and relate the percolation level-sets to other classical random graphs, in particular the Erdős-Rényi model (Section 3.1.4).

To do so, we refine some properties of [START_REF] Abächerli | Level-set percolation of the Gaussian free field on regular graphs I: regular trees[END_REF] on T d . In a work in progress [START_REF] Conchon-Kerjan | Supercritical GFF percolation on a regular tree : anatomy of the infinite cluster[END_REF], we study further the GFF on T d and the random walk on the level-sets.

Setting

In all this work, we fix an integer d ≥ 3. We denote T d the infinite d-regular tree rooted at an arbitrary vertex •, and G n a uniform d-regular random graph for n ≥ 1 (if d is odd, consider only even n). Let V n be its vertex set and π n be the uniform measure on V n , i.e. π n (x) = 1/n for every x ∈ V n .

Gaussian Free Field on regular trees

The GFF ϕ T d on T d is a centred Gaussian field (ϕ T d (x)) x∈T d indexed by the vertices of T d , and with covariances given by the Green function G T d : for all vertices x, y ∈ T d , we set

Cov(ϕ T d (x), ϕ T d (y)) = G T d (x, y). Recall that G T d (x, y) = E T d x   k≥0 1 {X k =y}  
where (X k ) k≥0 is a discrete-time SRW (Simple Random Walk) on T d . In general, we will denote

P G
µ the law of a SRW on a graph G with initial distribution µ.

Gaussian Free Field on finite graphs

If G n is connected, the zero-average GFF ψ Gn on G n is a centred Gaussian field (ψ Gn (x)) x∈Gn
indexed by the vertices of G n , and with covariances given by the zero-average Green function

G Gn on G n : for all x, y ∈ G n , we set Cov(ψ Gn (x), ψ Gn (y)) = G Gn (x, y) := E Gn x +∞ 0 1 {Xt=y} - 1 n dt
where (X t ) t≥0 is a continuous time SRW on G n started at x with Exp(1) independent jumptimes. Precisely, let (ζ i ) i≥1 be a sequence of independent exponential variables of parameter 1. Let (X k ) k≥0 be a SRW started at x, independent of (ξ i ) i≥1 . Then for all t ≥ 0, we define

X t := X k(t) , with k(t) := sup k≥0 k i=1 ζ i ≤ t.
The function G Gn is symmetric, finite and positive semidefinite. This ensures that ψ Gn is welldefined (see [START_REF] Abächerli | Local picture and level-set percolation of the gaussian free field on a large discrete torus[END_REF] for details, in particular Remark 1.2).

Two layers of randomness

Denote P ann and E ann the annealed law and expectation for the joint realization of G n and of ψ Gn on it. For a fixed realization of G n , denote P Gn and E Gn the quenched law and expectation.

Results

Define the level set E ≥h

ϕ T d := {x ∈ T d | ϕ T d (x) ≥ h}. Let C h • be the connected component of E ≥h ϕ T d containing the root •. Similarly, define the level sets E ≥h ψ Gn := {x ∈ G n | ψ Gn (x) ≥ h} for n ≥ 1. For i ≥ 1, let C (n) i be the i-th largest connected component of E ≥h ψ Gn .
In [START_REF] Sznitman | Coupling and an application to level-set percolation of the gaussian free field[END_REF], Sznitman showed that there exists a constant h > 0 such that

if h > h , η(h) := P T d (|C h • | = +∞) = 0, and if h < h , η(h) > 0. (3.1)
In [START_REF] Abächerli | Level-set percolation of the Gaussian free field on regular graphs I: regular trees[END_REF] (Theorems 4.3 and 5.1), Abächerli and Černý showed that if h > h , the size of C h • has exponential moments, and if h < h , C h

• has a positive probability to grow exponentially. In [START_REF] Abächerli | Level-set percolation of the Gaussian free field on regular graphs II: finite expanders[END_REF] (Theorems 3.1 and 4.1), they proved that P ann -w.h.p.

: if h > h , |C (n) 1 | = O(log n),
and if h < h , at least ξn vertices of E ≥h ψ Gn are in components of size at least n δ , for some constants δ, ξ > 0 depending on h. They even found deterministic conditions on G n , satisfied w.h.p., so that these events hold P Gn -w.h.p. (see the discussion in Section 3.1.4). Thus, in the supercritical case h < h , a positive proportion of the vertices is in at least "mesoscopic" components (there is no explicit lower bound for δ).

This paper focuses exclusively on the supercritical case. We prove the existence of a giant component:

Theorem 3.1.1. Let h < h . It holds: |C (n) 1 | n Pann -→ η(h), (3.2)
where Pann -→ stands for convergence in P ann -probability as n → +∞. Moreover, there exists K 0 > 0 such that

P ann K -1 0 log n ≤ |C (n) 2 | ≤ K 0 log n -→ n→+∞ 1. (3.3)
Note that by Markov's inequality, for any ε > 0 and any sequence of events (E n ) n≥1 such that

P ann (E n ) → 1, w.h.p. G n is such that P Gn (E n ) ≥ 1 -ε.
Thus, w.h.p. on G n , the conclusions of Theorem 3.1.1 hold with arbitrarily large P Gn -probability.

We also establish some structural properties of

C (n) 1 . Let C (n) be the 2-core of C (n)
1 , obtained by deleting recursively the vertices of degree 1 of

C (n) 1
and their edges. Let K (n) be the kernel n) where simple paths are contracted to a single edge, so that the vertices of K (n) are those of C (n) with degree at least 3.

of C (n) 1 , i.e. C (
Theorem 3.1.2. Global structure of C (n) 1 Fix h < h . There exist K 1 , K 2 > 0 such that |C (n) | n Pann -→ K 1 (3.4)
and

|K (n) | n Pann -→ K 2 . (3.5)
Moreover, there exists

K 3 > 0 such that if D (n) 1
is the diameter of C

(n)

1 , then

P ann (D (n) 1 ≤ K 3 log n) -→ n→+∞ 1. (3.6)
Last, there exists λ h > 1 such that for every ε > 0,

π 2,n ({(x, y) ∈ (C (n) 1 ) 2 , (1 -ε) log λ h n ≤ d C [n) 1 (x, y) ≤ (1 + ε) log λ h n}) Pann -→ 1, (3.7)
where π 2,n is the uniform measure on (C

(n) 1 ) 2 and d C (n) 1 the usual graph distance on C (n)
1 . In other words, the typical distance between vertices of

C (n) 1 is log λ h n.
We will see in Section 3.3 that λ h is the growth rate of C h • conditioned on being infinite. Say that a random graph G is the local limit of the random graph sequence (G n ) n≥1 if G n converges to G in distribution w.r.t to the local topology (see for instance the lecture notes of Curien [START_REF] Curien | Random graphs -the local convergence point of view[END_REF] for a precise definition). We prove that the local limit of

C (n) 1 is C h • conditioned to be infinite. Theorem 3.1.3. Local limit of C (n) 1
For every radius k ≥ 1, for every rooted tree T of height k, let

V (T ) n :={x ∈ C (n) 1 , B C (n) 1 (x, k) = T } and p T := P T d (B C h • (•, k) = T | |C h • | = +∞). Then |V (T ) n | |C (n) 1 | Pann -→ p T .

Discussion and open questions

GFF percolation versus bond percolation

The graph E ≥h ψ Gn undergoes the same phase transition as some classical bond percolation models for the size of the largest connected component. We draw a comparison with the Erdős-Rényi random graph (i.e. bond percolation on the complete graph), introduced by Gilbert in [START_REF] Gilbert | Random graphs[END_REF]: for a constant c > 0 and n ∈ N, ER(n, c/n) is the graph on n vertices such that for every pair of vertices x, y, there is an edge between x and y with probability c/n, independently of all other pairs of vertices. Erdős and Rényi [START_REF] Erdős | On the evolution of random graphs[END_REF] showed that the supercritical regime corresponds to c > 1 and the subcritical regime to c < 1. Theorems 3.1.1, 3.1.2 and 3.1.3 hold for ER(n, c/n) as n → +∞, for any fixed c > 1, the tree C h

• being replaced by a Galton-Watson tree whose offspring distribution is Poisson with parameter c, and λ h being replaced by c.

As for Bernoulli bond percolation on G n (each edge of G n is deleted with probability 1 -p, independently of the others), the same phase transition holds for the size of the largest connected component, the critical threshold being p = 1/(d -1) (Theorem 3.2 of [START_REF] Noga Alon | Percolation on finite graphs and isoperimetric inequalities[END_REF]).

The structure of C (n) 1

It was shown recently in [START_REF] Ding | Anatomy of the giant component: the strictly supercritical regime[END_REF] that the distribution of the giant component of ER(n, p/n) is continuous w.r.t. to a random graph which can be explicitly described. Its kernel is a configuration model whose vertices have i.i.d. degrees with a Poisson distribution (conditioned on being at least 3). In particular, it is an expander. The lengths of the simple paths in the 2-core are i.i.d. geometric random variables. See Theorem 1 of [START_REF] Ding | Anatomy of the giant component: the strictly supercritical regime[END_REF] for details. This implies a result analogous to Theorem 3.1.2 for ER(n, p/n).

We conjecture that the kernel K (n) is an expander for every h < h . The main obstacle to gathering information on its global structure is that if ψ Gn is revealed on a positive proportion of the vertices of K (n) (and hence of G n ), then it could affect substantially ψ Gn on the remaining vertices. In particular, if h > 0 is large enough, we could imagine that the average of ψ Gn on the discovered vertices is positive. But by (3.18), the average of the GFF on the remaining vertices would be negative, hence below the threshold h.

Deterministic regular graphs

The results of [START_REF] Abächerli | Level-set percolation of the Gaussian free field on regular graphs II: finite expanders[END_REF] and [START_REF] Noga Alon | Percolation on finite graphs and isoperimetric inequalities[END_REF] hold in fact for any deterministic sequence of large-girth expanders (conditions (I) and (II) in Proposition 3.2.1), which is w.h.p. the case for G n . Very recently, after a first preprint of our work, Černý [START_REF] Černý | Level-set percolation of the Gaussian free field on regular graphs III: giant component on expanders[END_REF] gave another proof of (3.2) that holds under these deterministic conditions. He also showed that

|C (n) 2 | = o(n) w.h.p.
His approach is very different, and uses notably a novel decomposition of the GFF as an infinite sum of fields with finite range interactions, introduced in [START_REF] Duminil-Copin | Equality of critical parameters for percolation of gaussian free field level-sets[END_REF] and [START_REF] Duminil-Copin | Existence of phase transition for percolation using the gaussian free field[END_REF].

In our proofs, averaging on the randomness of G n is a crucial ingredient to control the presence of cycles on large subgraphs of G n , and allows us to extend some arguments of [START_REF] Abächerli | Level-set percolation of the Gaussian free field on regular graphs II: finite expanders[END_REF], where ψ Gn is locally approximated by ϕ T d .

We conjecture that those deterministic conditions are not sufficient for (3.3) to hold. This was shown for the Bernoulli bond percolation in [START_REF] Krivelevich | Asymptotics in percolation on high-girth expanders[END_REF] (Theorem 2): for every a ∈ (0, 1), one can build a sequence (G n ) n≥1 satisfying (I) and (II) such that the second largest connected component has at least n a vertices (the second largest component first grows exponentially on a tree-like ball until it has a polynomial size, and then is "trapped" in zones where the expansion of the graph is close to an arbitrarily small constant).

Behaviour at criticality

Almost nothing is known about C h

• (even the value of h ), though one might conjecture that it is a.s. finite, as critical Galton-Watson trees. In the Erdős-Rényi model, the critical case is by far the most interesting. When p = 1 + Θ(n -1/3 ), the i-th largest connected component has size ς i n 2/3 , where ς i is an a.s. finite random variable (see the celebrated paper by Aldous [START_REF] Aldous | Brownian excursions, critical random graphs and the multiplicative coalescent[END_REF]).

Its structure is similar to a modified Brownian tree [START_REF] Addario-Berry | The continuum limit of critical random graphs[END_REF]. Hence, there is a finite but arbitrary large number of components of the biggest order, and some of their main characteristics, such as their cardinality, are random.

It would be interesting to look for the size and shape of |C (n) i | for h = h , or for an hypothetical critical window (h -ε n , h +ε n ) for a sequence (ε n ) n≥1 converging to 0 at an appropriate speed.

Proof outline

Our proofs rely on two main arguments:

1) An annealed exploration of E ≥h ψ Gn (Proposition 3.2.4), where the structure of G n is progressively revealed (there is a standard sequential construction of G n , see Section 3.2). Each newly discovered vertex is given an independent standard normal variable. Then ψ Gn is built via a recursive procedure, using these Gaussian variables (Proposition 3.2.3).

2) A comparison of ψ Gn and ϕ T d (Proposition 3.4.1) : on a tree-like subgraph T of G n , such that there are no cycles in G n at distance κ log log n of T for a large enough constant κ, there is a bijective map Φ between T and an isomorphic subtree of T d and a coupling of

ψ Gn and ϕ T d so that sup y∈T |ψ Gn (y) -ϕ T d (Φ(y))| ≤ log -1 n.
We stress the fact that we reveal ψ Gn only after having explored G n : if we reveal ψ Gn at a given vertex, it conditions the structure of G n and thus the pairings of the still unmatched half-edges, so that we cannot use the sequential construction any more to further explore the graph. Hence, during the exploration, we will need to build an approximate version of ψ Gn , depending on the Gaussian variables of 1). This makes some proofs tedious, in particular that of (3.3).

The base exploration

The exploration that we will perform in all proofs, with some modifications, is as follows: pick

x ∈ V n , and reveal its connected component C Gn,h x in E ≥h ψ Gn in a breadth-first way, as well as its neighbourhood up to distance a n = κ log log n. Until we meet a cycle, the explored zone is a tree T x , growing at least like C h+log -1 n

• , and at most like 1) to be infinite, with a growth rate λ h > 1 (Section 3. Conversely,

C h-log -1 n • by 2). On one hand, C h+log -1 n • has a probability η(h + log -1 n) = η(h) + o(
C h-log -1 n • has a probability 1 -η(h -log -1 n) = 1 -η(h) + o(1)
to be finite, and

with P ann -probability 1 -η(h) + o(1), |C Gn,h x | = o( √ n) (Proposition 3.5.4).
Proof of (3.2).

First, we show that for any two vertices x, y ∈ V n , there is a P ann -probability η(h) 2 + o(1) that they are connected in E ≥h ψ Gn . To do so, we explore C Gn,h

x and C Gn,h y , that we couple with independent copies of C h+log -1 n

•

, so that with probability η(h) 2 +o(1), ∂T x and ∂T y have Θ( √ n log -κ n)

vertices. The explorations from x and y are disjoint with probability 1 -o(1), since o( √ n) vertices have been explored. Then, we draw multiple paths between T x and T y (with an "envelope" of radius Θ(log log n) around each of them to allow the use of the approximation 2)), the joining balls (Section 3.6.1). The probability that E ≥h ψ Gn percolates through at least one of these paths is 1 -o(1).

Second, we prove by a second moment argument that P ann -w.h.p., the number of couples

(x, y) ∈ V 2 n such that y ∈ C Gn,h x is (η(h) 2 + o(1))n 2 (Lemma 3.6.4).
Third, knowing that |C Gn,h Proof of (3.3).

The most difficult part is the upper bound. We show that for K 0 large enough, for x ∈ V n ,

P ann (K 0 log n ≤ |C Gn,h x | ≤ K -1 0 n) = o(1/n),
and conclude by a union bound on x and a corollary of the proof of (3.2), namely that |C

(n) 2 |/n Pann -→ 0.
The greater precision o(1/n) requires three additional ingredients:

• the size of C h

• conditioned on being finite has exponential moments (Proposition 3.3.6), in particular,

P T d (|C h • | ≥ c log n, |C h • | < +∞) = o(1/n) for a large enough constant c;
• when exploring k vertices around x, there is a probability Θ(k 2 /n) that a cycle arises, so that we will need to handle at least one cycle to fully explore C Gn,h

x ;

• we need a better approximation of ψ Gn than log -1 n in 2): with probability at least Θ(1/n), we will meet too many vertices with an approximate value of ψ Gn that are in

[h -log -1 n, h + log -1 n]
, so that we can not tell whether they are in C Gn,h

x or not before the end of the exploration. To remedy this, we replace the "security radius" a n in 2) by some r n = Θ(log n), so that we approximate ψ Gn up to a difference n -Θ (1) .

Other proofs.

The proofs of Theorems 3.1.2 and 3.1.3 are based on slightly modified explorations, and are much simpler.

Plan of the rest of the paper

In Section 3.2, we review some basic properties of G n (structure, Green function and GFF). In Section 3.3, we study the GFF on T d . In Section 3.4, we establish a coupling between recursive constructions of the GFF on T d and on a tree-like neighbourhood of G n . In Section 3.5, we explore the connected component of a vertex in E ≥h ψ Gn . In Section 3.6, we prove (3.2). In Section 3.7, we prove (3.3). In Section 3.8, we prove Theorems 3.1.2 and 3.1.3.

Further definitions

In this paper, edges are non-oriented, and thus graphs are undirected. For any graph G, denote d G the standard graph distance on its vertex set V , and for every vertex x and R ≥ 0, let (II) for all x ∈ G n , B Gn (x, K 3 log n ) contains at most one cycle. Moreover, there exists K 4 > 0 such that w.h.p. on G n , it holds: for all x ∈ V n such that tx(B Gn (x, K 4 log log n )) = 0,

B G (x, R) := {y, d G (x, y) ≤ R} and ∂B G (x, R + 1) = B G (x, R + 1) \ B G (x, R). For any S ⊆ V , let similarly B G (S, R) := ∪ x∈S B G (x, R) and ∂B G (S, R + 1) = B G (S, R + 1) \ B G (S, R). If A is a subgraph of G with vertex set S, denote B G (A, R) = B G (S,
G Gn (x, x) - d -1 d -2 ≤ log -6 n. (3.8)
If moreover y is a neighbour of x,

G Gn (x, y) - 1 d -2 ≤ log -6 n. (3.9)
Say that a given realization of G n is a good graph when (I), (II), (3.8) and (3.9) hold. The equations (3.8) and (3.9) illustrate the fact that G Gn is close to G T d on a tree-like neighbourhood:

it is well-known that for all x, y ∈ T d ,

G T d (x, y) = (d -1) 1-d T d (x,y) d -2 . ( 3.10)
A quick computation can be found in [START_REF] Woess | Random Walks on Infinite Graphs and Groups[END_REF], Lemma 1.24.

By Proposition 1.1 of [START_REF] Abächerli | Level-set percolation of the Gaussian free field on regular graphs II: finite expanders[END_REF], (I) and (II) imply that for some K 5 , K 6 > 0 and for n large enough, for all x, y ∈ V n ,

|G Gn (x, y)| ≤ K 5 (d -1) d Gn (x,y) ∨ n -K 6 . (3.11)
Throughout this paper, we will often make binomial estimations, because the number of edges between two sets of vertices in G n is close to a binomial random variable, as highlighted in the Lemma below. We will use repeatedly the following classical inequalities: for n ≥ m ≥ 0 and p ∈ (0, 1), if Z ∼ Bin(n, p), one has

P(Z ≥ m) ≤ n m p m , P(Z ≤ m) ≤ n m (1 -p) m , n m ≤ n m m! ≤ n m . (3.12)
The following Lemma is an important consequence of the sequential construction of G n . Add its other endpoint v in W 0 , if it was not already in it. Let s be the number of times that

v ∈ W 1 . Suppose that m E + m + m 1 < n. Then s ≤ st. Bin m, m 1 n -(m E + m) , (3.13)
In particular, a) for any fixed k ∈ N, there exists C(k) > 0 so that for n large enough, if m E + m < n/2, Proof. Pick v ∈ W 0 , such that v has an unmatched half-edge e. There are at most m 1 vertices in W 1 , so that there are at most dm 1 unmatched half-edges that belong to its vertices. And the total number of unmatched half-edges is at least dn -2(|E| + m) ≥ d(n -m E -m). Thus, the probability that e is matched with a half-edge belonging to a vertex of W 1 is not greater than

P E (s ≥ k) ≤ C(k) m 1 m n k . ( 3
dm 1 d(n-m E -m) = m 1 n-(m E +m)
. The successive matchings are performed independently, and (3.13) follows.

Let Z ∼ Bin m,

m 1 n-(m E +m)
. By (3.12), for k ∈ N, we have

P E (Z ≥ k) ≤ m k m 1 n-(m E +m) k ≤ m k m 1 n/2 k ≤ 2 k k! m k 1 m k n k .
This yields (3.14). Moreover, if k → +∞ as n → +∞ and kn > 6(m

1 + m 0 + m E )m, by
Stirling's formula, we have that for n large enough: It is straightforward to adapt this when s counts the number of times that v was in W 0 (and there is no set W 1 ). Throughout this paper, we will refer to these equations without mentioning explicitly if we count the connections from W 0 to W 1 or from W 0 to itself.

P E (Z ≥ k) ≤ (2e+0.1)m 1 m kn k < 0.
Proof of Proposition 3.2.1. By Theorem 1 of [START_REF] Bollobás | The isoperimetric number of random regular graphs[END_REF] and the Cheeger bound, G n satisfies (I) w.h.p.

As for (II), fix K 3 > 0. For all x ∈ V n , one obtains B Gn (x, K 3 log n ) by proceeding to at most d(d -1) K 3 log n pairings of half-edges. If K 3 is small enough, d(d -1) K 3 log n < n 1/5 -1 so that by (3.14) with m 0 = 1, m E = 0, m = d(d -1) K 3 log n and k = 2, for n large enough:

P(tx(B Gn (x, K 3 log n )) ≥ 2) ≤ C(2) n 2/5 n 2 ≤ n -11/10
.

By a union bound on x ∈ V n , w.h.p. G n is such that for all x ∈ V n , tx(B Gn (x, K 3 log n )) ≤ 1.

We now establish (3.8). Note that U := B Gn (x, K 4 log log n ) and

W := B T d (•, K 4 log log n ) are isomorphic. Then G U Gn (x, x) = G W T d (•, •)
, where we let

G A Gn (y, z) := E Gn y [ T A k=0 1 {X k =z} ] for every y, z ∈ V n and A V n .
Recall that T A is the exit time of A by the SRW (X k ) k≥0 . Similarly for every B T d and y, z ∈ T d , we define

G B T d (y, z) := E T d y [ T B k=0 1 {X k =z} ]. (3.16)
On one hand, by the strong Markov property applied to the exit time T W ,

G W T d (•, •) = G T d (•, •) -E T d • [G T d (•, X T W )] = d-1 d-2 -E T d • [G T d (•, X T W )].
On the other hand, by Lemma 1.4 of [START_REF] Abächerli | Local picture and level-set percolation of the gaussian free field on a large discrete torus[END_REF], for all y, z ∈ V n and A V n :

G A Gn (y, z) = G Gn (y, z) -E Gn y [G Gn (z, X T A )] + E Gn y [T A ] n , (3.17) so that G U Gn (x, x) = G Gn (x, x) -E Gn x [G Gn (x, X T U )] + E Gn x [T U ] n
. Therefore,

G Gn (x, x) - d -1 d -2 ≤ |E T d • [G T d (•, X T W )]| + |E Gn x [G Gn (x, X T U )]| + E Gn x [T U ] n .
By (3.10) and (3.11), if K 4 is large enough, then for large enough n:

|E T d • [G T d (•, X T W )]| + |E Gn x [G Gn (x, X T U )]| ≤ log -7 n.
Note that T U is stochastically dominated by the hitting time H of K 4 log log n by a SRW (Z k ) k≥0 on Z starting at 0, whose transition probabilities from any vertex are d-1 d towards the right and 1/d towards the left. By Markov's exponential inequality, there exists a constant c > 0 such that for n large enough and every k > n 1/10 ,

P(H ≥ k) ≤ P(Z k ≤ K 4 log log n ) ≤ P(Z k ≤ ( d-2 d -1/100)k) ≤ e -ck .
Hence for n large enough,

E Gn x [T U ] ≤ E[H] ≤ n 1/10 + k≥n 1/10 ke -ck ≤ n 1/2 . Thus, G Gn (x, x) - d -1 d -2 ≤ log -7 n + n -1/2 ≤ log -6 n
for large enough n, and this yields (3.8). One proves (3.9) by the same reasoning.

GFF on G n

The name "zero-average" for the GFF on G n (or on any finite connected graph) comes from the fact that a.s.,

x∈Vn ψ Gn (x) = 0 (3.18)
since Var x∈Vn ψ Gn (x) = x,y∈Vn G Gn (x, y) = 0. This prevents the existence of a domain Markov property. However, there exists a recursive construction of ψ Gn :

Proposition 3.2.3 (Lemma 2.6 in [4]). Let A V n , x ∈ V n \ A. Write σ(A) := σ({ψ Gn (y), y ∈ A}). Let (X k ) k≥0 be a SRW on G n and let H A be the hitting time of A. Conditionally on σ(A), ψ Gn (x) is a Gaussian variable, such that E Gn [ψ Gn (x)|σ(A)] = E Gn x [ψ Gn (X H A )] - E Gn x [H A ] E Gn πn [H A ] E Gn πn [ψ Gn (X H A )] (3.19)
and

Var Gn (ψ Gn (x)|σ(A)) = G Gn (x, x) -E Gn x [G Gn (x, X H A )] + E Gn x [H A ] E Gn πn [H A ] E Gn πn [G Gn (x, X H A )]. (3.20)
Combining this Lemma and the sequential procedure to build G n , we obtain the following construction.

Proposition 3.2.4 (Joint realization of G n and ψ Gn ). A realization of (G n , ψ Gn ) is given by the following process. Let (ξ i ) i≥1 be a sequence of i.i.d. standard normal variables. A move consists in:

• choosing an unpaired half-edge e and matching it to another unpaired half-edge chosen uniformly at random (independently of (ξ i ) i≥1 ), or

• choosing x ∈ V n and k ∈ N such that ξ k has not yet been attributed, and attributing ξ k to

x.

At each move, the choice of e, x or k might depend in an arbitrary way on the previous moves, i.e. on the matchings and on the value of the normal variables attributed before, but not on the value of the remaining normal variables. Perform moves until all half-edges are paired, and every vertex x ∈ V n has received a normal variable, that we denote ξ x . Erase loops and replace each multiple edge by a single edge.

To generate ψ Gn , let x 1 , . . . , x n be the vertices of V n , listed in the order in which they received their normal variable. Let ψ Gn (x 1 ) := G Gn (x 1 , x 1 )ξ x 1 . For i = 2, . . . , n successively, define

A i := {x 1 , . . . , x i-1 }. Recall that we write σ(A i ) for σ({ψ Gn (y), y ∈ A i }). Let ψ Gn (x i ) := E Gn [ψ Gn (x i )|σ(A i )] + ξ x i Var(ψ Gn (x i )|σ(A i )).
It might be confusing that ψ Gn (x i ) appears on both sides of the equation. Note that the conditional expectation and variance on the RHS are σ(A i )-measurable random variables.

Proof. Clearly, the graph obtained after pairing all the half-edges is distributed as G n . ξ x 1 is a standard normal variable independent of the realization of G n . Finally, remark that for every i ≥ 2, ξ x i is a standard normal variable, independent of the realization of G n and of σ(A i ), so that we can conclude by Proposition 3.2.3.

Last, we prove that the maximum of |ψ Gn | on G n has a subexponential tail.

Lemma 3.2.5 (Tail for the maximum of |ψ Gn |). Suppose that max x∈Vn G Gn (x, x) ≤ K 5 .

Then for all ∆ > 0, if n is large enough,

P Gn max x∈Vn |ψ Gn (x)| ≥ log 2/3 n ≤ n -∆ . (3.21)
In particular, by Proposition 3. Proof. Let N ∼ N (0, K 5 ). If n is large enough, then for all x ∈ V n ,

P Gn |ψ Gn (x)| ≥ log 2/3 n ≤ P Gn |N | ≥ log 2/3 n ≤ 2 exp - log 4/3 n 4K 5 ≤ n -∆-1
by Markov's inequality applied to the function u → exp log 2/3 n 2K 5 u . By a union bound on all x ∈ V n , we get P Gn max x∈Gn |ψ Gn (x)| > log 2/3 n ≤ n -∆ .

The Gaussian Free Field on T d

In Section 3.3.1, we characterize C h

• as a branching process, with a recursive construction (Proposition 3.3.1). Then, in Section 3.3.2, we establish its exponential growth, conditionally on {|C h

• | = +∞}. The main results are Propositions 3.3.4, 3.3.6 and 3.3.8.

C h • as a branching process

There is an alternative definition of ϕ T d , starting from its value at • and expanding recursively to its neighbours. It shows that C h • is an infinite-type branching process, the type of a vertex x being ϕ T d (x). For U G, and x, y ∈ G, let

G U G (x, y) := E G x T U k=0 1 X k = y . (3.22) Define the field ϕ U G on G by ϕ U G (x) := ϕ G (x) -E G x [ϕ G (X T U )
] for all x ∈ G, T U being the exit time from G.

Proposition 3.3.2 (Domain Markov property). The field ϕ U

G is a Gaussian field, independent from (ϕ G (x)) x∈G\U , with covariances given by Cov(φ

U G (x), ϕ U G (y)) = G U G (x, y).
We apply it with G = T

z ) z∈T d , (Φ (2) z ) z∈T d ∈ R T d such that Φ (1) z ≤ Φ (2) z for all z ∈ T d , (Φ (1) z ) z∈T d ∈ S only if (Φ (2) z ) z∈T d ∈ S. Say that the event {ϕ T d ∈ S} is increasing if S is increasing. Lemma 3.3.3 (Conditional monotonicity). If E is an increasing event, then the map a → P T d a (E) is non-decreasing on R.
Proof. Let a 1 , a 2 ∈ R such that a 1 > a 2 . It suffices to give a coupling between a GFF ϕ

(1)

T d conditioned on ϕ (1)
T d (•) = a 1 and a GFF ϕ

(2)

T d conditioned on ϕ (2)
T d (•) = a 2 such that a.s., for every z ∈ T d , ϕ

(1)

T d (z) ≥ ϕ (2)
T d (z). To do this, let (ξ y ) y∈T d be i.i.d. standard normal variables, and define recursively ϕ

T d (z) = ϕ (2) T d (z) + (a 1 -a 2 )(d -1) -k . (1)

Exponential growth

Let Z h k := C h • ∩ ∂B T d (•, k) be the k-th generation of C h • .
We first prove the following: Proposition 3.3.4. There exists λ h > 1 such that

lim k→+∞ P T d (|Z h k | > λ k h /k 2 ) = η(h)
and

lim k→+∞ P T d (|Z h k | < kλ k h ) = 1.
Moreover, h → λ h is a decreasing homeomorphism from (-∞, h ) to (1, d -1).

We will need the following Lemma (whose proof is immediate from Propositions 3.1 and 3.3 of [START_REF] Sznitman | Coupling and an application to level-set percolation of the gaussian free field[END_REF] and Proposition 2.1 (ii) of [START_REF] Abächerli | Level-set percolation of the Gaussian free field on regular graphs I: regular trees[END_REF]), from which λ h originates. Let • be an arbitrary neighbour

of •. Let T + d be the cone from • out of •. Write C h,+ • := C h • ∩ T + d . For k ≥ 1, let Z h,+ k := C h,+ • ∩ ∂B T + d (•, k).
Lemma 3.3.5. Fix h < h . There exist λ h > 1 and a function χ h that is continuous with a positive minimum χ min on [h, +∞), that vanishes on (-∞, h) and such that

M h k := λ -k h x∈Z h,+ k χ h (ϕ T d (x))
is a martingale w.r.t. the filtration

F k := σ ϕ T d (x), x ∈ B T + d (•, k) , k ≥ 0,
and has an a.s. limit M h ∞ .

Proof of Proposition 3.3.4. We first establish that lim k→+∞ P

T d (|Z h k | > λ k h /k 2 ) = η(h). Clearly, lim sup k→+∞ P T d (|Z h k | > λ k h /k 2 ) ≤ P T d (|C h • | = +∞) = η(h). Conversely, denote E + = {|C h,+ • | = +∞} and E + k = {|Z h,+ k | ≥ λ k h /k 2 }. By Theorem 4.3 of [3], lim k→+∞ P T d (E + k ) = P T d (E + ) > 0. (3.23)
Hence, for any ε > 0, for k large enough,

P T d (E + k ) ≥ P T d (E + ) -ε. (3.24) Let T - d be the cone from • out of •, C h,- • := C h • ∩T - d , and Z h,- k := C h,- • ∩∂B T - d (•, k) for k ≥ 1. Let E -= {|C h,- • | = +∞} and E - k = {|Z h,- k | ≥ λ k h /k 2 }. Define E := {ϕ T d (•) ≥ h}∩{ C h,+ • is finite} and E k := {ϕ T d (•) ≥ h} ∩ { Z h,+ k = ∅}.
We have

P T d (E - k ∩ E k ) ≥ P T d (E -∩ E) -P T d (E -∩ E ∩ (E k ) c ) -P T d (E -∩ E ∩ (E - k ) c ). Define M h,- ∞ on C h,- • as M h ∞ on C h,+ • .
From the proof of Theorem 4.3 in [START_REF] Abächerli | Level-set percolation of the Gaussian free field on regular graphs I: regular trees[END_REF], we get that

P T d ({M h,- ∞ > 0} ∩ (E - k ) c ) → 0. And, by Proposition 4.2 in [3], P T d (E -∩ {M h,- ∞ = 0}) = 0, so that P T d (E -∩ (E - k ) c ) → 0. Moreover, (E k ) k≥0
is an increasing sequence of events and E = ∪ k≥1 E k , so that P T d (E ∩ (E k ) c ) → 0. Hence, for k large enough,

P T d (E - k ∩ E k ) ≥ P T d (E -∩ E) -2ε. (3.25) Note that {|C h • | = +∞} = E + (E -∩ E), so that P(E + ) + P(E -∩ E) = η(h). And for all k ≥ 2, E + k (E - k ∩ E k ) ⊆ {|Z h k | > λ k h /k 2 },
P T d (|Z h k | > λ k h /k 2 ) ≥ η(h) -3ε.
Since ε > 0 was arbitrary, lim k→+∞

P T d (|Z h k | > λ k h /k 2 ) = η(h). Now, we show that P T d (|Z h k | < kλ k h ) → 1. For all k ≥ 1, by definition of M h k , M h k ≥ χ h,min λ -k h Z h k .
From the proof of Proposition 3.3 in [START_REF] Sznitman | Coupling and an application to level-set percolation of the gaussian free field[END_REF],

M h ∞ is a.s. finite. Therefore, k -1 λ -k h Z h k → 0 a.s.
, and

P T d (|Z h,+ k | ≥ kλ k h /2) → 0.
In the same way,

P T d (|Z h,- k-1 | ≥ kλ k h /2) → 0. Since Z h k ⊆ Z h,+ k ∪ Z h,- k-1
, we are done. The last part of the Proposition comes directly from Propositions 3.1 and 3.3 in [START_REF] Sznitman | Coupling and an application to level-set percolation of the gaussian free field[END_REF].

Next, we establish finer results on the growth of C h

• . |C h • | has exponential moments: Proposition 3.3.6. There exists a constant K 7 > 0 such that as k → +∞,

max a≥h P T d a (k ≤ |C h • | < +∞) = o(exp(-K 7 k)). (3.26) Since {Z h k = ∅} ⊆ {|C h • | ≥ k},
we have the following straightforward consequence:

Corollary 3.3.7. For k large enough, for every a ≥ h,

P T d a (|C h • | = +∞) ≤ P T d a (Z h k = ∅) ≤ P T d a (|C h • | = +∞) + e -K 7 k .
In addition, there are large deviation bounds for the growth rate of Z h k :

Proposition 3.3.8. For every ε > 0, there exists C > 0 such that for every k ∈ N large enough,

max a≥h P T d a (k -1 log |Z h k | ∈ [log(λ h -ε), log(λ h + ε) + k -1 log χ h (a)] | Z h k = ∅) ≤ exp(-Ck). (3.27)
This also holds when replacing C h • by C h,+ • , and Z h k by Z h,+ k .

A crucial idea to prove Propositions 3.3.6 and 3.3.8 is to make a finite scaling, in order to get a branching process that is uniformly supercritical w.r.t. to the value of ϕ T d (•). Indeed, the fact λ h > 1 does not ensure that the expected number of children of • in T + d (or even in T d ) conditionally on ϕ T d (•) = a is more than one for every a ≥ h, in particular if a is small. However, due to the exponential growth of C h

• (and C h,+ • ) of Proposition 3.3.4, it turns out that for ∈ N large enough, even conditionally on ϕ T d (•) = h, the expected number of vertices in the -offspring of • is more than one, as stated in the Lemma below. Lemma 3.3.9. There exists ∈ N such that for every a ≥ h,

E T d a [|Z h |] ≥ E T d a [|Z h,+ |] ≥ E T d h [|Z h,+ |] > 1.
We will use it in the proofs of Propositions 3.3.6 and 3.3.8, looking at the branching process whose vertices are those of C h • at height 0, , 2 , etc.

Proof of Lemma 3.3.9.

Write E k := {|Z h,+ k | ≥ λ k h /k 2 }. By (3.23), there exists ε > 0 small enough such that for every k ≥ ε -1 , P T d (E k ) ≥ ε. For a 1 large enough, P T d (ϕ T d (•) ≥ a 1 ) < ε/2.
Note that E k is an increasing event, so that by Lemma 3.3.3, the map a → P T d a (E k ) is nondecreasing. Therefore, for every a ≥ a 1 and k

≥ M , if ν ∼ N (0, d-1 d-2 ) denotes the law of ϕ T d (•), P T d a (E k ) ≥ P T d a 1 (E k ) ≥ a 1 -∞ P T d b (E k )ν(db) ≥ P T d (E k ) -P T d (ϕ T d (•) ≥ a 1 ) ≥ ε/2.
From the construction of Proposition 3.3.1, it is straightforward that p := P T d h (• has one child z in C h,+ • , and ϕ T d (z) ≥ a 1 ) > 0.

Hence for ∈ N large enough,

E T d h [|Z h,+ |] ≥ pε 2 × λ h 2 > 1.
In addition, for every M ∈ R, {|{Z h,+ | ≥ M } is an increasing event. By Lemma 3.3.3, for every

a ≥ h, E T d a [|Z h,+ |] ≥ E T d h [|Z h,+ |]. Since Z h,+ ⊆ Z h a.
s., the conclusion follows.

Proof of Proposition 3.3.6. Fix a ≥ h, and let ∈ N be such that the conclusion of Lemma 3.3.9

holds. Let

F 1 := ∂B C h • (•, ). For j ≥ 1, if F j = ∅, choose an arbitrary vertex z j ∈ F j . Let O j be the -offspring of z j in C h
• and let F j+1 := O j ∪ F j \ {z j }. Thus, we explore C h • by revealing subtrees of height ≤ , so that at each step, we see at most (d -1) + . . .

+ (d -1) ≤ d +1 new vertices. Hence, if |C h
• | ≥ k for some k ∈ N, there will be at least k/d +1 steps before C h • is fully explored. By Lemma 3.3.3 (applied to {|Z h,+ | ≥ k} for every k ≥ 1), for every j ≥ 1, |F j | dominates stochastically a sum S j of j i.i.d. random variables of law ρ ,h -1, where

ρ ,h is the law of |Z h,+ | conditionally on ϕ T d (•) = h.
(3.28) Therefore,

P T d a (k ≤ |C h • | < +∞) ≤ +∞ j= k/d +1 P(S j ≤ 0).
But a variable of law ρ ,h -1 is bounded and has a positive expectation by Lemma 3.3.9, therefore there exist c, c > 0 such that P(S j ≤ 0) ≤ ce -c j for all j ≥ 1. Hence,

P T d a (k ≤ |C h • | < +∞) ≤ c +∞ j= k/d +1 e -c j ≤ c exp(-c k/d +1 ) 1 -e -c
and (3.26) follows.

Proof of Proposition 3.3.8. Let ε > 0. By definition of M h k and Lemma 3.3.5,

{|Z h,+ k | ≥ χ h (a)(λ h + ε) k } ⊆ {M h k ≥ λ -k h χ h,min χ h (a)(λ h + ε) k }
so that by Markov's inequality, for any a ≥ h,

P T d a (|Z h,+ k | ≥ χ h (a)(λ h + ε) k ) ≤ P T d a M h k ≥ χ h,min χ h (a) λ h + ε λ h k ≤ χ -1 h,min λ h λ h + ε k χ h (a) -1 E T d a [M h k ] ≤ χ -1 h,min λ h λ h + ε k .
This yields the upper large deviation for Z h,+ k (and for Z h k , using the facts that

Z h k ⊆ Z h,+ k ∪Z h,- k-1
and that |Z h,- k-1 | and |Z h,+ k-1 | have the same law). It remains to prove that for some C > 0 and k large enough,

max a≥h P T d a (k -1 log |Z h k | ≤ log(λ h -ε) | Z h k = ∅) ≤ exp(-Ck). (3.29)
We proceed in two steps. We first initiate the growth of C h • by showing that if Z n = ∅, the probability that |Z n | = o(n) decays exponentially with n, where is such that Lemma 3.3.9 holds. Then, if Z n has at least Θ(n) vertices, each of them has a positive probability to have a Kn-offspring of size at least λ Kn h /(Kn) 3 ≥ (λ h -ε) k with k := (K + )n and for a large enough constant K, independently of the others vertices. Hence the probability that

|Z k | ≤ (λ h -ε) k
decays exponentially with n, and thus with k.

First step. Recall the exploration of C h

• of the proof of Proposition 3.3.6, but perform it in a breadth-first way: reveal first the -offspring of •, then the -offspring of each vertex of Z h , then the -offspring of each vertex of Z h 2 , and so on. For n ≥ 1, if Z h n = ∅, let j + 1 be the first step at which we explore the offspring of a vertex of Z n . Note that j ≥ n. As in the proof of Proposition 3.3.6, there exist , c, c > 0 such that P(S i ≤ i) ≤ ce -c i for every i ≥ 1. Hence, for every n ≥ 1,

min a≥h P T d a (|Z h n | ≥ n |Z h n = ∅) ≥ 1 - i≥n ce -c i ≥ 1 - c 1 -e -c e -c n .
(3.30)

Second step. Let K be a positive integer constant, and let F be the set of vertices z ∈ Z h n such that the Kn-offspring of z has at least λ Kn h /n 3 vertices. This step mainly comes down to showing that the probability that F is empty decays exponentially with n.

Define the events

E n := {|Z h,+ Kn | ≥ λ Kn h /n 3 } and E n := {|Z h,+ Kn-1 | ≥ λ Kn h /n 3 }.
We first show that

p := min a≥h P T d a (E n ) > 0. ( 3
min a≥h P T d a (∃z ∈ Z h,+ 1 , ϕ T d (z) ≥ a 1 ) = P T d h (∃z ∈ Z h,+ 1 , ϕ T d (z) ≥ a 1 ) =: p > 0.
Hence p ≥ p p /4 and (3.31) is proved.

Note that |F | ≥ st.

Bin(|Z h n |, p ). Thus by (3.30), for n large enough,

min a≥h P T d a (|F | ≥ 1|Z h n = ∅) ≥ 1 -P T d a (|Z h n | ≤ n|Z h n = ∅) -(1 -p ) j ≥ 1 -ce -c n ,
up to changing the values of the constants c and c . Therefore,

max a≥h P T d a |Z h (K+ )n | < λ Kn h /n 3 | Z h n = ∅ ≤ ce -c j ≤ ce -c n .
If K is large enough, then for n large enough, λ Kn h /n 3 > (λ h -ε) (K+ )(n+1) , so that

max a≥h P T d a |Z h (K+ )n | < (λ h -ε) (K+ )(n+1) | Z h n = ∅ ≤ ce -c n .
We adjust the conditionning:

{Z h (K+ )n = ∅} ⊆ {Z h n = ∅}, and |Z h (K+ )n | < (λ h -ε) (K+ )(n+1) on {Z h n = ∅} \ {Z h (K+ )n = ∅}.
Therefore, there exists n 0 ≥ 1 such that for all n ≥ n 0 ,

max a≥h P T d a |Z h (K+ )n | < (λ h -ε) (K+ )(n+1) | Z h (K+ )n = ∅ ≤ c exp(-c (K + )(n + 1)), (3.32)
where the new value of c depends on the constants K and . This yields (3.29) for large enough multiples of K + . One can readily replace each Z h m by Z h,+ m in this reasoning (for any m ≥ 1), to get the same result for Z h,+ (K+ )n instead of Z h (K+ )n . It remains to show the result for non multiples of K + . Let k ≥ (K + )n 0 . Write k = (K + )n + m, with 0 ≤ m ≤ (K + ) -1. Note that on {Z h k = ∅}, Z h m has at least one vertex whose (k -m)-offspring is not empty. Hence

max a≥h P T d a |Z h k | < (λ h -ε) k | Z h k = ∅ ≤ max a≥h P T d a |Z h,+ k-m | < (λ h -ε) k | Z h,+ k-m = ∅ ≤ max a≥h P T d a |Z h,+ k-m | < (λ h -ε) (k-m)+(K+ ) | Z h,+ k-m = ∅ ≤ c exp(-c (K + )(n + 1)) ≤ ce -c k ,
where the third inequality comes from (3.32). Adapting this last computation for Z h,+ k is immediate. This concludes the proof of (3.27).

Approximate recursive construction of ψ G n

Let κ > 0 be a constant, and let

a n := κ log d-1 log n . (3.33)
The following statement is the main result of this section. It shows that a recursive construction of ψ Gn , under some assumptions on the subset A ⊆ V n of vertices where ψ Gn is already known, is very close to the construction of ϕ T d in Proposition 3.3.1. It is a crucial tool for comparing ψ Gn and ϕ T d in the exploration in the next section. It is analogous to Proposition 2.7 of [START_REF] Abächerli | Level-set percolation of the Gaussian free field on regular graphs II: finite expanders[END_REF],

where the assumptions on A are slightly different: they are suited to a deterministic d-regular graph satisfying (I) and (II), while ours will be adapted to an annealed exploration, where the randomness of G n plays a role. Our proof is very similar, but we feel that the general argument is sufficiently subtle and interesting to merit a full account.

Proposition 3.4.1. If the constant κ from (3.33) is large enough, then the following holds for n large enough. Assume that G n is a good graph as defined in Proposition 3.2.1, and that A ⊆ V n satisfies

• |A| ≤ n log -8 n,
• tx(B Gn (A, a n )) = tx(A), and

• max z∈A |ψ Gn (z)| ≤ log 2/3 n.
For every y ∈ ∂B Gn (A, 1), writing y for the unique neighbour of y in A, we have: In the proof, we will need the fact below, which is a consequence of Lemma 3.

E Gn [ψ Gn (y)|σ(A)] - 1 d -1 ψ Gn (y) ≤ log -3 n (3.
E Gn [ψ Gn (y)|σ(A)] = E Gn y [ψ Gn (X H A )] - E Gn y [H A ] E Gn πn [H A ] E Gn πn [ψ Gn (X H A )] ,
where (X j ) j≥0 is a SRW on G n . Write T y for B Gn (y, y, a n -1), which is a tree rooted at y by our assumptions on A and y (it consists of the paths of length a n -1 starting at y and not going through y).

Let ∂T y be the (a n -1)-offspring of y in T y . Let τ be the hitting time of ∂T y by (X j ). Then splitting the first term of the RHS above into two terms, we obtain

E Gn [ψ Gn (y)|σ(A)] = E Gn y [ψ Gn (X H A ) 1 H A ≤τ ] + E Gn y [ψ Gn (X H A ) 1 H A >τ ] - E Gn y [H A ] E Gn πn [H A ] E πn [ψ Gn (X H A )] . (3.36)
On {H A ≤ τ }, X H A = y. And, P Gn y (H A ≤ τ ) is the probability that a SRW on Z started at 1 with transition probabilities 1/d towards the left and (d -1)/d towards the right hits 0 before a n . It is classical that its difference with 1/(d -1) decays exponentially with a n (see for instance Theorem 4.8.9 in [START_REF] Durrett | Probability: Theory and examples[END_REF]). Hence, if κ is large enough,

P Gn y (H A ≤ τ ) - 1 d -1 ≤ log -6 n, (3.37)
and since max z∈A |ψ Gn (z)| ≤ log 2/3 n,

E Gn y [ψ Gn (X H A ) 1 H A ≤τ ] - 1 d -1 ψ Gn (y) ≤ log -4 n.
Therefore, to establish (3.34), it is enough to show that

E Gn y [ψ Gn (X H A ) 1 H A >τ ] - d -2 d -1 E Gn πn [ψ Gn (X H A )] ≤ 3 log -5 n (3.38) and d -2 d -1 - E Gn y [H A ] E Gn πn [H A ] E Gn πn [ψ Gn (X H A )] ≤ log -5 n. (3.39)
By the strong Markov property, letting p z := P Gn y (H A > τ, X H ∂Ty = z) for z ∈ ∂T y , we have

E Gn y [ψ Gn (X H A ) 1 H A >τ ] = z∈∂Ty p z E Gn z [ψ Gn (X H A )].
For all z ∈ ∂T y , by Lemma 3.4.2 (with s = a n ), if κ is large enough, then for large enough n,

P Gn z (H A < log 2 n) ≤ 1 -(1 -(d -1) -an ) log 2 n ≤ 2 log 2 n (d -1) -an ≤ log -6 n.
Therefore,

E Gn y [ψ Gn (X H A ) 1 H A >τ ] - z∈∂Ty p z z ∈Vn P Gn z (X log 2 n = z )E Gn z [ψ Gn (X H A )] ≤ P Gn z (H A < log 2 n) max z∈A |ψ Gn (z)| ≤ log -5 n.
By Corollary 2.1.5 of [START_REF] Saloff-Coste | Lectures on finite Markov chains[END_REF] and (I), P Gn z X log 2 n = z -π n (z ) ≤ e -λ Gn log 2 n ≤ 1 n 2 for n large enough and all z, z ∈ V n , so that

E Gn y [ψ Gn (X H A ) 1 H A >τ ] - z∈∂Ty p z E Gn πn [ψ Gn (X H A )] ≤ 2 log -5 n.
Finally, note that

z∈∂Ty p z -d-2 d-1 = P Gn y (H A > τ ) -d-2 d-1 ≤ log -6
n by (3.37), and this yields (3.38).

Let τ be the hitting time of ∂T y ∪ {y} by (X j ). By the strong Markov property again, we have

E Gn y [H A ] = E Gn y τ + z∈∂Ty P Gn y (X τ = z)E Gn z [H A ] . (3.40)
If (Z j ) j≥0 is a SRW on Z starting at 1 with transition probabilities 1/d towards the left and (d -1)/d towards the right, then τ has the law of the hitting time of {0, a n } by (Z j ). Hence, for all k ≥ 0, P y (τ ≥ k) ≤ P(Z k ≤ a n ). But Z j is the sum of j i.i.d. increments that are bounded a.s. and with mean (d -2)/d > 0. We apply the exponential Markov inequality and obtain that for some γ > 0 and n large enough, for all k ≥ log n, P(Z k ≤ a n ) ≤ exp(-γk). Thus,

E Gn y τ ≤ log n P Gn y (τ ≤ log n) + k≥log n k exp(-γk) ≤ 2 log n. ( 3
E Gn y [H A ] E Gn πn [H A ] - z∈∂Ty p z E Gn z [H A ] E Gn πn [H A ] ≤ 8 log n log 8 n ≤ log -6 n,
where p z := P Gn y (X τ = z). By (3.37), z∈∂Ty p z -d-2 d-1 ≤ log -6 n. Therefore,

E Gn y [H A ] E Gn πn [H A ] - d -2 d -1 ≤ 2 log -6 n + max z∈∂Ty E Gn z [H A ] E Gn πn [H A ] -1 .
To conclude the proof of (3.39), and thus of (3.34), it is enough to show that max z∈∂Ty

E Gn z [H A ] E Gn πn [H A ] -1 ≤ 5 log -6 n. (3.42)
We adapt for this the proof of Proposition 3.5 in [START_REF] Černý | Giant vacant component left by a random walk in a random d -regular graph[END_REF]. For all z ∈ ∂T y , we can write

E Gn z [H A ] ≤ E Gn z H A 1 H A <log 2 n + z ∈Vn P Gn z (X log 2 n = z , H A ≥ log 2 n)(E Gn z [H A ] + log 2 n
), hence using Corollary 2.1.5 of [START_REF] Saloff-Coste | Lectures on finite Markov chains[END_REF] and the fact that π n is uniform on V n ,

E Gn z [H A ] ≤ log 2 n + z ∈Vn π n (z ) + e -λ Gn log 2 n E Gn z [H A ] ≤ log 2 n + (1 + ne -λ Gn log 2 n )E Gn πn [H A ]. (3.43)
Recall that E Gn πn [H A ] ≥ log 8 n/4. For n large enough, we have by (3.43):

E Gn z [H A ] E Gn πn [H A ] ≤ 4 log 2 n log 8 n + 1 + ne -λ Gn log 2 n ≤ 1 + 5 log -6 n. (3.44)
Conversely,

E Gn z [H A ] ≥ z ∈Vn P Gn z (X log 2 n = z )E Gn z [H A ] - z ∈Vn P Gn z (X log 2 n = z , H A ≤ log 2 n)E Gn z [H A ] ≥ z ∈Vn (π n (z ) -e -λ Gn log 2 n )E Gn z [H A ] -P Gn z (H A ≤ log 2 n) sup z ∈Vn E Gn z [H A ] ≥(1 -ne -λ Gn log 2 n )E Gn πn [H A ] -P Gn z (H A ≤ log 2 n) sup z ∈Vn E Gn z [H A ].
(3.43) and (3.44) are in fact true for all z ∈ V n , so that for large enough n,

E Gn z [H A ] E Gn πn [H A ] ≥1 -ne -λ Gn log 2 n -P Gn z (H A ≤ log 2 n)(1 + 5 log -6 n) ≥1 -2 log -6 n,
provided that κ is large enough: by Lemma 3.4.2 applied as below (3.39), we have

P Gn z (H A ≤ log 2 n) ≤ log -6 n for every z ∈ ∂T y .
Here lies the main difference with Proposition 3.5 of [START_REF] Černý | Giant vacant component left by a random walk in a random d -regular graph[END_REF], where there might be more cycles in B Gn (A, a n ) than in A (i.e. the trees planted on the boundary of A might intersect). One has to use a weaker version of Lemma 3.4.2, and the lower bound becomes 1 -|A| log -∆ n, ∆ being any positive constant (κ has to be large enough w.r.t. ∆).

Thus, the proof of (3.34) is complete (note that the required lower bounds on κ given by Lemma 3.4.2 are uniform in y and A).

We prove (3.35) in the same fashion. Note in particular that by (3.20),

Var Gn (ψ Gn (y)|σ(A)) = G Gn (y, y) -E Gn y [G Gn (y, X H A )] + E Gn y [H A ] E Gn πn [H A ] E Gn πn [G Gn (y, X H A )]
so that we get, noticing that H A ≥ τ (the hitting time of ∂T y ∪ {y}) a.s.,

Var Gn (ψ Gn (y)|σ(A)) - d d -1 ≤ G Gn (y, y) -E Gn y [G Gn (y, X H A ) 1 H A =τ ] - d d -1 + E Gn y [G Gn (y, X H A ) 1 H A >τ ] - E Gn y [H A ] E πn [H A ] E Gn πn [G Gn (y, X H A )] (3.45)
We have

E Gn y [G Gn (y, X H A ) 1 H A =τ ] = E Gn y [G Gn (y, X τ ) 1 H A =τ ] = E Gn y [G Gn (y, X τ )] -E Gn y [G Gn (y, X τ ) 1 H A >τ ] .
Remark that

|E Gn y [G Gn (y, X τ ) 1 H A >τ ] | ≤ max z∈∂Ty |G Gn (y, z)| ≤ log -5 n
if κ is large enough, by (3.11). Now, by (3.17) applied to the tree T y (note that T Ty = τ ), we get

|G Gn (y, y) -E Gn y [G Gn (y, X τ )] -G Ty Gn (y, y)| ≤ E Gn y [τ ] n By (3.41), E Gn y [τ ] n = O(log -5 n), therefore, the first term of the RHS of (3.45) is |G Ty Gn (y, y) -d d-1 | + O(log -5 n). But T y is isomorphic to B := B T + d (•, a n -1), so that G Ty Gn (y, y) = G B T d (•, •) = G T d (•, •) -P T d • (T B = •)G T d (•, •) -P T d • (T B = •)G T d (•, z)
for any z ∈ T + d at distance a n -1 of •, by cylindrical symmetry of T + d . By (3.10), if κ is large enough:

G Ty Gn (y, y) = d-1 d-2 -P T d • (T B =•) d-2 + O(log -5 n).
By the same reasoning as that leading to (3.37),

P T d • (T B = •) = 1 d-1 + O(log -5 n) for κ large enough, hence |G Ty Gn (y, y) -d d-1 | = O(log -5 n).
All in all, we get that the first term of the RHS of (3.45) is O(log -5 n).

One applies to the second term of the RHS of (3.45) the same reasoning as that for (3.38) and

(3.39). In particular, since G n is a good graph, (3.11) yields max x,y∈Vn |G Gn (x, y)| ≤ K 5 , an upper bound easier to use than max z∈A |ψ Gn (z) | ≤ log 2/3 n.

Exploration of ψ G n around a vertex 3.5.1 Successful exploration

In this section, we prove that with P ann -probability arbitrarily close to η(h) as n → +∞,

|C Gn,h x | ≥ n 1/2 log -κ-6 n, C Gn,h
x being the connected component of a given vertex x in E ≥h ψ Gn (Proposition 3.5.1), and κ the constant defined in (3.33).

To do so, we explore a tree-like neighborhood T x of x in E ≥h ψ Gn . We reveal T x generation by generation, and couple it with a realization of C h+log -1 n • ⊆ T d that is independent of the pairing of the half-edges of G n . By Proposition 3.2.4, a realization of ψ Gn is given by a recursive construction with the same normal variables as those of the realization of ϕ T d . When: At every step of the exploration, T x will be a tree rooted at x, T x its respective counterpart in T d , rooted at •, and Φ an isomorphism from T x to T x . At step k, we will reveal the k-th generation of T x and T x .

• that realization of C h+log -1 n • is infinite (which happens with probability η(h + log -1 n) η(h)),
Precisely, the exploration from x consists of the following steps:

• at step 0, T x = {x} and T x = {•}. Reveal the pairings of the half-edges of B Gn (x, a n ).

Stop the exploration if tx(B

Gn (x, a n )) > 0 or if ϕ T d (•) < h + log -1 n.
• at step k ≥ 1, reveal the edges of B Gn (T x , a n + 1) that were not known at step k -1. Let O k-1 be the set of the vertices of T x of height k -1. Stop the exploration if at least one of the following conditions holds:

C1 a cycle appears in B Gn (T x , a n + 1),

C2 |O k-1 | ≥ n 1/2 b n , C3 O k-1 = ∅ (i.
T d is below h + log -1 n. Any number near a vertex v is ϕ T d (Φ(v)).
If the exploration is stopped at some step k, at which only C2 is met, say that it is successful.

In this case, by Proposition 3.2.4, we can sample ψ Gn as follows: we reveal the remaining pairings of half-edges in G n . We set

ψ Gn (x) = ϕ T d (•). For all k, i ≥ 1, if A k,i = {x} ∪ {x ,j |( , j) ≺ (k, i)}
where ≺ is the lexicographical order on N 2 , let

ψ Gn (x k,i ) = E Gn [ψ Gn (x k,i )|σ(A k,i )] + ξ k,i Var(ψ Gn (x k,i )|σ(A k,i )). (3.47)
Note that the conditional expectation and variance of the RHS are σ(A k,i )-measurable random variables, so that this definition makes sense, even if ψ Gn (x k,i ) appears on both sides of the equation.

Let S(x) := {the exploration from x is successful}. We prove the following: Proposition 3.5.1.

P ann (S(x) ∩ {T x ⊆ C Gn,h x }) -→ n→+∞ η(h). ( 3
n 1/2 (d -1) -an log -6 n log λ h n ≤ n 1/2 (d -1) -an log -4 n
vertices, so that

|R x | ≤ n 1/2 (d -1) -an log -4 n × (1 + (d -1) + . . . + (d -1) an+1 ) ≤ n 1/2 log -3 n.
In order to prove Proposition 3.5.1, we first show that C h+log -1 n

• either has an exponential growth at rate > √ λ h with probability close to η(h), or dies out before reaching height log λ h n with probability close to 1 -η(h). Although the proof is slightly technical, it relies merely on Proposition 3.3.4 and on the continuity of the maps h → λ h and h → η(h ).

Lemma 3.5.3. Let F (n) k := {n 1/2 b n ≤ |Z h+log -1 n k | ≤ dn 1/2 b n } and F (n) k := {Z h+log -1 n k-1 = ∅} for k ≥ 1. Let k 0 := inf{k ≥ 1, F (n) k or F (n) k happens}, F * (n) := {k 0 ≤ log λ h n} ∩ F (n) k 0 and F * (n) := {k 0 ≤ log λ h n} ∩ F (n) k 0 .
Then, as n → +∞:

P ann (F * (n) ) → η(h) and P ann (F * (n) ) → 1 -η(h). (3.49)
Proof. Remark first that by construction, P ann acts like P T d on events that only depend on ϕ T d .

Note that for every n ≥ 1,

F * (n) ∩ F * (n) = ∅, implying P ann (F * (n) ) + P ann (F * (n) ) ≤ 1. Hence it is enough to prove that lim inf n→+∞ P ann (F * (n) ) ≥ η(h) (3.50) and lim inf n→+∞ P ann (F * (n) ) ≥ 1 -η(h) (3.51) Let ε ∈ (0, η(h)). Let δ > 0 be such that |η(h + δ) -η(h)| ≤ ε
P ann ∃k ≤ log λ h n, |Z h+log -1 n k | > n 1/2 b n ≥ lim inf n→+∞ P ann ∃k ≤ log λ h n, |Z h+δ k | > n 1/2 b n ≥ lim inf n→+∞ P ann |Z h+δ log λ h n | > n
P ann ∃k ≤ log λ h n, |Z h+log -1 n k | > n 1/2 b n ≥ η(h + δ) ≥ η(h) -ε. (3.52)
Since each vertex of T d has at most d children, we have

|Z h+log -1 n k | ≤ d|Z h+log -1 n k-1 | determin- istically for all k ≥ 1. Hence, letting k := inf{k ≥ 0, |Z h+log -1 n k | ≥ n 1/2 b n } when this set is non-empty, F (n) k holds, so that {∃k ≤ log λ h n, |Z h+log -1 n k | > n 1/2 b n } ⊆ ∪ k≤log λ h n F (n) k . Thus lim inf n→+∞ P ann (F * (n) ) ≥ lim inf n→+∞ P ann ∪ k≤log λ h n F (n) k ≥ η(h) -ε
by (3.52), and this shows (3.50).

For n ≥ e 1/δ , C h+δ

• ⊆ C h+log -1 n • ⊆ C h • . Note that for n ≥ 1, P ann (F * (n) ) ≥ P ann Z h+log -1 n log λ h n -1 = ∅ -P ann {Z h+log -1 n log λ h n -1 = ∅} ∩ {∃k ≥ 1, |Z h+log -1 n k | ≥ n 1/2 b n } ≥ P ann Z h log λ h n -1 = ∅ -P ann {|C h+log -1 n • | < +∞} ∩ {∃k ≥ 1, |Z h+log -1 n k |≥ n 1/2 b n } ≥ P ann Z h log λ h n -1 = ∅ -P ann |C h+log -1 n • | < +∞ | ∃k ≥ 1, |Z h+log -1 n k | ≥ n 1/2 b n .
The first term of the RHS converges to 1 -η(h) as n → +∞. For any k ≥ 1 and for any

v ∈ B T d (•, k), denoting T v the possible subtree from v in C h+log -1 n • (if v ∈ Z h+log -1 n k
) and

C • (h, δ) the connected component of • in ({•} ∪ E ≥h+δ ϕ T d ) ∩ T + d , P ann (|T v | < +∞ |v ∈ Z h+log -1 n k ) ≤ P T d h (|C • (h, δ)| < +∞)
by Lemma 3.3.3, independently of the other vertices of Z h+log -1 n k . Thus,

P ann (|C h • | < +∞ | ∃k ≥ 1, |Z h k | ≥ n 1/2 b n ) ≤ P T d h (|C • (h, δ)| < +∞) n 1/2 bn .
Therefore, it only remains to prove that

P T d h (|C • (h, δ)| < +∞) < 1. By Lemma 3.3.3, the map a → P T d a (|C • (h, δ)| = +∞) is non-decreasing. By (3.23), +∞ h+δ P T d a (|C • (h, δ)| = +∞)ν(da) > 0,
where we recall that ν is the density of ϕ T d (•). Hence for some a 1 ≥ h + δ, for every a ≥ a 1 ,

P T d a (|C • (h, δ)| = +∞) ≥ P T d a 1 (|C • (h, δ)| = +∞) > 0.
And, there exists p > 0 such that

P T d h (• has one child z such that ϕ T d (z) ≥ a 1 ) ≥ p.
Therefore,

P T d h (|C • (h, δ)| = +∞) ≥ p P T d a 1 (|C • (h, δ)| = +∞) > 0, and P T d h (|C • (h, δ)| < +∞) < 1. (3.51) follows.
Proof of Proposition 3.5.1. We first establish that C1 happens with P ann -probability o(1). Then, if there is no cycle in B Gn (T x , a n ), we can apply Proposition 3.4.1, to bound the difference between ϕ T d and ψ Gn . By Remark 3.5.2, at most dn 1/2 log -3 n matchings of half-edges are performed during the exploration. By (3.14) with k = m 0 = 1, m E = 0 and m ≤ dn 1/2 log -3 n, the probability to create at least one cycle during these matchings is less than log -1 n for large enough n. Therefore, P ann (C1 happens) → 0.

(3.53)

Note that if C1 does not happen, then on F * (n) , (resp. F * (n) ), C3 (resp. C2) is satisfied, but not C4. Moreover, on F * (n) , (resp. F * (n) ), C2 (resp. C3) does not hold, so that P ann (S(x)) ≥ P ann (F * (n) ) -P ann (C1 happens)

and, since F * (n) ∩ F * (n) = ∅:

P ann (S(x)) ≤ P ann ((F * (n) ) c ) + P ann (C1 happens) ≤ 1 -P ann (F * (n) ) + P ann (C1 happens).
Thus, by (3.49) and (3.53),

P ann (S(x)) → η(h). (3.54)
Now, we compare ψ Gn with ϕ T d (supposing that the exploration is over and that S(x) holds).

Note that by C4, T x has a maximal height log λ h n so that by the triangle inequality,

{T x ⊆ C Gn,h x } ⊆ {∃y ∈ T x , |ψ Gn (y) -ϕ T d (Φ(y))| ≥ log -1 n} ⊆ ∪ y∈Tx E(y),
where

E(x) := {|ψ Gn (x) -ϕ T d (Φ(x))| ≥ log -3 n} and E(y) := {|ψ Gn (y) -ϕ T d (Φ(y))| ≥ |ψ Gn (y) -ϕ T d (Φ(y))| + 2 log -3 n} for y = x.
Suppose that G n is a good graph. For x k,i ∈ T x \ {x}, we can apply Proposition 3.4.1 on the event E

(n) k,i := {max y ∈A k,i |ψ Gn (y )| < log 2/3 n }, (note that |A k,i | ≤ n 1/2 by Remark 3.5.2 and that tx(B Gn (A k,i , a n )) = tx(A k,i ) by C1). Writing y = x k,i and ξ = ξ Φ(x k,i ) , we get for n large enough:

|ψ Gn (y) -ϕ T d (Φ(y))| ≤ ψ Gn (y) -ϕ T d (Φ(y)) d -1 + log -3 n + Var Gn (ψ Gn (y)|σ(A)) - d -1 d ξ and Var Gn (ψ Gn (y)|σ(A)) - d -1 d ≤ d -1 d -log -4 n - d -1 d ≤ 11 10 d -1 d d 2(d -1) log -4 n ≤ log -4 n. Let E (y) := E(y) ∩ E (n)
k,i . We have

P Gn (E (y)) ≤ P(|ξ| log -4 n ≥ log -3 n) ≤ n -3 (3.55)
by the exponential Markov inequality used as in the proof of Lemma 3.2.5. Moreover, by (3.8), if κ is large enough, we obtain in the same manner:

P Gn (E(x)) ≤ P Gn (|ξ • | log -4 n ≥ log -3 n) ≤ n -3 . ( 3
P Gn (T x ⊆ C Gn,h x ) ≤ P Gn (∪ y∈Tx E(y)) ≤ n -5/2 + n -2 ≤ n -1 ,
so that by Proposition 3.2.1 and (3.54):

P ann (S(x) ∩ {T x ⊆ C Gn,h x }) → η(h).

Aborted exploration

For x ∈ V n , the lower exploration is the exploration of Section 3.5.1, modified by replacing (3.57)

h + log -1 n by h -log -1 n, so that we compare T x and C h-log -1 n • . If it is stopped at some step k at which only C3 is met, say that it is aborted. Write A(x) := {the lower exploration from x is aborted} ∩ {C Gn,h x ⊆ T x }.
The proof follows from a direct adaptation of Lemma 3.5.3 and Proposition 3.5.1. Note in particular that P ann (C1 happens) = o(1), and that P ann (A(x)

) = P ann (Z h-log -1 n k 0 -1 = ∅) + o(1) = 1 -η(h) + o(1).

Existence of a giant component

In Section 3.6.1, we prove that two vertices x, y ∈ V n are in the same connected component of E ≥h ψ Gn with P ann -probability -→ n→+∞ η(h) 2 (Proposition 3.6.2). Then in Section 3.6.2, we use a second moment argument to get concentration and to show (3.2).

Connecting two successful explorations

Let us describe our strategy to establish Proposition 3.6.2. We perform explorations as in Section 3.5.1 from x and y. If they are both successful and do not meet (which happens with probability η(h) 2 ), we develop disjoint balls, denoted "joining balls", from ∂T x to ∂T y (Section 3.6.1). Each of them is rooted at a vertex of ∂T x , hits ∂T y at exactly one vertex, and has a "security radius" of depth a n around its path from ∂T x to ∂T y (see Figure 3). The construction of the joining balls only depends on the structure of G n , and not on the values of ψ Gn . Then, we realize ψ Gn on T x , T y and those balls (Section 3.6.1). If they are all disjoint and tree-like, once we have revealed ψ Gn on T x and T y , this security radius allows us to apply Proposition 3.4.1 to approximate ψ Gn on the paths from ∂T x to ∂T y by ϕ T d .

Let us explain with a back-of-the-envelope computation how the joining balls allow to connect T x and T y in E ≥h ψ Gn . Since |T y | n 1/2 b n by C2, the probability that for a given z ∈ ∂T x , exactly one of the vertices at distance γ log d-1 log n (and no vertex at distance < γ log d-

1 log n ) from z is in ∂T y is P(Bin((d -1) γ log d-1 log n , n 1/2 bn dn ) = 1) log γ n × n -1/2 b n .
And there are n 1/2 b n vertices in ∂T x , hence we can expect that that the number of joining balls is at least

n 1/2 b n × log γ n × n -1/2 b n = b 2
n log γ n, provided that we can control some undesirable events (such as an intersection between balls, or a cycle in a ball). This is the purpose of Lemma 3.6.1.

Moreover, we know that for large r ∈ N and v ∈ ∂B T d (•, r),

P T d (v ∈ C h • ) is of order (λ h /(d- 1
)) r , by Proposition 3.3.8. Taking r = γ log d-1 log n, the probability that E ≥h ψ Gn percolates from ∂T x to ∂T y through a given joining ball is log γ(log d-1 λ h -1) n, if we can approximate ψ Gn by ϕ T d .

For γ large enough w.r.t κ (recall (3.46) and (3.33), and recall that λ h > 1),

b 2 n log γ n × log γ(log d-1 λ h -1) n ≥ log -2κ-13 n × log γ log d-1 λ h n >> 1,
so that with high probability, E ≥h ψ Gn will percolate through at least one joining ball from ∂T x to ∂T y .

The joint exploration

For x, y ∈ V n , write x h ↔ y if y ∈ C Gn,h
x . Let (ξ z,v ) z∈{x,y},v∈T d be an array of i.i.d. standard normal variables independent from everything else. Define the joint exploration from x and y as the exploration from x (with the (ξ x,v )'s), then the exploration from y (with the (ξ y,v )'s), as in Section 3.5.1, with the additional condition C5 the exploration is stopped as soon as R x ∩ R y = ∅, where R x (resp. R y ) is the set of vertices seen during the exploration from x (resp. from y), as defined in Remark 3.5.2. Note that the families (ξ x,v ) v∈T d and (ξ x,v ) v∈T d generate two independent copies of ϕ T d .

If both explorations are successful and C5 does not happen (denote S(x, y) this event), we add the following steps to the joint exploration. Let γ > 0 and

a n := γ log d-1 log n .
(3.58)

Denote z 1 , . . . , z |∂Tx| the vertices of ∂T x . For j = 1, 2, . . . , |∂T x | successively, build B * (z j , a n ) the subgraph of G n obtained as follows (see Figure 3 for an illustration). Write

B * j := ∪ j <j B * (z j , a n ) and Q j := R x ∪ R y ∪ B * j , (3.59)
so that Q j is the set of vertices seen in the exploration before the construction of B * (z j , a n ).

Let initially B * (z j , a n ) be the subtree from z j of height a n in the tree B Gn (T x , a n ) (in blue in Figure 3). For k ≤ a n , write B * (z j , k) := B * (z j , a n ) ∩ B Gn (z j , k). If B * (z j , a n ) ∩ B * j = ∅, say that j is spoiled, and the construction of B * (z j , a n ) stops.

Else, for k = a n , a n + 1, . . . , a n -2a n -2 successively, while tx(B * (z j , k) ∪ Q j ) = tx(Q j ) (i.e. no cycle has been discovered) and B * (z j , k) ∩ B Gn (T y , a n ) = ∅, add to B * (z j , a n ) the neighbours of B * (z j , k) and the corresponding edges (in red in Figure 3). If for some k ∈ {a n , . . . , a n -2a n -2},

tx(B * (z j , k) ∪ Q j ) > tx(Q j ) (i.e. at least one cycle is discovered) or B * (z j , k) ∩ B Gn (T y , a n ) = ∅, the construction of B * (z j , a n ) stops.
If the construction has not been stopped for some k < a n -2a n -1, add the neighbours of B * (z j , a n -2a n -1) to B * (z j , a n ) (also in red in Figure 3). If

|B * (z j , a n -2a n ) ∩ B Gn (T y , a n )| = 1, the construction of B * (z j , a n ) stops.
Else, let v j (0) be the unique vertex of B * (z j , a n -

2a n ) ∩ B Gn (T y , a n ). If tx((B * (z j , a n -2a n ) ∪ Q j ) \ {v j (0)}) > tx(Q j ), the construction of B * (z j , a n ) stops.
Else, for k = a n -2a n , . . . , a n -1 successively, while tx(B * (z j , k)∪Q j ) = tx(B * (z j , a n -2a n )∪Q j ), add the neighbours of B * (z j , k) to B * (z j , a n ) (in green in Figure 3). Then, the construction of B * (z j , a n ) is completed. In this case only, and if

tx(B * (z j , a n ) ∪ Q j ) = tx(B * (z j , a n -2a n ) ∪ Q j ),
say that B * (z j , a n ) is a joining ball. In other words, we obtain a joining ball if, revealing the offspring up to generation a n of z j , the (a n -2a n ) offspring of z j intersects ∂T y at a unique vertex v j (0), and no cycle is discovered in the whole construction (except when B * (z j , a n ) reaches ∂B Gn (T y , a n ) at v j (0), if T x and T y were already connected in Q j by B * (z j , a n ) for some j < j). Proof. Denote F * (n) (x) (resp. F * (n) (y)) the event F * (n) for x (resp. y). Remark that the realization of F * (n) (x) (resp. F * (n) (y)) only depends on the version of ϕ T d defined by (ξ x,v ) v∈T d (resp. by (ξ y,v ) ,v∈T d ) and not on the pairings of G n . Hence, F * (n) (x) and F * (n) (y) are independent. As in the proof of Lemma 3.5.3, we get that

P ann (F * (n) (x) ∩ F * (n) (y)) = P ann (F * (n) (x))P ann (F * (n) (y)) → η(h) 2 and P ann (F * (n) (x) ∪ F * (n) (y)) → 1 -η(h) 2 .
Moreover, P ann (C1 or C5 happens) → 0. Indeed, by Remark 3.5.2, less than 2dn We proceed in two steps: in step 1, we control the number of spoiled vertices, and the number of vertices of ∂B Gn (T y , a n ) that are hit when building the B * (z j , a n )'s (if a large proportion of those vertices are in Q j , then it significantly affects the probability that B * (z j , a n ) is a joining ball). In step 2, we estimate the probability that for a given j, B * (z j , a n ) is a joining ball, provided that the bounds of step 1 hold. This gives a binomial lower bound for |J|.

Step 1. Note that by Remark 3.5.2 and (3.33

), |∂B Gn (T x , a n )| + |∂B Gn (T y , a n )| ≤ 2n 1/2 log -1 n.
Note also that for every j ≤ |∂T x |, B * (z j , a n ) contains less than (d -1) a n ≤ log γ n half-edges.

Hence:

at every moment of the exploration, less than n 1/2 log γ n half-edges have been seen. (3.61) Let

B * := ∪ j≤|∂Tx| B * (z j , a n ). (3.62)
To reveal the edges of B * , one proceeds to at most n 1/2 log γ n pairings of half-edges by (3.61).

Any pairing that results in an edge e between some B * (z j , k) and B Gn (T y , a n ) then leads to at

most 1 + (d -1) + . . . + (d -1) 2an ≤ 3(d -1) 2an ≤ log 2κ+1 n ≤ log γ-1 n
vertices of B * (z j , a n ) ∩ B Gn (T y , a n ), since the construction of B * (z j , a n ) stops if such an edge happens at distance less than a n -2a n of z j (and recall that we choose γ > 3κ + 18 > 2κ + 2).

Thus, by (3.15) with k = log 2γ+1 n , m < n 1/2 log γ n and m 1 + m 0 + m E < n 1/2 log γ n (due to 3.61), for n large enough:

P ann (S(x, y) ∩ {|B Gn (T y , a n ) ∩ B * | ≥ log 3γ n}) ≤ 0.99 log 2 n ≤ n -3 . (3.63)
Let N be the total number of spoiled vertices. By (3.15) with the same parameters,

P ann (S(x, y) ∩ {N ≥ log 3γ n}) ≤ n -3 . (3.64)
Step 2. Recall the definition of B * j from (3.59). For j ≤ m, denote

S j := S(x, y) ∩ {|B Gn (T y , a n ) ∩ B * j | ≤ log 3γ n} ∩ {z j is not spoiled}. (3.65)
Suppose that for every j ≥ 1,

P ann (B * (z j , a n ) is a joining ball | S j ) ≥ n -1/2 log γ-2κ-10 n. (3.66)
On E := {|B Gn (T y , a n ) ∩ B * | < log 3γ n} ∩ {N < log 3γ n}, the number of j's such that S j holds is at least

|∂T x | -log 3γ n ≥ n 1/2 log -κ-7 n
by (C2) and (3.46). Thus, if Z ∼ Bin n 1/2 log -κ-7 n , n -1/2 log γ-2κ-10 n , P ann S(x, y) ∩ {|J| ≤ log γ-3κ-18 n} ≤ P(Z ≤ log γ-3κ-18 n) + P ann (S(x, y) ∩ E c ).

For large enough n, P ann (S(x, y) ∩ E c ) = o(n -2 ) by (3.63) and (3.64). Moreover, one checks easily (using γ > 3κ + 18 and (3.12)) that for n large enough:

max 0≤k≤ log γ-3κ-18 n P(Z = k) ≤ 1/n.
This yields (3.60). Hence, it only remains to prove (3.66).

Remark that P ann (B * (z j , a n ) is a joining ball | S j ) ≥ p 1 p 2 p 3 where:

• p 1 := P ann (E 1 |S j ) and E 1 := S j ∩ {no cycle is created and no connection to Q j is made when revealing B * (z j , a n -2a n -1)},

• p 2 := P ann (E 2 |E 1 ) where E 2 := E 1 ∩ {exactly one edge connects B * (z j , a n -2a n -1) and

D := ∂B Gn (T y , a n )\{B Gn (B Gn (T y , a n )∩B * j , 2a n )} }∩{no cycle is created and no connection to ∂B Gn (T y , a n ) ∪ B *
j is made when revealing the other edges of B * (z j , a n -2a n )},

• p 3 := P ann (E 3 |E 2 ) where E 3 := E 2 ∩{no cycle is created and no connection to ∂B Gn (T y , a n )∪ B * j is made when revealing the remaining edges of B * (z j , a n )}.

This definition of D guarantees that B * (z j , a n ) will not intersect a previously realized joining ball when growing the subtree from v j (0) in B Gn (T y , a n ).

(3.14) with k = 1, m 0 , m ≤ log γ n and m E , m 1 ≤ n 1/2 log γ n due to (3.61) yields for n large enough:

p i ≥ 1 -C(1) log γ n max(n 1/2 log γ n, 2 log γ n) n ≥ 1 -n -1/3 (3.67)
for i ∈ {1, 3}. Therefore, p 1 p 3 ≥ 1/2 for n large enough.

On E 1 , reveal the pairings of the half-edges of ∂B * (z j , a n -2a n -1) one by one. E 2 holds if:

• one given half-edge is matched to a half-edge of D, which happens with probability at least |D| dn , and

• each other half-edge is matched to a half-edge that had not been seen before (by (3.61),

for each half-edge this happens with probability at least 1

-n 1/2 log γ n dn-n 1/2 log γ n ≥ 1 -n 1/2 log γ n n ).
Since ∂B * (z j , a n -2a n -1) has (d -1)|∂B * (z j , a n -2a n -1)| unpaired half-edges,

p 2 ≥ (d -1)|∂B * (z j , a n -2a n -1)| |D| dn 1 -n 1/2 log γ n n |∂B * (z j ,a n -2an-1)|-1
By (3.33) and (3.58), one checks easily that on E 1 ,

log γ-2κ-1 n ≤ |∂B * (z j , a n -2a n -1)| ≤ log γ n,
and that on S j (defined in (3.65)),

P ann (E j |S (x, y) ∩ E c n ) ≥ log (γ-2κ)(K 8 /2-1)-K 9 κ n ≥ log γ(K 8 /3-1) n
if γ/κ is large enough, which yields (3.70).

Part 1: proof of (3.71).

Since |A j,k,i | ≤ n 2/3 and tx(B Gn (A j,k,i , a n )) = tx(A j,k,i ) for all k, i ≥ 0, we can apply Proposition 3.4.1 as below (3.54) to bound the difference between ψ Gn on C z j and ϕ T d on an isomorphic subtree of T d , with the following coupling: ϕ T d (•) := ψ Gn (z j ), and then ϕ T d is defined as in Proposition 3.3.1 via (ξ j,k,i ) k,i≥0 . Recall that on S (x, y), we have that ψ Gn (z j ) ≥ h+(log -1 n)/2.

By Proposition 3.3.8, for any δ > 0 and for large enough n,

min a≥h+(log -1 n)/2 P T d a |Z h+(log -1 n)/2,+ a n -2an | ≥ (λ h -δ) a n -2an ≥ pP T d (E + ) 2
where P T d (E + ) > 0 (recall (3.23)) and

p := min a≥h+(log -1 n)/2 P T d a (∃v ∈ B T + d (•, 1), ϕ T d (z) ≥ h + 1) > 0.
Note in particular that for δ > log -1 n 2 such that λ h+δ > λ h -δ (such δ exists by Proposi-

tion 3.3.4, if n is large enough), Z h+δ ,+ a n -2an ⊆ Z h+(log -1 n)/2,+ a n -2an
. Proposition 3.4.1 yields then

P ann |∂C z j | ≥ (λ h -δ) a n -2an S (x, y) ∩ E c n ≥ pP(E + ) 2 + o(1) ≥ pP(E + ) 3 .
By cylindrical symmetry of B T + d (•, a n -2a n ), we even have

P ann v(0) ∈ ∂C z j S (x, y) ∩ E c n ≥ pP(E + ) 3 (λ h -δ) a n -2an |∂B T + d (•, a n -2a n )| ≥ pP(E + ) 3 λ h -δ d -1 a n -2an
.

Since K 8 = log d-1 ((1 + λ h )/2), taking δ small enough yields (3.71).

Part 2: proof of (3.72).

Denote v j (1), . . . , v j (a n -1) the vertices from v j (0) to v j (a n ) on the path P j . Remark that it suffices to prove that there exists a constant K 9 > 0 such that for n large enough, for every k ∈ {1, . . . , a n },

P ann ψ Gn (v j (k)) ≥ h| (S (x, y) ∩ E c n ∩ {ψ Gn (v j (k -1)) ≥ h}) ≥ (d -1) -K 9 . (3.73)
In the notation of (3.68),

v j (k) = y j,k+a n -2an,1 for 1 ≤ k ≤ a n . Write A k := A j,k+a n -2an,1 .
Suppose that for n large enough and all k ∈ {1, . . . , a n }, on S (x, y) ∩ E c n ∩ {ψ Gn (v j (k -1)) ≥ h}:

E Gn [ψ Gn (v j (k))|σ(A k )] > -|h| -1 (3.74)
and

Var Gn (ψ Gn (v j (k))|σ(A k )) > 1 d -1 . (3.75)
Then (3.73) holds with

K 9 := -log d-1 P(Y ≥ (|h| + |h|+1 d-1 )/ √ d -1 ),
where Y ∼ N (0, 1). Thus, it is enough to establish (3.74) and (3.75).

For k ≥ 1, note that by construction of B * (z j , a n ), v j (k -1) and v j (a n ) are the only vertices of ∂A k at distance less than a n of v j (k). Let (X s ) s≥0 be a discrete time SRW started at v j (k), and τ := inf{s ≥ 0, d Gn (v j (k), X s ) ≥ a n }. Write H for the hitting time of A k by (X s ). Letting

a 1 := P Gn v j (k) (X H = v j (k -1), H < τ ) and a 2 := P Gn v j (k) (X H = v j (a n ), H < τ ),
we get as in the proof of Proposition 3.4.1 that for large enough n, for every realization of G n and ψ

Gn (A k ) in S (x, y) ∩ E c n ∩ {ψ Gn (v j (k -1)) ≥ h}: E Gn [ψ Gn (v j (k))|σ(A k )] > a 1 ψ Gn (v j (k -1)) + a 2 ψ Gn (v j (a n )) -log -1 n. Since 0 ≤ a 1 + a 2 ≤ 1 and min(ψ Gn (v j (k -1)), ψ Gn (v j (a n ))) ≥ h ≥ -|h|, (3.74) follows.
Split V := Var Gn (ψ Gn (v j (k))|σ(A k )) as follows:

V = G Gn (v j (k), v j (k)) -E Gn v j (k) G Gn (v j (k), X H ) 1 {H<τ } -E Gn v j (k) G Gn (v j (k), X H ) 1 {H≥τ } + E Gn v j (k) [H] E Gn πn [H] E Gn πn [G Gn (v j (k), X H )] .
By (3.11), (3.8) and (3.9), if κ is large enough, for n large enough,

G Gn (v j (k), v j (k))-a 1 G Gn (v j (k), v j (k -1))-a 2 G Gn (v j (k), v j (a n ))> d -1 d -2 - a 1 + a 2 d -2 -log -1 n ≥ 1 d -2 -log -1 n.
As below (3.45), we get that

E Gn v j (k) G Gn (v j (k), X H ) 1 {H<τ } -a 1 G Gn (v j (k), v j (k -1)) -a 2 G Gn (v j (k), v j (a n )) ≤log -1 n
and that

E Gn v j (k) G Gn (v j (k), X H ) 1 {H≥τ } - E Gn v j (k) [H] E Gn πn [H] E Gn πn [G Gn (v j (k), X H )] ≤ log -1 n.
These three inequalities imply that for large enough n, for every realization of G n and ψ Gn (A k ) in S (x, y) ∩ E c n ∩ {ψ Gn (v j (k -1)) ≥ h}:

Var Gn (ψ Gn (v j (k))|σ(A k )) ≥ 1 d-2 -3 log -1 n > 1 d-1 .
This shows (3.75) and the proof is complete.

Average number of connections in E ≥h ψ Gn

Write x h ↔ y if x and y are in the same connected component of E ≥h ψ Gn , for x, y ∈ V n . In this section, we prove (3.2) of Theorem 3.1.1 via an argument on the number of pairs of vertices such that x h ↔ y. Let S n be the set of pairs of distinct x, y ∈ V n such that x h ↔ y. Let A n ⊆ V n be the set of vertices x such that |C Gn,h

x | ≤ n 1/2 .
We first suppose that the following two Lemmas hold, and show (3.2). Then, we derive them from Propositions 3.5.4 and 3.6.2, using a second moment argument.

Lemma 3.6.3. For every ε > 0, lim n→+∞ P ann

(|A n | ≥ (1 -η(h) -ε)n) = 1. Lemma 3.6.4. For every ε > 0, lim n→+∞ P ann (|S n | ≥ (η(h) 2 /2 -ε)n 2 ) = 1. Proof of (3.2). Fix ε > 0. Remark that every connected component of E ≥h ψ Gn is either included in A n , or does not intersect A n . Let ( γ (n) i ) i≥1 (resp. (γ (n) i ) i≥1
) be the sizes of the connected components in A n (resp. not in A n ), listed in decreasing order of size.

Let E n := {|A n | ≥ (1 -η(h) -ε)n} ∩ {|S n | ≥ (η(h) 2 /2 -ε)n 2 }. On E n , we have that i≥1 γ (n) i ( γ (n) i -1) + i≥1 γ (n) i (γ (n) i -1) = 2S n ≥ η(h) 2 n 2 -2εn 2 .

Moreover, we have by definition of

A n : i≥1 γ (n) i ( γ (n) i -1) ≤ i≥1 γ (n) i √ n ≤ n 7/4 .
Thus, for n large enough,

γ (n) 1 (|V n | -|A n |) ≥ i≥1 γ (n) i (γ (n) i -1) ≥ (η(h) 2 -3ε)n 2 . But |V n | -|A n | ≤ (η(h) + ε)n, so that γ (n) 1 ≥ η(h) 2 -3ε η(h) + ε n ≥ ((η(h) -4η(h) -1 ε)n. Since γ (n) 1
is the cardinality of a set included in V n \ A n , one has γ

(n) 1 ≤ (η(h) + ε)n. Note that for n large enough, γ (n) 1 > √ n ≥ γ (n) 1 . Therefore, γ (n) 1 = |C (n)
1 |, and we have

((η(h) -4η(h) -1 ε)n ≤ |C (n) 1 | ≤ (η(h) + ε)n
on E n . By Lemmas 3.6.3 and 3.6.4, lim n→+∞ P ann (E n ) = 1. Since ε was arbitrary, the proof is complete.

Remark 3.6.5. Note that we have

|C (n) 2 | = max(γ (n) 2 , γ (n) 1 ). Since on E n , γ (n) 1 ≤ √ n and γ (n) 2 ≤ |V n | -|A n | -γ (n) 1 ≤ (1 + 4η(h) -1 )εn, we get that |C (n) 2 |/n Pann -→ 0.
Proof of Lemma 3.6.3. Let A n be the set of vertices such that their lower exploration (Section 3.5.2) is aborted. By Remark 3.5.2, for n large enough, A n ⊆ A n , and it is enough to prove the Lemma for A n instead of A n .

Let ε ∈ (0, 1). By Proposition 3.5.4, for n large enough and every x ∈ V n , we have

|P ann (x ∈ A n ) -(1 -η(h))| ≤ ε. (3.76)
We claim that for n large enough, for all distinct x, y ∈ V n ,

|Cov ann (1 x∈A n , 1 y∈A n )| ≤ 4ε. (3.77) Indeed, Cov ann (1 x∈A n , 1 y∈A n ) = P ann (x, y ∈ A n ) -P ann (x ∈ A n )P ann (y ∈ A n ).
Then by (3.76),

we have

|P ann (x ∈ A n )P ann (y ∈ A n ) -(1 -η(h)) 2 | ≤ 2ε + ε 2 ≤ 3ε.
Perform successively the lower explorations from x and then from y as in Section 3.5.2 (with the additional condition C5). We get P ann (C5 happens) = o(1) in the same way than (3.53).

Then, revealing ψ Gn on R x ∪ R y and comparing it to ϕ T d as below (3.54), we obtain

|P ann (x, y ∈ A n ) -(1 -η(h)) 2 | ≤ ε.
This shows (3.77).

We now apply Bienaymé-Chebyshev's inequality:

P ann (|A n | ≤ (1 -η(h) -2ε 1/4 )n) ≤ P ann (| |A n | -E ann [|A n |] | ≥ ε 1/4 n) ≤ 1 √ εn 2   x,y∈Vn Cov ann (1 x∈A n , 1 y∈A n )   ≤ n + n(n -1)4ε √ εn 2 ≤ 5 √ ε
for n large enough. Since ε can be taken arbitrarily small, the proof is complete.

Proof of Lemma 3.6.4. Let ε ∈ (0, 1). Denote S * n the set of pairs x, y ∈ V n such that S * (x, y) holds. Since S * n ⊆ S n , it is enough to prove the Lemma for S * n instead of S n . First, by Proposition 3.6.2,

E ann [|S * n |] ≥ (η(h) 2 -ε) n(n-1) 2 ≥ (η(h) 2 /2 -ε)n 2
for large enough n. Second, we claim that for n large enough and all distinct x, y, w, t ∈ V n ,

|Cov ann (1 S * (x,y) , 1 S * (w,t) )| ≤ 2ε. (3.78)
Remark that if this holds, then we can conclude by a second moment computation as in the proof of Lemma 3.6.3. We have union bound, it is enough to prove that for x ∈ V n , P ann ({|C Gn,h

x | = O(log n)} ∪ {|C Gn,h x | = Θ(n)}) = o(1/n). (3.81)
Let us sketch this exploration in the lines below.

First phase (Section 3.7.2). We explore the connected component C of x ∈ V n in the set We replace a n of (3.33) by a "security radius" r n = Θ(log n). Adapting Proposition 3.4.1 (Lemma 3.7.7), this allows us in Section 3.7.5 to bound the difference between ψ Gn and ψ Gn by n -a , so that for every connected component

{y ∈ V n , ψ Gn (y) ≥ h -n -a }
C of C \ M, either C ⊆ C Gn,h x or C ∩ C Gn,h x = ∅.
If we kept a n , ψ Gn would approximate ψ Gn only with precision log -Θ(1) n. With probability Θ(1/n), there would be too much vertices y such that | ψ Gn (y) -h| ≤ log -Θ(1) n, hence for which we cannot know by anticipation whether they will be in C Gn,h

x or not.

Moreover, we do not have P(C1 happens) = O(1/n) as soon as the number of vertices explored goes to infinity with n. We will need to accept the possible occurrence of one cycle. When this happens, we have to define ψ Gn in a slightly different manner. In Section 3.7.5, we need a variant of Lemma 3.7.7 to control the difference between ψ Gn and ψ Gn (Lemma 3.7.8).

Second phase (Section 3.7.3). If (i) happens for every component C , the exploration is over. For each C such that (ii) happens, we explore its K log n remaining vertices, this time in a fashion similar to Section 3.5.1. Each of these explorations has a probability bounded away from 0 to be successful. If K is large enough, with probability at least 1 -o(1/n), at least one of these explorations is successful, and has a boundary of size Θ(n 1/2 b n ).

Third phase(Section 3.7.4). For every C such that (ii) happens, we show that the successful exploration of the second phase is connected to a positive proportion of the vertices of V n , via an adaptation of the joint exploration in Section 3.6.1. This yields (3.81).

Lower bound

In this section, we prove the existence of K 0 > 0 such that

P ann (|C (n) 2 | ≥ K -1 0 log n) → 1. (3.82)
To show the existence of x ∈ V n such that the size of its connected component is exactly of order log n requires an exploration in which we compare C Gn,h

x to both C h+log -1 n • and C h-log -1 n • .
Then, the proof strategy simply consists in performing explorations from different vertices of V n , one after another, until one of them is successful. Proposition 3.3.6 hints that the probability that C h • has size c log n should be of order n -f (c) , where f is an unknown function of c > 0, with f (c) -→ c→0 0. Hence, for K -1 0 = c small enough, performing at most n 1/10 explorations will be enough.

The exploration. Let K > 0. For x ∈ V n , we modify the lower exploration of Section 3.5.2, replacing C2 by |B

C h-log -1 n • (•, k -1)| ≥ 2K log n.
If at some step k, the lower exploration is stopped exclusively because of C3, and if

|B C h+log -1 n • (•, k -1)| ≥ K log n, say that it is success- ful.
Pick x 0 ∈ V n . Denote S 0 the set of vertices seen during its lower exploration (i.e. vertices having at least one half-edge paired during the exploration). For 0 ≤ i ≤ n 1/10 , if the lower exploration from x i is not successful, let S i be the set of vertices seen in the lower explorations of x 0 , . . . , x i . Pick x i+1 ∈ V n \ S i and perform its lower exploration, stopping it if a vertex of S i is seen, in which case it is not successful. Let E i,n := {the lower exploration from x i is successful} and

E n := ∪ n 1/10 i=0 E i,n .
Suppose that the following result holds:

Lemma 3.7.1. If K is small enough, then lim n→+∞ P ann (E n ) = 1.
On E n , let i 0 ≥ 1 be such that the lower exploration from x i 0 is successful. Applying Proposition 3.4.1 as below (3.54), we get that

P ann (C h+log -1 n • ⊆ Φ(C Gn,h x i 0 ) ⊆ C h-log -1 n • ),
where Φ is an isomorphism from C Gn,h x i 0 to T d . In this case, K log n ≤ |C Gn

x i 0 | ≤ 2K log n. With high P ann -probability, |C (n) 1 | > 2K log n by (3.
2), so that C Gn,h

x i 0 = C (n)
1 . This yields (3.82) with K = K -1 0 . It remains to establish the Lemma.

Proof of Lemma 3.7.1. Clearly, it is enough to show that for K small enough, for n large enough and every 1 ≤ i ≤ n 1/10 ,

P ann (E i,n | ∩ i-1 j=0 E c j,n ) ≥ n -1/11 , (3.83)
since it would imply that

P ann (E n ) ≥ 1 -(1 -n -1/11 ) n 1/10 -→ n→+∞ 1.
For n ≥ 1 and 1 ≤ i ≤ n 1/10 , we have

E i,n ⊇ E line,n \ ({C1 happens} ∪ E i,n,meet ),
where

E i,n,meet :={the lower exploration from x i meets S i-1 },E line,n :={C h+log -1 n • =C h-log -1 n • =L K log n }
and for any k ≥ 1, L k is a "line" subtree of T d , i.e. it is rooted at •, has k + 1 vertices and total height k.

During each exploration, less than n 1/10 vertices and half-edges are seen for n large enough.

Thus |S i | ≤ n 1/5 . Hence, for i ≥ 1, by (3.14) with k = 1, m ≤ n 1/10 , m 0 = 1 and m E , m 1 ≤ n 1/5 , for n large enough:

Since we suppose that no cycle arises, B Gn (y 1 , y 1 , ) is a tree, that we root at y 1 . If y 1 = y = x, replace B Gn (y 1 , y 1 , r n + ) and B Gn (y 1 , y 1 , ) by B Gn (x, r n + ) and B Gn (x, ) respectively.

We construct T y (y 1 ), the subtree of B Gn (y 1 , y 1 , ) in {z, ψ Gn (z) ≥ h + n -a }. We start with T y (y 1 ) = {y 1 }.

For k = 1, 2, ... -1 successively, denote y k,1 , . . . , y k,m the children of the (k -1)-th generation of T y (y 1 ). Let (ξ y 1 ,k,i ) k,i≥0 be an array of i.i.d. variables of law N (0, 1), independent of everything else. Set

ψ Gn (y k,i ) := 1 d -1 ψ Gn (y k,i ) + d d -1 ξ y 1 ,k,i . (3.85) Add y k,i to T y (y 1 ) if ψ Gn (y k,i ) ≥ h + n -a
, and give a mark to y k,i (and thus add it to M) if

h -n -a ≤ ψ Gn (y k,i ) < h + n -a .
Finally, include T y (y 1 ) in T y , add the vertices of ∂T y (y 1 ) to N y and take y 1 away from ∂T y . The step is then over.

If |∂T y | ∈ [1, K log n], the subexploration is finished. Say that it is fertile if |∂T y | > K log n,
and infertile else (hence if ∂T y = ∅).

If M = ∅, the exploration from y is finished.

II -Suppose now that a unique cycle C arises in the subexploration of y, when revealing the pairings of B Gn (y 1 , y 1 , r n + ) in the step from y 1 , for some y, y 1 ∈ V n . Let m := |C| be the number of vertices in C. There are two cases.

Case 1: When C is discovered, there are already k consecutive vertices y 1 , . . . , y k of C where ψ Gn has been defined, for some 1 ≤ k ≤ m -1. Reveal B Gn (C, r n ). Denote z 1 , . . . , z m-k the remaining vertices of C, such that z 1 = y 2 is a neighbour of y 1 , and z i is a neighbour of z i-1 for i ≥ 2. Give a mark to z 1 , . . . , z m-k and y 1 . Take y 1 away from ∂T y . If y k was in ∂T y for some y whose subexploration was performed previously, take it away from that set. Let u k , . . . , u 1 , v 1 , . . . , v m-k be the vertices of C m , listed consecutively. Let U :=G m \{u 1 , . . . , u k }, (X j ) j≥0 a SRW on G m and recall that T U is the exit time of U . Define

α := P Gm v 1 (X T U = u 1 , T U <+∞), β := P Gm v 1 (X T U = u k , T U <+∞)1 {k>1} and γ := E Gm v 1 [ T U -1 j=0 1 {X j =v 1 } ],
and let (ξ i ) i≥1 be a family of i.i.d standard normal variables, independent of everything else.

Define

ψ Gn (z 1 ) := α ψ Gn (y 1 ) + β ψ Gn (y k ) + √ γξ 1 .
Then for i ≥ 2, define recursively

ψ Gn (z i ) := α m-(k+i-1) ψ Gn (z i-1 ) + β m-(k+i-1) ψ Gn (y k ) + γ m-(k+i-1) ξ i (3.86)
where we set

U i := G m \ {u 1 , . . . , u k , v 1 , . . . , v i-1 }, α m-(k+i-1) := P Gm v i (X T U i = v i-1 , T U i < +∞), β m-(k+i-1) := P Gm v i (X T U i = u k , T U i < +∞) and γ m-(k+i-1) := E Gm v i [ T U i -1 j=0 1 {X j =v i } ].
Case 2: ψ Gn has not been defined on any vertex of C. There exists a unique path of consecutive vertices y 1 , . . . , y j for some j ≥ 2 such that y j ∈ C, and for 2 ≤ i ≤ j -1, ψ Gn (y i ) has not been defined and y i ∈ C. Reveal B Gn ({y 1 . . . , y j-1 } ∪ C, r n ). Give a mark to the vertices of We define ψ Gn on {y 1 , . . . , y j-1 } ∪ C in a way similar to Case 1. Let (ξ i ) i≥1 be a sequence of i.i.d. standard normal variables, independent of everything else. For i = 1, 2, . . . , j, set

ψ Gn (y i ) := α j-i ψ Gn (y i-1 ) + γ j-i ξ i , (3.87)
where α j-i := P Gm z (H {z } < +∞) and γ j-i := E Gm z [

H {z } -1 l=0 1 {X l =z} ], z, z being two neigh- bours in G m such that z (resp. z ) is at distance j -i (resp. j -i + 1) of C m .
Then, define ψ Gn on C as in Case 1 with k = 1.

After the discovery of C. Resume the subexplorations as before, starting by the subexploration interrupted when C was discovered. For any marked vertex y: when proceeding to the step from y 1 in the subexploration from y, do not reveal the edges of B Gn (y 1 , y 1 , r n + ) in the direction of any already marked neighbour of y 1 .

III -If a second cycle arises, the exploration from x is over, and is not successful.

If at some point, D1 M = ∅, D2 ψ Gn has been defined on at most K log n vertices, and D3 at most one cycle has been discovered, say that the first phase of the exploration from x is successful. Denote S 1 (x) this event. If all subexploration trees were infertile, then the exploration from x is over, and said to be successful.

Denote S 1,stop (x) this event.

Proposition 3.7.2. Fix K, a > 0. If K is large enough, then for large enough n and every

x ∈ V n , P ann (S 1 (x)) ≥ 1 -n -5/4 .
Proof. In a nutshell, the argument is as follows: by our choice of and Lemma 3.7.3, the same reasoning as that in the beginning of Proposition 3.3.6 shows that the size of a subexploration tree has a bounded expectation (as n → +∞) with exponential moments. Hence if K > 0 is large enough, with probability ≥ 1-n -2 , the first K log n subexplorations encompass less than K 2 log n vertices. The total number of vertices seen is at most K 2 log n×(d-1) 2rn ≤ n 3/10 . In that case, Lemma 3.2.2 entails that we see at most one cycle, with probability at least 1 -n -4/3 . This allows to control |M|, since a cycle brings less than 2d r n = Θ(log n) marked vertices.

Without cycles, each vertex where ψ Gn is defined has a chance O(n -a ) to get a mark. Hence for K large enough, with probability at least 1 -n -5/4 , less than K log n vertices get a mark in the first K log n explorations, ensuring that the exploration from x is finished -and successful.

Let E 1,n be the event that two cycles are discovered before n 3/10 vertices have been seen during the exploration from x. By (3.14) with k = 2, m 0 = 1, m E = 0 and m ≤ n 3/10 ,

P ann (E 1,n ) ≤ n -4/3 . (3.88)
Suppose that we perform the subexploration from some vertex y. Let y 1 ∈ ∂T y , with y 1 = y. If no cycle arises when revealing the (r n + )-offspring of y 1 , then |∂T y (y 1 )| -1, the increment of |∂T y | during that step, dominates stochastically ρ ,h,δ -1, by definition of ρ ,h,δ at Lemma 3.7.3, as soon as n -a < δ. If a cycle arises (which happens at most once on E c 1,n ), at most two vertices are marked and taken away from ∂T y . After j steps (if the subexploration is not over),

|∂T y | ≥ st.
S j-1 -2, where S j-1 is the sum of j -1 i.i.d. random variables of law ρ ,h,δ -1, hence taking values in the bounded interval [0, d ], and with a positive expectation by Lemma 3.7.3 (the -1 in j -1 comes from the fact that we do not include the step from y, and the +2 from the possibility that two vertices can be taken away from ∂T y if there is a cycle). Hence P ann (the subexploration from y lasts more than j steps) ≤ P(S j-1 ≤ K log n + 2) By the exponential Markov inequality, there exist constants c, c > 0 such that for every j ≥ 2K(E[ρ ,h,δ ] -1) -1 log n, P(S j-1 ≤ K log n + 2) ≤ ce -c j . Now, let K > 0 and focus on the first K log n subexplorations (or on all subexplorations if there are less than K log n of them). Let N be the total number of steps during those subexplorations. On E c 1,n , N is stochastically dominated by a sum S of K log n i.i.d. variables of some law µ (independent of n) such that

for every j ≥ 2K(E[ρ ,h,δ ] -1) -1 log n, µ([j, +∞)) ≤ ce -c j .
Hence, letting E 2,n := {N ≥ K 2 log n}, by the exponential Markov inequality,

P ann (E 2,n ∩ E c 1,n ) ≤ P ann (S ≥ K K log n ) ≤ (E[e c Y /2 ]e -c K µ/2 ) K log n
for large enough n and for Y ∼ µ. Taking K large enough, we have for n large enough:

P ann (E 2,n ∩ E c 1,n ) ≤ n -2 . (3.89)
In a step of a subexploration, less than (d -1) 2rn ≤ n 1/10 new vertices are seen. When a cycle C is revealed, there are at most 2r n new vertices in C (and on the path leading to C, in Case 2), so that less than 3r n (d -1) rn ≤ n 1/10 new vertices are seen. Therefore, on E c 1,n ∩ E c 2,n , less than n 3/10 vertices are seen during the first K log n subexplorations.

Moreover, in a step of a subexploration, less than d vertices are added to the subexploration tree. Hence if K > K 2 d , on E c 2,n , D2 holds. We now estimate the total number of vertices that will receive a mark. When a cycle C appears, our construction implies that less than 3r n vertices receive a mark. When performing m steps in a subexploration, the number of marked vertices obtained is stochastically dominated by a binomial random variable Bin(m, d n -a ). Indeed, at each step, we reveal ψ Gn (and thus ϕ

P T d (ϕ T d (y) ∈ [h -n -a , h + n -a ] | ϕ T d (y) = a ) ≤ 2n -a 2πd/(d -1) ≤ n -a .
Hence, if E 3,n is the event that more than 3r n + K 2 log n marks are given during the first K log n subexplorations and if Z ∼ Bin( K 2 log n , C 2 d n -a ),

P ann (E c 1,n ∩ E c 2,n ∩ E 3,n ) ≤ P Z ≥ K 2 log n ≤ K 2 log n K 2 log n (C 2 d n -a ) K 2 log n ,
thus by(3.12), for large enough n,

P ann (E c 1,n ∩ E c 2,n ∩ E 3,n ) ≤ K 2 log n C 2 d n -a K 2 log n ≤ n -2 . (3.90) Taking K > 1, on E c 1,n ∩ E c 2,n ∩ E c 3
,n , less than 1 + 3r n + K 2 log n ≤ K log n vertices receive a mark during the first K log n subexplorations, so that M = ∅ after at most K log n subexplorations. Therefore, on E c 1,n ∩ E c 2,n ∩ E c 3,n , conditions D1, D2 and D3 hold. By (3.88), (3.89) and (3.90), for large enough n

P ann (E c 1,n ∩ E c 2,n ∩ E c 3,n ) ≥ 1 -2n -2 -n -4/3 ≥ 1 -n -5/4 ,
and this concludes the proof.

We state here the variant of Lemma 3.3.9. Let δ ∈ [0, h -h). For ≥ 1, write Z for the -th

generation of the connected component of • in ({•} ∪ E ≥h+δ ϕ T d ) ∩ T + d . Let ρ ,h,δ be the law of |Z | conditionally on ϕ T d (•) = h, so that ρ ,h = ρ ,h,0 (recall (3.28)).
The following result is a straightforward consequence of Lemma 3.3.9 (applied to h + δ instead of h) and from the fact

that P T d h (∃v ∈ ∂B(•, 1), ϕ T d (v) ≥ h + δ) > 0. Lemma 3.7.3. For every δ ∈ [0, h -h), if is large enough, E[ρ ,h,δ ] > 1.

Second phase

If S 1,stop (x) holds, i.e. all subexploration trees are infertile, the exploration is over. In this Section, we suppose that S 1 (x)\S 1,stop (x) holds. For every fertile tree, we perform an exploration similar to that of Section 3.5.1 from each vertex of its boundary, and show that with probability at least 1-n -6/5 , at least one exploration per fertile tree is successful, and hence has a boundary of size Θ(n 1/2 b n ) (Proposition 3.7.4). We illustrate this second phase in Figure 6 below.

Let T 1 , . . . T m be the fertile subexploration trees for some positive integer m. For every q ∈ {1, . . . , m}, denote y q,1 , y q,2 , . . . the vertices of ∂T q . For q = 1, 2, . . . successively, we perform the explorations from y q,i , i ≥ 1 as defined in Section 3.5.1 (using an array of independent standard normal variables (ξ y q,i ,k,j ) k,j≥0 ). If y is the j-th vertex of the k-th generation in the exploration tree of y q,i , let recursively

ψ Gn (y) := ψ Gn (y) d -1 + d d -1 ξ y q,i ,k,j ,
so that ψ Gn plays the role of ϕ T d on T d in Section 3.5.1. We implement two modifications:

• we do not explore towards y q,i , the parent of y q,i in T q (hence we identify ψ Gn to the GFF on a subtree of T + d instead of T d ), and

• we do not stop the exploration if ψ Gn (y q,i ) < h + log -1 n (we only know a priori from the first phase that ψ Gn (y q,i ) ≥ h + n -a ), and

• we stop the exploration if it meets a vertex already discovered in the first phase or during the previous exploration of some y q ,j (thus with q < q, or q = q and j < i).

A vertex y q,i whose exploration is successful is back-spoiled if one vertex of its exploration is seen later during the exploration of y q ,j . Let E 4,n := {∃q ≤ m, all successful explorations of y q,1 , y q,2 , . . . , are back-spoiled} On S 2 (x) := (S 1 (x) \ S 1,stop (x)) ∩ E c 4,n , say that the second phase is successful.

Proposition 3.7.4. If K of Proposition 3.7.2 is large enough, then for n large enough and every x ∈ V n ,

P ann (S 2 (x) ∪ S 1,stop (x)) ≥ 1 -n -6/5 (3.91)
Proof. Say that y q,i is spoiled if it is met during the previous exploration of some y q ,j . Define E 5,n := {at least 1000 log n vertices are spoiled or back-spoiled}. We claim that for n large enough,

P ann (E 5,n ) ≤ n -2 , (3.92)
and for some constant K 10 > 0 (that only depends on d and h), for each of these vertices y q,i , P ann (the exploration from y q,i is successful | y q,i is not spoiled) ≥ K 10 .

On E c 5,n , if (3.93) holds, there are at least (K -1000) log n non-spoiled vertices on each ∂T q (and at most (K + 1) log n since each step of a subexploration brings less than log n vertices to ∂T q ), where K was defined in the beginning of Section 3.7.2. And, if more than 1000 log n vertices of each ∂T q are successful, one of them will be successful and not back-spoiled, thus fulfilling the requirement of S 2 (x).

If K is large enough, for n large enough, the probability that no more than 1000 log n explorations from ∂T q are successful is at most

(K+1) log n 1000 log n (1 -K 10 ) (K-1000) log n ≤ n (K+1) K+1 1000 1000 (K-999) K-999 (1 -K 10 ) K-1000 log n = o(n -3 )
by Stirling's formula. Hence by a union bound on 1 ≤ q ≤ m, noticing that m ≤ 2K /K for n large enough by D2, we have:

P ann ((S 2 (x) ∪ S 1,stop (x)) c ) ≤ P ann (S 1 (x) c ) + P ann (E c 5,n ) + mn -3 ≤ 2n -5/4
by Proposition 3. areas correspond to the explorations of the y q,i 's in the second phase. y 2,2 is spoiled by y 1,2 . y 3,2 is back-spoiled by y 3,3 . Each of the subxploration trees will be either included in C Gn,h x or have no common vertex with C Gn,h x , depending on the value of ψ Gn on the marked vertices.

Proof of (3.92). Note that by D2, by Remark 3.5.2 and by (3.84), less than n 1/2 log -1 n vertices and half-edges have been seen in the first two phases. (3.94) In particular, less than n 1/2 log -1 n edges are built during second phase. Since there are at most K log n y q,i 's by D2, each new edge has a probability at most

K log n/(n -n 1/2 log -1 n) ≤ 2K n -1 log n
to spoil a vertex. Thus, the number of spoiled vertices is stochastically dominated by a random variable Z ∼ Bin(n 1/2 log -1 n, 2K n -1 log n). For n large enough,

P(Z ≥ 10) ≤ n 1/2 log -1 n 10 2K log n n 10 ≤ n 5 log 20 n n 10 ≤ n -3 .
Moreover, by (3.15) with k = 999 log n and m 0 , m 1 , m E , m ≤ n 1/2 log -1 n due to (3.94), P ann (more than 999 log n vertices are back-spoiled) ≤ n -3 .

(3.92) follows.

Proof of (3.93). By (3.94) and (3.14) with k = 1 and m 0 , m 1 , m E , m ≤ n 1/2 log -1 n, for n large enough, P ann (a cycle is created during the exploration from y q,i ) ≤ log -1 n.

The law of ψ Gn on the exploration tree from y q,i is that of ϕ T d on an isomorphical subtree of T + d (and not T d , since we do not explore towards y q,i ), with ϕ

T d (•) = ψ Gn (y q,i ). Denote C n • the connected component of • in E ≥h+log -1 n,+ ϕ T d ∪ {•},
and Z k its k-th generation for every k ≥ 0. Then P ann (the exploration from y q,i is successful|y q,i is not spoiled) ≥ p n -log -1 n,

where p n := min b≥h+n -a P T d b (∃k ≤ log λ h n, |Z k | ≥ n 1/2 b n ) (recall that ψ Gn (y q,i ) ≥ h + n -a ). Let δ ∈ (0, h -h).
Clearly, there exists p > 0 such that for n large enough,

min b≥h+n -a P T d b (∃v ∈ Z 1 , ϕ T d (v) ≥ h + δ) > p .
For ε > 0 small enough so that log

d-1 (λ h+δ -ε) ≥ (3 log d-1 λ h )/4 (such ε exists by continuity of h → λ h , Proposition 3.3.4), for n ∈ N, p n := min b≥h+δ P T d b (∃k ≤ log λ h n -1, |Z h+log -1 n,+ k | ≥ n 1/2 b n ) ≥ min b≥h+δ P T d b (|Z h+δ,+ log λ h n -1 | ≥ n 1/2 b n ) ≥ min b≥h+δ P T d b (|Z h+δ,+ log λ h n -1 | ≥ (λ h+δ -ε) log λ h n -1 ).
By Proposition 3.3.8, lim inf n→+∞ p n =: p > 0. Since p n ≥ p p n for all n ≥ 1, P ann (the exploration from y q,i is successful|y q,i is not spoiled) ≥ p n -log -1 n ≥ p p 2 for n large enough, and we can take K 10 = p p 2 . This shows (3.93).

Proposition 3.7.5. For n large enough, we have P ann (S 2 (x)∩{∃(J, q), there are less than log γ-3κ-18 n J-joining balls from B q }) ≤ n -4 (3.96)

and if K 11 is large enough, then for large enough n:

P ann (S 2 (x) ∩ {s 0 ≤ log n}) ≤ n -4 . (3.97)
Proof. Proof of (3.96). We adapt the proof of Lemma 3.

P ann (S 2 (x) ∩ { |B * ∩ (∪ s 0 j=1 B Gn (T z i j , a n ))| ≥ log 3γ n}) ≤ n -5 (3.99)
Let N be the number of vertices of ∪ m q=1 B q that are spoiled, i.e. the vertices z such that

B * (z, a n ) = B Gn (z, z, a n ) is hit by a previously constructed B * (z , a n ). (3.64) becomes P ann (S 2 (x) ∩ {N ≥ log 3γ n}) ≤ n -5 . (3.100)
In addition, by (3.15)

with k = log 2 n, m 0 = 1, m 1 , m E , m ≤ n 1/2 log -1/2 n, P ann (E 6,n ) ≤ n -5 , (3.101)
where E 6,n := S 2 (x) ∩ more than log 2 n vertices are w-back-spoiled . Then, define

S i,q := S 2 (x) ∩ E c 6,n ∩ {|(∪ s 0 j=1 B Gn (T w i j , a n )) ∩ B * i,q | ≤ log 3γ n} ∩ {z i,q is not spoiled}.
It is straightforward to adapt the proof of (3.66) to get that for every 1 ≤ J ≤ s 0 , the probability that B * (z i,q , a n ) is a J-joining ball, conditionally on S i,q , is at least n -1/2 log γ-2κ-10 n. On

S 2 (x) ∩ {N ≤ log 3γ n} ∩ E c 6,n , at least n 1/2 b n -log 2 n -log 3γ n ≥ n 1/2 log -κ-7 n
z i,q 's are neither spoiled nor w-back-spoiled by (3.46). As in the end of the proof of Lemma 3.6.1, if γ > 3κ + 18, we get that for n large enough: for every (J, q) ∈ ({1, . . . , s 0 } ∩ {1, . . . , m}) and

Z ∼ Bin(n 1/2 log -κ-7 n , n -1/2 log γ-2κ-10 n):
P ann (S 2 (x) ∩ {there are less than log γ-3κ-18 n J-joining balls from B q }) and for every y, we let A y := {z ∈ V n , ψ Gn (z) was defined before ψ Gn (y)} and

≤ P ann S 2 (x) ∩ ({ |B * ∩ (∪ s 0 j=1 B Gn (T z i j , a n ))| ≥ log 3γ n} ∪ {N ≥ log 3γ n} ∪ E 6,n ) + P(Z ≤ log γ-3κ-18 n) ≤ 3n -5 + n max
ψ Gn (y) := E Gn [ψ Gn (y)|σ(A y )] + ξ y Var(ψ Gn (y)|σ(A y )),
for every y ∈ R ψ 1 , where ξ y is the normal variable used when defining ψ Gn (y),

• revealing ψ Gn on R ψ 3 , and finally on

V n \ (R ψ 1 ∪ R ψ 2 ∪ R ψ 3 ). Let E 7,n := {G n is a good graph} ∩ {max z∈Vn |ψ Gn (z)| ≤ log 2/3 n}, S ψ 1 (x) := S 1 (x) ∩ {sup y∈R ψ 1 |ψ Gn (y) -ψ Gn (y)| ≤ n -a /2}, S ψ 2 (x) := S 2 (x) ∩ S ψ 1 (x) ∩ {sup y∈R ψ 2 |ψ Gn (y) -ψ Gn (y)| ≤ (log -1 n)/2}, and
S ψ 3,i (x) := S ψ 2 (x) ∩ {∀q ∈ {1, . . . , m} at least log n vertices of D i are connected to T q in E ≥h ψ Gn } for every i ≥ 1.
Suppose that for a > 0 (defined in the beginning of Section 3.7.2) small enough, and for n large enough:

P ann ((S 1 (x) \ S ψ 1 (x)) ∩ E 7,n ) ≤ n -3 , (3.105)
P ann ((S 2 (x) \ S ψ 2 (x)) ∩ E 7,n ) ≤ n -3 , (3.106)
and for every 1 ≤ i ≤ r -1,

P ann ((S 2 (x) \ S ψ 3,i (x)) ∩ E 7,n ) ≤ n -3 . (3.107)
Letting S ψ 1,stop (x) := S 1,stop (x) ∩ S ψ 1 (x), (3.105), (3.106), 3.107) and (3.91) imply that

P ann ((S ψ 1,stop (x) ∪ (∩ r-1 i=1 S ψ 3,i (x)) ) ∩ E 7,n ) ≥ 1 -n -7/6 . (3.108)
On S ψ 1,stop (x) ∪ (∩ r-1 i=1 S ψ 3,i (x)), we have the following alternative:

• either C Gn,h
x contains a subexploration tree T q whose exploration was fertile, the exploration from y q is successful and connected to at least log n vertices of every

D i , 1 ≤ i ≤ r-1 in E ≥h ψ Gn ; • or C Gn,h
x contains no such tree, and C Gn,h

x ⊆ R ψ 1 , so that |C Gn,h

x | ≤ K 0 log n, where we take K 0 ≥ K , and where K is the constant of Proposition 3.7.2.

Note that the second case comprises S ψ 1,stop (x) but is a priori not included in it: there could exist fertile subexploration trees not connected to x in E ≥h ψ Gn if ψ Gn is below h on the appropriate marked vertices.

In the first case, C Gn,h x contains at least log n vertices of each D i , 1 ≤ i ≤ r -1, so that by (3.94) and (3.95) for n large enough:

|C Gn,h x | ≥ (r -1) log n ≥ log n n -|R 1,2 | -|D r | K 11 log n ≥ n 2K 11 .
Letting thus K 12 := (2K 11 ) -1 , for every n large enough, we have on S ψ 1,stop (x) ∪ (∩ r-1 i=1 S ψ 3,i (x)):

|C Gn,h x | ≤ K 0 log n or |C Gn,h x | ≥ K 12 n.
By (3.108) and a union bound on all x ∈ V n ,

P ann (K 0 log n ≤ |C (n) 2 | ≤ K 12 n) ≤ n -1/6 + P ann (E c 7,n ).
By Remark 3.6.5, |C

2 | ≤ K 12 n P ann -w.h.p. By Proposition 3.2.1 and Lemma 3.2.5, we have P ann (E c 7,n ) → 0 so that

P ann (|C (n) 2 | ≤ K 0 log n) → 1, yielding (3.3)
. Hence, it remains to show (3.105), (3.106) and 3.107).

The field ψ Gn on the first phase: proof of (3.105) Proposition 3.7.6. Let a, K 0 , be such that Proposition 3.7.2 holds with K = K 0 , and such that the conclusion of Lemmas 3.7.7 and 3.7.8 hold. Then for n large enough, (3.105) holds.

To prove Proposition 3.7.6, we need two variants of Proposition 3.4.1. The first consists in replacing the "security radius" a n = Θ(log log n) by r n = Θ(log n).

Lemma 3.7.7. If a > 0 is small enough, then the following holds for n large enough. Assume that G n is a good graph, that A ⊆ V n satisfies

• |A| ≤ n 2/3 ,
• tx(B Gn (A, r n )) = tx(A), and

• max z∈A |ψ Gn (z)| ≤ log 2/3 n.
For every y ∈ ∂B Gn (A, 1), writing y for the unique neighbour of y in A, we have:

E Gn [ψ Gn (y)|σ(A y )] - 1 d -1 ψ Gn (y) ≤ n -2a (3.109)
and

Var Gn (ψ Gn (y)|σ(A y )) - d d -1 ≤ n -2a . (3.110)
Proof. We follow the argument of Proposition 3.4.1, with a few adjustments.

First, the bounds in (3.37) and (3.38) are in fact e -can for some constant c > 0, and one can replace a n by r n .

Second, concerning the proof of (3.39), note that (3.41) still holds for large enough n since for k ≥ 0.3 log n and a constant γ > 0 that depends neither on k nor on n,

P(Z k ≤ r n ) ≤ P(Z k ≤ 0.05 log n) ≤ P(Z k ≤ d-2 2d k) ≤ e -γk .
The condition |A| ≤ n 2/3 implies that E πn [H A ] ≥ n 1/3 /4, so that we can adapt the argument below (3.40), as well as the proof of (3.42).

The second variant of Proposition 3. For k ≥ 0, let z k be a vertex at distance k of the cycle C m in G m , and z k be a neighbour

of z k at distance k + 1 of C m . Note that B Gm (z k , z k , r n ) (the subgraph of G m obtained by
taking all paths of length r n starting at z k and not going through z k ) contains C m if and only

if k ≤ r n -m/2 .
Lemma 3.7.8. If a > 0 is small enough, then the following holds for n large enough (uniformly

in m ≥ 3).
Assume that G n is a good graph, that A ⊆ V n is such that

• |A| ≤ n 2/3 ,
• A is a tree, and

• max z∈A |ψ Gn (z)| ≤ log 2/3 n.
Case 1. Let y ∈ V n , and suppose that -y has a neighbour y in A, -for some 1 ≤ k < m, there exists y in A, a path P of length m -k from y to y whose only vertex in A is y, and a path P in A of length k -1 from y to y, so that C := P ∪ P ∪ (y, y)

is a cycle of length m (and y = y if k = 1), and

-tx(B Gn (A ∪ C, r n )) = 1. Then E Gn [ψ Gn (y)|σ(A)] -α k ψ Gn (y) -β k ψ Gn ( y) ≤ n -2a (3.111)
and

Var Gn (ψ Gn (y)|σ(A)) -γ k ≤ n -2a . (3.112)
Case 2. Let y ∈ V n , and suppose that -y has a unique neighbour y in A,

-for some 1 ≤ k ≤ r n -m/2 , B Gn (y, y, r n ) is isomorphic to B Gm (z k , z k , r n ), and
-tx(B Gn (A ∪ P ∪ C, r n )) = 1
, where C is the cycle in B Gn (y, y, r n ) and P the path from y to C.

Then E Gn [ψ Gn (y)|σ(A)] -α k ψ Gn (y) ≤ n -2a (3.113)
and

Var Gn (ψ Gn (y)|σ(A)) -γ k ≤ n -2a . (3.114)
Proof. We will only show (3.113) and (3.114). The other proofs are very similar and left to the reader.

We start with the proof of (3.113). We follow the proof scheme of (3.34). Let τ be the hitting

time of ∂B Gn (y, y, r n ) \ {y} by a SRW (X k ) k≥0 . Note that {H A ≤ τ } ⊆ {X H A = y}.
We write

E Gn [ψ Gn (y)|σ(A)] =P Gn y (X H A = y, H A ≤ τ )ψ Gn (y) + E Gn y [ψ Gn (X H A )1 H A >τ ] - E Gn y [H A ] E Gn πn [H A ] E Gn πn [ψ Gn (X H A )].
Since B Gn (y, y, r n ) and B Gm (z k , z k , r n ) are isomorphic,

P Gn y (X H A = y, H A ≤ τ ) = P Gm z k (X H ∂B Gm (z k ,z k ,rn) = z k ) =: α k .
Since

{X H ∂B Gm (z k ,z k ,rn) = z k } ⊆ {H {z k } < +∞}, we have α k ≤ α k . Reciprocally, on {H {z k } < +∞} \ {X H ∂B Gm (z k ,z k ,rn) = z k }, a SRW starting at distance r n of C m has to reach C m .
As in the proof of (3.37), a comparison with a biased SRW on Z shows that this happens with a probability O(e -crn ) for some constant c > 0 uniquely depending on d and we get that if a small enough, then for large enough n, |P Gn y (X

H A = y, H A ≤ τ ) -α k | ≤ n -3a . It remains to establish E Gn y [ψ Gn (X H A )1 H A >τ ] - E Gn y [H A ] E Gn πn [H A ] E Gn πn [ψ Gn (X H A )] ≤ n -3a . (3.115)
To do so, one adapts the proofs of (3.38) and (3.39) exactly as in the proof of Lemma 3.7.7: if

H A > τ , (X k ) k≥0 leaves B Gn (A ∪ C ∪ P, r n ) before hitting A. By Lemma 3.4.2 with s = r n , if
a is small enough, (X k ) does not hit A within the next log 2 n steps with probability at least

(1 -(d -1) -rn ) log 2 n ≥ 1 -2 log 2 n (d -1) -rn ≥ 1 -n -4a .
Then, we use Corollary 2.1.5 of [START_REF] Saloff-Coste | Lectures on finite Markov chains[END_REF] as six lines above (3.40): after log 2 n steps, the fact that G n is an expander forces the empirical distribution of X k to be very close to the uniform distribution π n . (3.113) follows.

For (3.114), we follow the proof scheme of (3.35). If τ is the exit time of B Gn (y, y, r n ), we have

Var Gn (ψ Gn (y)|σ(A)) -γ k ≤ G Gn (y, y) -E Gn y [G Gn (y, X H A ) 1 H A =τ ] -γ k + E Gn y [G Gn (y, X H A ) 1 H A >τ ] - E Gn y [H A ] E πn [H A ] E Gn πn [G Gn (y, X H A )] . (3.116)
For y = x, we can apply Lemma 3.7.8 (if ψ Gn (y) is defined in Case 1 or Case 2) or Lemma 3.7.7

(in the other cases), so that

P ann (E(y)) ≤ P ann (n -2a |ξ y | ≥ n -a log -2 n -n -2a ) ≤ P ann (|ξ y | ≥ n a/2 ) ≤ n -4
by the exponential Markov inequality, and P ann (E(x)) ≤ n -4 by the same argument, where we set E(x) := {| ψ Gn (x) -ψ Gn (x)| ≥ n -a log -2 n} ∩ E n . Hence, we get by a union bound on y ∈ R ψ 1 that for large enough n,

P ann (E n ) ≤ n -4 K 0 log n ≤ n -3 .
The conclusion follows.

The field ψ Gn on the second phase: proof of (3.106)

It is enough to show that for each y q,i whose exploration in the second phase is successful,

P ann {sup z∈Ty q,i \{y q,i } | ψ Gn (z) -ψ Gn (z)| ≥ log -1 n 2 } ∩ E 7,n ≤ n -4 ,
where T y q,i is its exploration tree, and to conclude by a union bound on y q,i . This follows from a straightforward adaptation of the reasoning below (3.54). Note that the n -3 in the RHS of (3.55) can be replaced by any polynomial in n.

The field ψ Gn on the third phase: proof of (3.107)

By symmetry, it is enough to consider the case i = 1. Following readily the argument of the proof of (3.70), we get that the probability that E ≥h ψ Gn percolates through a given J-joining ball is at least log γ(K 8 /3-1) n, for any J. By (3.96) and (3.97), and a union bound on every couple (J, q) ∈ {1, . . . , s 0 } × {1, . . . m},

P ann ((S 2 (x) \ S ψ 3,1 (x)) ∩ E 7,n ) ≤ 2n -4 + s 0 mP(Z = 0),
where Z ∼ Bin( log γ-3κ-18 n , log γ(K 8 /3-1) n). If κ and γ/κ are large enough, then for n large enough,

P(Z = 0) = (1 -log γ(K 8 /3-1) n) log γ-3κ-18 n ≤ n -4 ,
and (3.107) follows.

Properties of C

(n) 1

3.8.1 The local limit: proof of Theorem 3.1.3

Proof of Theorem 3.1.3. The proof mimics the reasoning of Lemma 3.6.3. Let k ≥ 0 and let T be a rooted tree of height k, with no vertex of degree more than d. Let x ∈ V n . We perform an 3.8.2 The core and the kernel: proof of (3.4) and (3.5)

We now prove (3.4) and (3.5) of Theorem 3.1.2, starting with (3.4). Let K 1 (resp. K 2 ) be the probability that • has at least 2 (resp. 3) children with an infinite offspring in C h • , under P T d . Suppose that for x, y ∈ V n ,

P ann (x ∈ C (n) ) -→ n→+∞ K 1 (3.120)
and Proof of (3.120). For x ∈ V n , we perform the exploration in Section 3.5.1 from x, replacing C2 by the following condition: for every neighbour v of x, stop exploring the subtree from v at step k + 1 if the k-offspring of v has at least n 1/2 b n vertices. Stop the exploration if this happens for at least two neighbours of x. In this case, say that the exploration is successful.

Cov ann (1 x∈C (n) , 1 y∈C (n) ) -→ n→+∞ 0. ( 3
We adapt easily the proofs of Lemma 3.5.3 and Proposition 3.5.1 to show that for x ∈ V n ,

P ann (the exploration from x is successful) -→ n→+∞ K 1 . (3.122)
Indeed, K 1 is the probability that the realization of ϕ T d to which we couple ψ Gn is such that • has at least two children with an infinite offspring. Then, as in Lemma 3.5.3, there is a probability 1 -o(1) that the offspring of these children grows at an exponential rate close to λ h (Proposition 3.3.8). Thus, letting F

v,k := ∪ 1≤j≤k {the j-offspring of v has at least n 1/2 b n vertices} for every child v of •, and

F (n) core,k := ∪ v 1 ,v 2 children of • (F (n) v 1 ,k-1 ∩ F (n) v 2 ,k-1 ), we get that P ann (∪ 1≤k≤ log λ h n F (n) core,k ) -→ n→+∞ K 1 and P ann (∪ 1≤k≤ log λ h n {at most one child of • has a non-empty k-offspring}) -→ n→+∞ 1 -K 1 .
As for (3.53), we see that we do not meet any cycle with P ann -probability 1 -o(1). This yields (3.122).

If the exploration from x is successful, let x 1 , x 2 be two children of x such that the exploration subtrees T x 1 and T x 2 from x 1 and x 2 satisfy min(|∂T

x 1 |, |∂T x 2 |) ≥ n 1/2 b n .
Then, let K > 0 and let w 1 , . . . , w K log n ∈ V n be vertices that have not been met in the exploration from x. Proceed to their w-exploration as described in the first part of the construction in Section 3.7.4. By Remark 3.5.2, o( √ n) vertices are seen during the exploration from x and the w-explorations.

Applying (3.14) with k = 1, m 0 , m 1 , m E , m = o( √ n),
we get that with P ann -probability 1-o(1), none of the w-explorations intersects the exploration from x, and no w i is spoiled or back-spoiled.

As in (3.104), we get that for each w i , its w-exploration has probability at least η(h)/2 to be w-successful. Hence with P ann -probability at least 1 -(1 -η(h)/2) K log n = 1 -o(1), there exists i 0 such that the w-exploration from w i 0 is successful. Denote ∂T w i 0 the boundary of its exploration tree. Write S core (x) :={the exploration from x is successful}∩ {∃i 0 ≥ 1, the w-exploration from w i 0 is successful}.

We have just shown that lim inf

n→+∞ P ann (S core (x)) ≥ lim inf n→+∞ P ann (the exploration from x is successful) ≥ K 1 . (3.123)
Next, we grow joining balls from ∂T x 1 to ∂T w i 0 , and then from ∂T x 2 to ∂T w i 0 . We proceed as in the second part of the construction in Section 3.7.4 with m = 2 and s 0 = 1. Similarly to (3.96), we get that

P ann (S core (x) ∩ (S core,joining,1 ∪ S core,joining,2 )) = o(1),
with S core,joining,i := {there are less than log γ-3κ-18 n joining balls from

∂T x i to ∂T w i 0 } for i = 1, 2.
Finally, we reveal ψ Gn on the exploration from x, on the w-exploration from w i 0 , on the joining balls from ∂T x 1 to ∂T w i 0 , and on the joining balls from ∂T x 2 to ∂T w i 0 , in that order. Denote S core,connect the event that there exists a joining ball B 1 from ∂T x 1 to ∂T w i 0 and another B 2 from ∂T x 2 to ∂T w i 0 such that min y∈Tx 1 ∪Tx 2 ∪Tw i 0 ∪B 1 ∪B 2 ψ Gn (y) ≥ h. As in the proof of Proposition 3.6.2, we get that P ann (S core (x) ∩ S c core,connect ) = o(1). Hence lim inf n→+∞ P ann (x, x 1 , x 2 , z are in a cycle of C Gn x and |C Gn x | ≥ n 1/3 ) ≥ lim inf n→+∞ P ann (S core (x)).

If

|C (n) 2 | < n 1/3 , this cycle is in C (n)
1 , and thus in C (n) so that by (3.3), for large enough n and any x ∈ V n , one has by (3.123):

lim inf n→+∞ P ann (x ∈ C (n) ) ≥ lim inf n→+∞ P ann (S core (x)) ≥ K 1 .
Reciprocally, for x ∈ V n , turn the exploration into a lower exploration, replacing h + log -1 n by h -log -1 n (as in Section 3.5.2). Say that the lower exploration from x is aborted if for some k ≤ log log n, at most one child of x has a non-empty (k -1)-offspring. Let A core (x) := {the lower exploration from x is aborted}. We get as in the proof of (3.122):

P ann (A core (x)) -→ n→+∞ 1 -K 1 . (3.124)
Moreover, revealing ψ Gn on T x , we can apply Proposition 3.4.1 as below (3.54) to get that

P ann (A core (x) ∩ {C Gn,h x ∩ B Gn (x, log log n ) ⊆ T x }) -→ n→+∞ 1 -K 1 . (3.125)
For each neighbour y of x, denote C y its connected component in C Gn x \ {x}. If the exploration is aborted and C Gn,h x ∩ B Gn (x, log log n ) ⊆ T x , then x has at most one neighbour y such that C y ∪ {x} is not a tree. Hence x ∈ C (n) . Thus, lim inf n→+∞ P ann (x ∈ C (n) ) ≥ 1 -K 1 and (3.120) follows.

Proof of (3.121). By (3.120), for x, y ∈ V n ,

P ann (x ∈ C (n) )P ann (y ∈ C (n) ) -→ n→+∞ K 2 1 .
It remains to show that

P ann (x, y ∈ C (n) ) -→ n→+∞ K 2 1 .
Perform the exploration from x as in the beginning of the proof of (3.120), then do the same from y (and stop the latter if it reaches a vertex of the exploration from x). Since o( √ n) vertices are revealed during these explorations (see Remark 3.5.2), then by (3.14), the probability that the exploration from y meets that of x is o(1). Thus by (3.122), P ann (the explorations from x and y are both successful)

-→ n→+∞ K 2 1 .
Then, let x 1 , x 2 (resp. y 1 , y 2 ) be the children of x (resp. y) whose exploration is successful.

We complete the exploration in a fashion similar to that above (3.123). Let K > 0 and let w 1 , . . . , w K log n ∈ V n be vertices that have not been met in the explorations from x and y, and proceed to their w-exploration. If there exists i 0 ≥ 1 such that the w-exploration from w i 0 is successful, build joining balls from ∂T x 1 , ∂T x 2 , ∂T y 1 and ∂T y 2 to T w i 0 . Finally, reveal ψ Gn on T x , on T y , on T w i 0 and on the joining balls from ∂T x 1 , ∂T x 2 , ∂T y 1 and ∂T y 2 , in that order. As in the proof of (3.123) and below, we get that lim inf

n→+∞ P ann (x, y ∈ C (n) ) ≥ K 2 1 . (3.126)
Conversely, if we perform the lower explorations from x and y as defined in the end of the proof of (3.120), we easily get that P ann (∃z ∈ {x, y}, the lower exploration from z is aborted)

-→ n→+∞ 1 -K 2 1 .
Then, we reveal ψ Gn on T z . Following the reasoning below (3.54), we get that P ann -w.h.p.,

B C Gn,h z (z, log log n ) ⊆ T z ,

and thus

P ann ({the lower exploration from z is aborted} ∩ {z ∈ C (n) }) = o(1).
This yields

lim inf n→+∞ P ann (∃z ∈ {x, y}, z ∈ C (n) ) ≥ 1 -K 2 1 .
Together with (3.126), this concludes the proof.

This reasoning can be readily adapted to prove (3.5), with a modification of the exploration (requiring that at least three children of x have a successful exploration).

3.8.3

The typical distance: proof of (3.7)

The proof of (3.7) goes as that of (3.2), with a slight modification of the explorations of Section 3.5. Those explorations were indeed stopped after at most log λ h n steps. But since around √ n vertices were explored, and since the growth rate of C h • is close to λ h , we can expect that a successful exploration lasts in fact (1/2 + o(1)) log λ h n steps. Then, connecting two such explorations as in Proposition 2.5.16 (adding an additional distance Θ(log log n) = o(log n)) yields the diameter (since reciprocally, explorations lasting less steps will be too small to be connected).

Proof of (3.7). Fix ε ∈ (0, 1). We start by the upper bound. In the exploration of Section 3.5.1, replace C4 by the condition: stop the exploration if k ≥ (1/2+ε/3) log λ h n. By Proposition 3.3.8, as n → +∞:

P T d |Z h (1/2+ε/3) log λ h n | > n 1/2 b n |C h+log -1 n • | = +∞ → 1.
Thus, Propositions 3.5.1, 3.6.2 and Lemma 3.6.4 remain unchanged. Note that if x, y are connected in E ≥h ψ Gn via the successful explorations from x and y and the joining balls from ∂T x to ∂T y , then for n large enough:

d E ≥h ψ Gn (x, y) ≤ 2(1/2 + ε/3) log λ h n + a n ≤ (1 + ε) log λ h n.
Then, Lemma 3.6.4 implies that

P ann (E 1,n ∩ E 2,n ∩ E 3,n ) -→ n→+∞ 1, where E 1,n := {|{(x, y) ∈ V 2 n , d E ≥h ψ Gn (x, y) ≤ (1 + ε) log λ h n}| ≥ (η(h) 2 -ε)n 2 }.
We have to check that only o(n 2 ) of the couples (x, y) described in E 1,n are not in C

(n)

1 , and that |C (n) 1 |/n is indeed close to η(h). Note that by (3.2) and (3.3),

P ann (E 1,n ∩ E 2,n ∩ E 3,n ) -→ n→+∞ 1, where E 1,n := {|{(x, y) ∈ V 2 n , d E ≥h ψ Gn (x, y) ≤ (1 + ε) log λ h n}| ≥ (η(h) 2 -ε)n 2 }, E 2,n := {∀i ≥ 2, |C (n)
i | ≤ K 0 log n} and E 3,n := {(η(h) -ε)n ≤ |C (n) 1 | ≤ (η(h) + ε)n}. On E 2,n , we have |{(x, y) ∈ V 2 n \ (C (n) 1 ) 2 , d E ≥h ψ Gn (x, y) ≤ (1 + ε) log λ h n}| ≤ n 3/2 , so that on E 1,n ∩ E 2,n , |{(x, y) ∈ (C (n) 1 ) 2 , d C (n) 1 (x, y) ≤ (1 + ε) log λ h n}| ≥ (η(h) 2 -2ε)n 2 . Thus, on E 1,n ∩ E 2,n ∩ E 3,n : π 2,n {(x, y) ∈ (C (n) 1 ) 2 , d C (n) 1 (x, y) ≤ (1 + ε) log λ h n} ≥ η(h) 2 -2ε (η(h) + ε) 2 ≥ 1 - 2ε + 2η(h)ε + ε 2 (η(h) + ε) 2 ≥ 1 - (3 + 2η(h)) η(h) 2 ε.
P ann (E ab (x, y) ∩ E x ∩ E y ) = 1 -o(1), with E x :={B C Gn,h x (x, (1/2-ε/2) log λ h n )⊆ T x } and E y :={B C Gn,h y (y, (1/2-ε/2) log λ h n )⊆ T y }. On E ab (x, y) ∩ E x ∩ E y , if x, y ∈ C (n) 1 , then d C (n) 1 (x, y) ≥ (1 -ε) log λ h n.
Therefore, for every x, y ∈ V n , P ann (E x,y ) = o(1), with

E x,y := {x, y ∈ C (n) 1 , d C (n) 1 (x, y) < (1 -ε) log λ h n}.
Similarly, for all distinct x, y, z, t ∈ V n , we get that P ann (E x,y ∩ E z,t ) = o(1), so that Cov ann (1 Ex,y , 1 Ez,t ) = o(1).

Thus by Bienaymé-Chebyshev's inequality,

P ann (E 3,n ∩ {|{(x, y) ∈ V 2 n , d C (n) 1 (x, y) ≤ (1 -ε) log λ h n}| ≥ εn 2 }) -→ n→+∞ 1.
For ε > 0 small enough and n large enough, on

E 3,n ∩ {|{(x, y) ∈ V 2 n , d C (n) 1 (x, y) ≤ (1 -ε) log λ h n}| ≥ εn 2 }, π 2,n {(x, y) ∈ (C (n) 1 ) 2 , d C (n) 1 (x, y) ≤ (1 -ε) log λ h n} ≤ 2ε η(h) 2 .
This concludes the proof of (3.7).

Thus, it remains to establish (3.127). Note first that P ann (C1 happens) = o(1) by (3.14) with k = 1, m 0 = 1, m E = 0 and m = o( √ n) by Remark 3.5.2. Therefore, it is enough to prove that

P T d (|B C h-log -1 n • (•, (1/2 -ε/2) log λ h n + a n )| < n 1/2-ε/8 ) → 1,
since if this event happens, less than n 1/2-ε/10 vertices and half-edges have been seen at step

(1/2 -ε/2) log λ h n . By (3.33), it suffices to show that

P T d (|B C h-log -1 n • (•, (1/2 -ε/2) log λ h n )| < n 1/2-ε/7 ) → 1, (3.128)
To do so, we first prove that for n large enough and every log log n ≤ k ≤ (1/2 -ε/2) log λ h n,

P T d |Z h-log -1 n k | ≥ n 1/2-ε/6 ≤ e -Ck + n -3 . (3.129) Let δ > 0 such that λ h ≤ λ h-δ ≤ λ h + ε/
P T d a |Z h-log -1 n k | ≥ n 1/2-ε/5 χ h-δ (a) ≤ P T d a |Z h-δ k | ≥ n 1/2-ε/5 χ h-δ (a) ≤ P T d a log |Z h-δ k | ≥ (1/2 -ε/5)λ h log λ h n + log χ h-δ (a) ≤ P T d a log |Z h-δ k | ≥ (λ h-δ -ε/10) 1/2 -ε/5 1/2 -ε/2 k + log χ h-δ (a) ≤ P T d a log |Z h-δ k | ≥ (λ h-δ -ε/10)(1 + 3ε/5)k + log χ h-δ (a) ≤ P T d a log |Z h-δ k | ≥ λ h-δ (1 -ε/10)(1 + 3ε/5)k + log χ h-δ (a) ≤ P T d a log |Z h-δ k | ≥ λ h-δ (1 + ε/5)k + log χ h-δ (a) ≤ e -Ck .
By Proposition 2.1 of [START_REF] Abächerli | Level-set percolation of the Gaussian free field on regular graphs II: finite expanders[END_REF], there exists c > 0 such that for all h ≤ h and a ≥ d -1, one has χ h (a) ≤ ca 1-log d-1 λ h ≤ ca. Since χ h is continuous on [h, +∞) (Lemma 3.3.5), we have for n large enough max h≤a≤log 2 n χ h-δ (a) < n ε/30 , so that

P T d |Z h-log -1 n k | ≥ n 1/2-ε/6 ≤ e -Ck + P T d (ϕ T d (•) ≥ log 2 n)
Using the exponential Markov inequality as in Lemma 3.2.5, we get

P T d (ϕ T d (•) ≥ log 2 n) ≤ n -3 .
This yields (3.129). Then, for n large enough, this implies

P T d (|B C h • (•, (1/2 -ε/2) log λ h n )| ≥ n 1/2-ε/7 ) ≤ P T d (∃k ∈ [log log n, (1/2 -ε/2) log λ h n], |Z h-log -1 n k | ≥ n 1/2-ε/6 ) ≤ (1/2-ε/2) log λ h n k= log log n (e -Ck + n -3 )
≤ 1/ log log n.

(3.128) and the conclusion follow.

3.8.4

The diameter: proof of (3.6)

Recall that D (n) 1 is the diameter of C (n)
1 . In Section 3.7, we have in fact proven that there exists a constant K 13 > 0 such that for

E n := {∀x ∈ C (n) 1 , |B C (n) 1 (x, K 13 log n )| ≥ K 12 n}, we have P ann (E n ) -→ n→+∞ 1.
Namely, one can take

K 13 = K 0 + 3log -1 λ h .
Hence it is enough to show that on E n , D

(n) 1 ≤ 6K -1
12 K 13 log n, which will imply (3.6). We do this by a short deterministic argument.

Let x 1 ∈ C (n) 1 . If ∂B C (n) 1 (x 1 , 2 K 13 log n + 1) = ∅, then D (n) 1 ≤ 4K 13 log n + 2. Else, let x 2 ∈ ∂B C (n) 1 (x 1 , 2 K 13 log n + 1). For i = 1, 2, we have |B C (n) 1 (x i , K 13 log n )| ≥ K 12 n and B C (n) 1 (x i , K 13 log n ) ⊆ B C (n) 1 (x 1 , 4 K 13 log n ). Moreover, B C (n) 1 (x 1 , K 13 log n ) ∩ B C (n) 1 (x 2 , K 13 log n ) = ∅. Thus, we have |B C (n) 1 (x 1 , 4 K 13 log n )| ≥ 2K 12 n. For i ≥ 2, if ∂B C (n) 1 (x 1 , (3i -4) K 13 log n + 1) = ∅, then D (n) 1 ≤ 2(3i -4)K 13 log n + 2. Else, let x i+1 ∈ ∂B C (n) 1
(x 1 , (3i -4) K 13 log n + 1). As for i = 2, we get that

|B C (n) 1 (x 1 , (3i -2) K 13 log n )| -|B C (n) 1 (x 1 , (3i -5) K 13 log n )| ≥ |B C (n) 1 (x i , K 13 log n)| ≥ K 12 n.
But there are only n vertices in V n . Therefore, ∂B C (n) 1 (x 1 , (3i 0 -4) K 13 log n + 1) = ∅ for some

i 0 ≤ K -1
12 , and

D (n) 1 ≤ 6K -1 12 K 13 log n.
This shows (3.6).

where ν 1 -ν 2 T V := sup S⊆Ω (ν 1 (S) -ν 2 (S)) is the total variation distance between the probability measures ν 1 and ν 2 on Ω. The worst-case mixing time t max (ε) := sup{t x (ε), x ∈ Ω} is often the quantity of main interest. Other distances than the total variation distance might be considered. A straightforward computation shows that t → P t (x, •) -π T V is a non-increasing function, so that the definition of mixing time is relevant.

For a sequence of Markov chains (Ω n , P n , π n ) n≥0 , there is cutoff when for all ε, ε ∈ (0, 1),

t (n) max (ε)/t (n)
max (ε ) → 1 as n → +∞. While the cutoff phenomenon remains far from being completely understood, first examples of it were given in the 1980s for different random walks on finite groups (see [START_REF] Diaconis | Generating a random permutation with random transpositions[END_REF] or [START_REF] Aldous | Shuffling cards and stopping times[END_REF] on the symmetric group) or on spaces that can be factored into a n-product of a base space (such as

Z n 2 in [9]
), and this direction is still investigated nowadays (see for instance [START_REF] Hermon | Cutoff for almost all random walks on abelian groups[END_REF], [START_REF] Hermon | Cutoff for Mixing Times on Random Abelian Cayley Graphs[END_REF] on random Cayley graphs of abelian groups).

A class of graphs where random walks mix fast and where cutoff is expected are the expander graphs. These are sequences (G n ) n≥1 of graphs whose size goes to infinity (say G n has n vertices) and whose isoperimetric constant is bounded away from 0: there exists c > 0 independent of n such that for any subset S of at most n/2 vertices of G n , |∂S| ≥ c|S|, where ∂S ⊆ S c is the set of vertices adjacent to vertices of S. The very accessible survey of Hoory, Linial and Widgerson [START_REF] Hoory | Expander graphs and their applications[END_REF] provides a good overview of the study of these graphs. This expansion property entails the existence of a spectral gap (this implication is called the "Cheeger bound", see [START_REF] Alon | Eigenvalues, expanders and superconcentrators[END_REF] for instance): the second largest eigenvalue of the transition matrix P n of the SRW on G n is bounded away from the largest one as n → +∞. It is classical that this spectral gap implies in turn that the SRW on G n mixes in O(log n) steps.

The simplest expander model is the random d-regular graph (i.e. G n (d) is chosen uniformly among graphs with n vertices having all degree d). Friedman [START_REF] Bordenave | A new proof of Friedman's second eigenvalue Theorem and its extension to random lifts[END_REF] proved in 2002 that w.h.p., G n (d) almost achieves the largest possible spectral gap, while Lubetzky and Sly [START_REF] Lubetzky | Cutoff phenomena for random walks on random regular graphs[END_REF] proved in 2008 that the SRW and the NBRW (Non-Backtracking Random Walk, i.e. a SRW conditioned at each step on not going back along the edge it has just crossed) on G n (d) admit a cutoff.

Several papers followed on cutoffs for other sparse graphs: see for instance [START_REF] Berestycki | Random walks on the random graph[END_REF] for the SRW on the largest component of a supercritical Erdős-Rényi random graph, [START_REF] Ben | Cutoff for nonbacktracking random walks on sparse random graphs[END_REF] for the NBRW on a configuration model, or [START_REF] Hermon | Universality of cutoff for graphs with an added random matching[END_REF] for a generic perspective.

Very recently, there has been increasing interest in mixing times on dynamical graphs (typically, edges are re-sampled at random at a given rate), when the mixing time profile is already well-known on a static version of the graph (for instance [START_REF] Avena | Random walks on dynamic configuration models:a trichotomy[END_REF], [START_REF] Sousi | Cutoff for random walk on dynamical Erdős-Rényi graph[END_REF] and [START_REF] Caputo | Mixing time of PageRank surfers on sparse random digraphs[END_REF]).

A natural way of combining the "product of a base space" and the "expanding sparse graph" perspectives for cutoff is to consider random walks on random n-lifts of a fixed graph G, that we define now.

Random walks on weighted graphs

For a multigraph G (multiple edges and multiple loops are allowed), denote V G its vertex set and E G its edge set. Every edge e ∈ E G gives rise to two opposite oriented edges. Denote -→ E G the set of oriented edges of G. For each -→ e ∈ -→ E G , note -→ e -1 its opposite. We study weighted random walks by giving to each -→ e ∈ -→ E G a nonnegative weight w( -→ e ), so that

• every e ∈ E G has at least one orientation with positive weight,

• for all u ∈ V G , the sum of the weights of the oriented edges going out of u is 1.

We define the random walk (RW) on G as a discrete-time Markov Chain (X t ) t≥0 on V G with transition matrix P G such that for all u, v ∈ V G ,

P G (u, v) = 1 2 1 {u=v} + 1 2 -→ e :u→v w( - → e ),
where " -→ e : u → v" means that the initial vertex of -→ e is u and its end vertex is v.

Random lifts

Fix now a finite multigraph

G. A n-lift of G is a graph G n with vertex set V Gn := V G × [n],
and edge set E Gn as follows: fix for each e ∈ E G an arbitrary ordering (u, v) of its endpoints and a permutation σ e ∈ S n , and draw the edges {u i , v σe(i) } for all 1 ≤ i ≤ n (see Figure 1). Say that u (resp. v) is the type of u i (resp. v σ(i) ) and that e is the type of {u i , v σ(i) }.

When the σ e 's are uniform independent permutations, G n is a random n-lift of G. The graph structure of random lifts has been studied since the early 2000s (see [START_REF] Amit | Random lifts of graphs: edge expansion[END_REF], [START_REF] Amit | Random lifts of graphs: independence and chromatic number[END_REF], [START_REF] Amit | Random lifts of graphs[END_REF] and [START_REF] Linial | Random lifts of graphs: perfect matchings[END_REF]). In particular, it is proved in [START_REF] Amit | Random lifts of graphs: edge expansion[END_REF] that random n-lifts are expanders w.h.p. as n goes to infinity, as long as G has at least two cycles. More recently, spectral properties of lifts have been investigated (see for instance [START_REF] Addario | The spectrum of random lifts[END_REF], [START_REF] Drier | Minors in lifts of graphs[END_REF], [START_REF] Lubetzky | Spectra of lifted ramanujan graphs[END_REF]): Bordenave [START_REF] Bordenave | A new proof of Friedman's second eigenvalue Theorem and its extension to random lifts[END_REF] generalized Friedman's theorem to the NBRW on random n-lifts of a finite graph, then Bordenave and Collins [START_REF] Bordenave | Eigenvalues of random lifts and polynomials of random permutation matrices[END_REF] established a similar result for the SRW. Bordenave and Lacoin [START_REF] Bordenave | Cutoff at the entropic time for random walks on covered expander graphs[END_REF] proved that if the RW associated to G is reversible, and if the invariant measure is uniform, then the RW on G n admits a cutoff, with a mixing time in h -1 log n + o(log n) steps, for some constant h (the "entropy") depending on G.

Results

We characterize all (finite) irreducible graphs G such that there is w.h.p. cutoff for the random walk on a random n-lift of G, and we prove that the cutoff window is of order √ log n.

We introduce the following assumptions:

A.

(G n ) n≥1 of n-lifts of G, lim inf n→+∞ t (n) min (ε) -h -1 log n √ log n ≥ σΦ -1 (ε), (4.4)
where t

(n) min (ε) := min x∈Vn t

(n)

x (ε) is the best-case ε-mixing time (ie, the shortest mixing time among all possible starting vertices).

Assumption A.2 is necessary in Theorem 4.1.1: remark that if G has at most one oriented cycle, inf x∈Vn π n ({y ∈ V n , there is no oriented path from x to y}) is w.h.p. bounded away from 0.

We conjecture that the lower bound of Proposition 4.1.2 is optimal, so that the cutoff window would have a Gaussian profile. This was established for the SRW on the random d-regular graph G n (d) in [START_REF] Lubetzky | Cutoff phenomena for random walks on random regular graphs[END_REF]. In this case, there is randomness for the speed of the walk (since it can backtrack), but the degree of the vertices met by the walk is constant. Conversely, for the NBRW on the configuration model whose cutoff window also exhibits this behaviour [START_REF] Ben | Cutoff for nonbacktracking random walks on sparse random graphs[END_REF], there is randomness for the degrees met by the walk, but not for the speed. In our setting, as for the SRW on the configuration model [START_REF] Berestycki | Random walks on the random graph[END_REF], both the environment and the speed of the walk might vary, and the result of this combination is not clear. Theorem 4.1.1 is also true for a lazy random walk on V n with any holding probability α ∈ (0, 1), i.e. for all u, v ∈ V n , the transition matrix is

P (α) n (u, v) = α1 {u=v} +(1-α) -→ e :u→v w( - → e )
(hence, the RW we defined in Section 4.1.1 and that we will study throughout this paper is lazy with holding probability 1/2). This gives a new value of the entropy:

h α = h 2(1 -α) . (4.5)
The question whether this holds in the case α = 0 remains unsolved, see Appendix 2 (Section 4.1.7) for a discussion.

One might also investigate to what extent Theorem 4.1.1 holds when G changes with n. It is proven in [START_REF] Lubetzky | Cutoff phenomena for random walks on random regular graphs[END_REF] that there is still cutoff for the RW on G n (d) if d = n o (1) .

Examples

Our setting is very general, since it includes lifts of any finite Markov chain with positive holding probability. We highlight two special cases below.

Random walks on the d-regular random graph

We can recover an approximate version of the result of [START_REF] Lubetzky | Cutoff phenomena for random walks on random regular graphs[END_REF] for the RW on d-regular graphs, 3). This is exactly the value of h in Theorem 1 of [START_REF] Lubetzky | Cutoff phenomena for random walks on random regular graphs[END_REF] for the SRW on G n (d). Their theorem states in addition that the cutoff window is of order √ log n with a Gaussian profile, hence corresponding to our lower bound.

when

Cutoff for non-Ramanujan graphs

It was recently proven that on every sequence of d-regular weakly Ramanujan graphs, the SRW admits a cutoff [START_REF] Lubetzky | Cutoff on all ramanujan graphs[END_REF] (a sequence G n of d-regular graphs is said to be weakly Ramanujan whenever for all ε > 0, every eigenvalue of the adjacency matrix of G n is either ±d or in

[-2 √ d -1 -ε, 2 √ d -1 + ε]
for n large enough). This was even extended to graphs having n o (1) eigenvalues anywhere in (-d + ε , d -ε ) for an arbitrary ε > 0.

Theorem 4.1.1 gives an alternative proof for the existence of sequences of non weakly Ramanujan graphs having a cutoff. Indeed, take for G a connected d-regular graph which is not Ramanujan.

One computes easily that all eigenvalues of G are also eigenvalues of G n , so that G n is not weakly Ramanujan.

Tools and reasoning

The graph we study has locally few cycles, so that the behaviour of the RW on G n is closely linked to that of a RW on its universal cover (T G , •), the infinite rooted tree obtained from G by "unfolding" all its non-backtracking paths starting at a given distinguished vertex •, the root. A non-backtracking path is an oriented path ( -→ e 1 , . . . , -→ e m ) such that -→ e i+1 = -→ e -1 i for all i ≤ m -1. Two vertices in T G are neighbours if and only if the non-backtracking path of one of them is the non-backtracking path of the other without its last edge, see Figure 2 for an example.

We will write abusively T G instead of (T G , •) when the root is irrelevant. Figure 2: the first levels of the universal cover of G in Figure 1 (not all weights are on the picture), rooted at u. Note that two non-backtracking paths start from u, since the blue edge in G gives rise to two oriented edges opposite to each other. This object, also called "periodic tree", has been thoroughly studied since the 1990s. We postpone a precise history to Section 4.1.3. Our main references are an article on trees with finitely many cone types [START_REF] Nagnibeda | Random walks on trees with finitely many cone types[END_REF], and another on regular languages [START_REF] Lorenz | Asymptotic entropy of random walks on regular languages over a finite alphabet[END_REF].

u u u v x x x v v u v x u v x 0
Let (X t ) t≥0 be a RW on (T G , •) starting at the root. A key observation is that this RW is transient. This is intuitive, since (T G , •) has an exponential growth by A.2 (that is, for some C > 1 and all R large enough, there are more than C R vertices at distance R from the root) and has a "regular" structure for the RW (A.1 guarantees the existence of an invariant measure).

Thus, we can define its loop-erased trace, or "ray to infinity" ξ := (ξ t ) t≥0 , ξ t being the last vertex visited by the RW at distance t of the root.

Let W t := -log W (X t ) where for x ∈ (T G , •), W (x) = P(x ∈ ξ). The following central limit theorem sums up almost all the information we need on T G .

Theorem 4.1.3 (CLT for the weight on T G ). There exist two constants h T G > 0, σ T G ≥ 0, only depending on T G , such that

W t -h T G t √ t law → N (0, σ 2 T G ), (4.6)
with the convention that N (0, 0) = δ 0 is the Dirac distribution in 0.

The proof relies on the regularity of the structure of T G , that allows us to cut the trajectory of (X t ) t≥0 into i.i.d. intervals between regeneration times at oriented edges of a certain fixed type.

This gives us almost directly a proof of Proposition 4.1.2. The proof of Theorem 4.1.1 proceeds in three steps: a ) we couple (X t ) t≥0 to a RW (X t ) t≥0 on G n , imitating the analogous construction in [START_REF] Berestycki | Random walks on the random graph[END_REF] for the configuration model. This coupling is viable as long as (X t ) does not meet cycles.

We can ensure this almost until the mixing time, for most starting points of the RW in G n . We stress that the RW on (T G , •) is strongly "localized" around ξ: Proposition 4.1.4 (Ray localization).

∀R, t ≥ 1, P(ξ ∩ B(X t , R) = ∅) ≤ C 1 exp(-C 2 R),
where for all y ∈ (T G , •) and r ≥ 0, B(y, r) is the set of vertices y such that there is an oriented path of length ≤ r from y to y . This crucial observation allows us to reveal a limited number of edges while coupling RWs on (T G , •) and G n , hence reducing the probability to meet a cycle. This leads to an "almost mixing" of the RW on G n after

h -1 log n + O( √ log n) steps: the mass of P t n (X 0 , •) is concentrated on values of order n -1 e O( √ log n) for some t = h -1 log n + O( √ log n).
Corollary 4.1.5 (Almost mixing). Let ε, K > 0 and a < 0. If -a and K are both large enough (depending on G and ε), then for n large enough, with probability at least 1 -ε,

G n is such that for all x ∈ V n , ν n (V n ) ≥ 1 -ε, where ν n (x ) := P t n n (x, x ) ∧ exp(K √ log n) n
for all x ∈ V n , and

t n := h -1 log n + a √ log n.
b ) As in [START_REF] Berestycki | Random walks on the random graph[END_REF], a spectral argument relying on the good expanding properties of random lifts (generalizing a little the result of [START_REF] Amit | Random lifts of graphs: edge expansion[END_REF]) allows us to make the last jump until the mixing time. We emphasize the fact that this spectral property holds even if the RW on G is not reversible.

c ) Finally, we extend the mixing to every starting point for the RW in G n , proving that (X t ) t≥0 quickly reaches a vertex to which we can apply a). We adapt the technique in [START_REF] Ben-Hamou | Comparing mixing times on sparse random graphs[END_REF], which was originally designed for the configuration model considered in [START_REF] Berestycki | Random walks on the random graph[END_REF].

Plan

We start with basic but essential properties in Section 4.1.2. We study the universal cover of

Basic properties

Three Lemmas

The next property is an essential tool for building G n while exploring it via a walk on its vertices.

It is analogous to the classical construction of a configuration model.

An oriented edge in a rooted tree is upward if the height of its initial vertex is smaller than the height of its end vertex, downward else. Its height is the height of its initial vertex. An oriented path of upward (resp. downward) edges is an upward (resp. downward) path. In a tree, there is at most one edge between two vertices x, y. We will denote {x, y}, (x, y) and (y, x) the corresponding edge and oriented edges.

A little history and plan of Section 4.1.3

Lyons [START_REF] Lyons | Random walks and percolation on trees[END_REF] and Takacs [START_REF] Takacs | Random walk on periodic trees[END_REF] have studied RWs on rooted periodic trees arising from simple graphs, with weights corresponding to the SRW with a positive or negative bias towards the root. Thus, this intersects our setting only when the bias vanishes, the RW being a SRW (hence, it is reversible). The more general case of trees with finitely many cone types has been studied by Nagnibeda and Woess [START_REF] Nagnibeda | Random walks on trees with finitely many cone types[END_REF] in 2002: these rooted weighted trees have finitely many subtrees up to isomorphism (periodic trees obviously have finitely many cone types, whereas the converse is not true). Their work relies on a fine understanding of the Green function initiated in [START_REF] Nagnibeda | Green functions on trees with finitely many cone types[END_REF],

which satisfies a finite system of (non-linear) equations, due to the repetitive structure of the tree. Among others, they give a transience criterion for the RW in terms of the eigenvalues of a matrix associated to the tree, and obtain a CLT for the rate of escape (or speed, i.e. lim t→+∞ he(X t )/t) when it exists) in the transient case. Similar formulas for Green functions were derived around the same time by Lalley [START_REF] Steven | Random walks on regular languages and algebraic systems of generating functions[END_REF] in the broader setting of regular languages.

Gilch [START_REF] Lorenz | Asymptotic entropy of random walks on regular languages over a finite alphabet[END_REF] later gave a formula for the entropy of the RW on regular languages, i.e. a law of large numbers for (log P k (X 0 , X k )) k≥0 where P is the transition matrix of the RW.

We extend slightly some of those results in the setting of periodic trees. We first give in Section A crucial argument from [START_REF] Gaudillière | A dirichlet principle for non reversible markov chains and some recurrence theorems[END_REF] is that whether the RW is reversible or not is irrelevant as soon as the RW is irreducible and there exists an invariant measure (in our case, π). Hence, we can apply a classical transience criterion for reversible RWs (see [START_REF] Lyons | A simple criterion for transience of a reversible Markov chain[END_REF]), noticing that |B(•, R)| grows exponentially with R by A.2.

Thus, the loop-erased trace (ξ t ) t≥0 of the RW is an infinite injective path. It is a Markov chain on (T G , •) with an easy description (Proposition 4.1.15). We can then define the entropic weight of a vertex x ∈ V T G as the probability W (x) that x is in ξ if the RW starts at •. The regularity of the structure of (T G , •) gives us a CLT for (log W (ξ t )) t≥0 (Corollary 4.1.16). Note that in the analogous Proposition 3 of [START_REF] Berestycki | Random walks on the random graph[END_REF] (in this case, the graph is locally a Galton-Watson tree), there is only a law of large numbers and a domination for the variance, and this prevents already to determine the profile of the cutoff window.

It remains to prove, in Section 4.1.3, that the RW on (T G , •) has a positive speed. Precise estimates on the Green function obtained by Lalley [START_REF] Steven | Random walks on regular languages and algebraic systems of generating functions[END_REF] and Nagnibeda and Woess [START_REF] Nagnibeda | Random walks on trees with finitely many cone types[END_REF] show in particular that for fixed x, y ∈ T G , P n T G (x, y) decays exponentially with n. We prove that there exist random times (τ i ) with exponential moments (Proposition 4. 1.19) such that in (T G , •), he(X τ i ) ≥ i and he(X t ) > he(X τ i ) for all t > τ i . Moreover, the trajectory of the RW after τ i is independent of the trajectory until τ i . Hence we can decompose the RW into i.i.d. excursions whose durations have exponential moments. This regularity allows us to prove that (he(X t )) t≥0 and (log W (X t )) t≥0 admit a CLT with nonzero mean (Theorem 4.1. ), see Woess [START_REF] Woess | Random walks and periodic continued fractions[END_REF] for a proof and a detailed study of this one-cycle case. In a nutshell, one can compute that the probability that a RW on G starting at the initial vertex x of -→ e 1 runs through C before running through

C -1 is w( - → e 1 ) × . . . × w( - → e m ) w(e 1 ) × . . . × w( - → e m ) + w( - → e -1 m ) × . . . × w( - → e - 1 1 )
.

If (X t ) t≥0 is a RW on (T G , •), the law of its visits to the copies of x is that of a RW on Z (where each integer represents a copy of x, and each interval between two consecutive integers represents a copy of C) with transition probabilities for all i ∈ Z. Therefore, it is recurrent if and only if the above ratio is equal to 1/2, i.e. (i) ∀i, j ∈ Ω, ν i,j = -ν j,i , (ii) there exists i 0 ∈ Ω such that for all i ∈ Ω, j∈Ω ν i,j =

p(i, i + 1) = 1 -p(i, i -1) = w( - → e
1 if i = i 0 , 0 else, (iii) i,j∈Ω ν 2 i,j Π(i)P(i,j) < ∞,
then the Markov chain is transient (here, we use the convention 0/0 = 0). Looking at conditions (i) and (ii), we can interpret u as a current flow entering at i 0 and spreading to infinity through an electrical network. The condition (iii) states that the kinetic (or electric) energy of the flow is finite.

In our setting, we consider the flow generated by a symmetric NBRW starting at •, i.e. moving at each step to a uniform random child of its current position (hence not regarding the weights w( -→ e )): for all x ∈ (T G , •) of height R ≥ 2, let x 1 be x's parent and x 2 be x 1 's parent, and let

ν x 1 ,x = ν x 2 ,x 1 deg(x 1 ) -1 ,
and set ν x,x 1 = -ν x 1 ,x . For every child x of •, let ν •,x = -ν x,• = 1/deg(•). For all x, y ∈ V T G such that none is the parent of the other, let ν x,y = 0.

Then (i) holds obviously. (ii) is also straightforward. As for checking (iii), note that ν x 1 ,x is the probability that a symmetric NBRW started at • visits x. Hence the sum of the transitions from one generation to the next is always 1, that is, for all R ≥ 0,

x∈∂B(•,R),y∈∂B(•,R+1) ν x,y = 1.
Remark that there exists ε > 0, only depending on G, such that for all neighbours x, y ∈ V T G , π(x)P T G (x, y) > ε. Therefore,

x,y∈V T G ν 2 x,y π(x)P T G (x, y) < 2ε -1 R≥0    (x,y)∈∂ -→ B (•,R) ν 2 x,y    ≤ 2ε -1 R≥0 max (x,y)∈∂ -→ B (•,R) ν x,
ν x 1 ,x ≤ K(1/2) R/| -→ E G | (4.7)
for some positive constant K.

Hence s R ≤ K(1 -w min ) R/| -→ E G | for all R ≥ 0, so that (s R ) R≥0
decays geometrically and the transience is proved. This concludes the proof.

Suppose now that A.4 does not hold. G can be decomposed into a core c(G) satisfying A.4

and "branches" attached to it. c(G) is constructed as follows: erase all vertices x of G such that deg(x) = 1 (call them leaves), and delete the edge attached to x. Perform this process again on the resulting multigraph, and so on, until no more vertex is erased. Denote G the graph obtained by this algorithm. For every oriented edge -→ e of G , change its weight to w( -→ e )/w (x),

Proof. Since the RW on T G is transient, there exists an oriented edge (x 1 , x 2 ) ∈ -→ E T G such that a RW started at x 1 has a probability p > 0 to visit x 2 at its first step and to never come back to

x 1 . Note that this holds with the same value of p if one replaces x 1 (resp. x 2 ) by any x 1 ∈ V T G (resp. any x 2 ∈ V T G ) such that (x 1 , x 2 ) has the same label as (x 1 , x 2 ). Let -→ e be this label.

We claim that A.1, A. 3). Note that we have {(X t ) follows ( -→ e 1 , . . . , -→ e m ) and then remains in

T Xm } ⊆ {∀t ≥ 1, X t ∈ V x }.
Hence the Markov Property of the RW on T G implies that the first m such vertices by increasing height, and y i the child of x i that is on p. Let E i (resp. E i ) be the event that (X t ) t≥0 hits x i (resp. y i ). By Lemma 4.1.14, after hitting x i , the RW has a probability at least C 0 to escape through the child of x i that is not on p, and to never hit y i . Hence by the strong Markov property,

P(∀t ≥ 1, X t ∈ V x ) ≥ (w min ) m × p ≥ w | -→ E G | min p,
P(E i |E i ) ≤ 1 -C 0 . Note that E m ⊆ E m ⊆ E m-1 ⊆ . . . ⊆ E 1 ⊆ E 1 . Therefore, P(∃t > 0, X t = x) ≤P(E m ) ≤P(∩ m i=1 (E i ∩ E i )) ≤P(E 1 ) × m-1 i=1 P(E i |E i )P(E i+1 |E i ) × P(E m |E m ) ≤(1 -C 0 ) m .
The conclusion follows.

As a corollary, we can prove Proposition 4.1.4:

Proof of Proposition 4.1.4. Note that ξ ∩ B(X t , R) = ∅ implies that there exists y ∈ B(X t , R)

such that for some s > t, X s = y, so that we can apply the previous Proposition.

CLTs for the RW on the universal cover

For x, y ∈ V T G such that w(x, y) > 0, let ŵ(x, y) := P(y ∈ ξ|X 0 = x) be the probability that the ray to infinity of a RW (X t ) t≥0 started at x goes through a given neighbour y of x. Note that this quantity only depends on the label of (x, y) (denote it -→ e ), so that one might define ŵ( -→ e ) = ŵ(x, y). Let G be G with weights ( ŵ( -

→ e )) -→ e ∈ -→ E G instead of (w( - → e )) -→ e ∈ -→ E G . Define the Non Backtracking Random Walk (NBRW) on G as a RW (Z t ) t≥0 on - → E G , such that for all - → e 1 , - → e 2 ∈ - → E G and t ≥ 0, P(Z t+1 = - → e 2 |Z t = - → e 1 ) =        ŵ( -→ e 2 ) 1-ŵ( -→ e -1 1 ) if the end vertex of - → e 1 is the initial vertex of - → e 2 and - → e 2 = - → e -1 1 , 0 else. (4.9) Lemma 4.1.14 ensures that for all - → e ∈ - → E G , ŵ( - → e ) > 0 if w( - → e ) > 0.
In particular, by our assumptions on G, and the claim in the proof of Lemma 4.1.14, the NBRW associated to G is irreducible, so that it has a unique invariant probability measure π.

Proposition 4.1.15. (Theorem F' in [START_REF] Nagnibeda | Random walks on trees with finitely many cone types[END_REF]) Let (X t ) be a RW on T G with X 0 = •, and let ρ t := (ξ t , ξ t+1 ) be the t-th upward edge of its ray to infinity ξ. Then (ρ t ) t≥0 is a NBRW on T G .

Its first oriented edge is chosen among those with initial vertex •, with probability given by ŵ.

It remains now to derive a similar result for (log W (X t )) t≥0 , which requires in particular to prove that the RW has a positive speed. Since (X t ) t≥0 is transient, θ i := sup{t ≥ 0 | he(X t ) = i} is a.s. well-defined and finite for all i ∈ N. Let θi := θ i+1 -θ i be the time between the last visits of the walk at level i and at level i + 1, for i ≥ 1. Let θ0 := θ 1 . Note that ξ i = X θ i . We have the following corollary of Theorem 2.5 in [START_REF] Steven | Random walks on regular languages and algebraic systems of generating functions[END_REF] (or Theorem C in [START_REF] Nagnibeda | Random walks on trees with finitely many cone types[END_REF]):

Proposition 4.1.17. There exist C 3 , C 4 > 0 such that for all n ≥ 0 and x ∈ (T G , •),

P n T G (x, x) ≤ C 3 exp(-C 4 n). (4.12)
We deduce the following:

Corollary 4.1.18. There exists C 5 > 0 such that for all i, n ≥ 0, and for all -

→ e ∈ - → E G such that P((X θ i -1 , X θ i ) has label - → e ) > 0, P( θi ≥ n| (X θ i -1 , X θ i ) has label - → e ) ≤ C 5 exp(-C 4 n) (4.13) when i > 0 and P( θ0 ≥ n) ≤ C 5 exp(-C 4 n).
Proof. For i = 0, this is a direct application of the above Proposition 4.1.17. Now, for i ≥ 1, note that the law of (X t ) t≥θ i is that of a RW started at X θ i conditioned on making its first step not towards the parent of X θ i , and then never coming back to X θ i after the start. If A denotes the event that this conditioning happens, then P(A) ≥ w min C 0 by Lemma 4.1.14. Therefore, (4.12) implies that for all n ≥ 1,

P( θi ≥ n) ≤ m≥n P(X θ i +m = X θ i +1 ) ≤ 1 w min C 0 m≥n C 3 exp(-C 4 m) ≤C 5 exp(-C 4 n) for C 5 ≥ C 3 w min C 0 .
We say that t ∈ N is an exit time if the oriented edge (X t , X t+1 ) is upward, has label -→ e * and if X s = X t for all s ≥ t + 1 (recall that -→ e * ∈ -→ E G was arbitrarily picked at the beginning of Section 4.1.3). For i ≥ 1, let τ i be the i-th exit time and -→ e i the corresponding exit edge. Note that the -→ e i 's are exactly the edges of type -→ e in ρ. Let τ 0 := 0 and τ i := τ i+1 -τ i for i ≥ 0 be the i-th renewal interval. Remark that Proposition 4.1.15 implies that τ i < ∞ for all i ≥ 1 a.s.

Let i := (X t ) τ i +1≤t≤τ i+1 be the i-th excursion between two exit times. Let x * be the end vertex of the oriented edge of label -→ e * starting at •. Let i be the projection of i on (T x * , x * ), the subtree of (T G , •) rooted at x * : for every i ≥ 1, there is an isomorphism φ i from (T X τ i +1 , X τ i +1 )

to (T x * , x * ) since X τ i +1 and x * have the same label. Let X t := φ i (X t ) for t ≥ τ i + 1 and define i := (X t ) τ i +1≤t≤τ i+1 . Proof. Note that for all i ≥ 1,

(X τ i +t ) t≥1 law = ( X t ) t≥0 , (4.15)
where ( X t ) is a RW on (T G , •) starting at x * and conditioned on never visiting •. Hence the variables i all have the same distribution.

As for the independence, note that for all i ≥ 1, conditionally on τ 1 , . . . , τ i and X 0 , . . . , X τ i , (4.15) holds. In particular, i is independent from the sigma-algebra σ ( 0 , . . . , i-1 ). From this, we deduce that the i 's are together independent.

We now prove (4.14). Remark that conditionally on the label of the oriented edge (X θ i -1 , X θ i ), θi is independent of σ ( θk ) for all t > 0 by Markov's inequality. Note that for t < C 4 , E[exp(tZ 1 )] is finite, and that for

t → 0, E[exp(tZ 1 )] = j≥0 t j E[Z j 1 ] j! = 1 + Et + O(t 2
). Thus, there exists t 0 > 0 such that 0 < r < 1 where r := E[exp(t 0 Z 1 )] exp(-2Et 0 ), and we have P is irreducible, so that a NBRW on G will go through -→ e after a geometrical time, independently of its starting position).

(θ n/2E ≥ n) ≤ r n/2E . Let ϑ := inf{k ≥ 1, ρ k has type - → e }.
Thus, there exist constants α, β > 0 such that for all m ≥ 0, P( -→ e 1 , -→ e 2 ∈ {ρ 1 , . . . , ρ m }) ≤ α exp(-βm), implying that

P(τ 2 ≥ θ m ) ≤ α exp(-βm).
Hence, for all i ≥ 0, since the ( τ i ) i≥1 are i.i.d.,

P( τ i ≥ n) ≤ P( τ 0 + τ 1 ≥ n) ≤ P(θ n/2E ≥ n) + P(τ 2 ≥ θ n/2E ) ≤ r n/2E + α exp(-β n/2E ).
This concludes the proof.

Recall that W t := -log W (X t ) is the log-weight of the RW. We now prove Theorem 4.1.3.

Proof of Theorem 4.1.3. Let W

( ) i := log W (X τ i ) -log W (X τ i+1 ) for every i ≥ 1,
:= E[W ( ) 1 ]. Clearly, W ( )
1 > 0 a.s., so that h w > 0. Now, cutting the trajectory of the RW into excursions between exit edges, we have

W t = -log W (X τ 1 ) -log W (X t ) + log W (X τr t ) + rt-1 i=1 W ( ) i ,
where r t := max{i ≥ 0, τ i ≤ t} for all t ≥ 0. By Proposition 4.1.19 again, log W (X τ 1 ) + log W (X t ) -log W (X τr t ) = o( √ t) with high probability, so that by Slutsky's Lemma, it is enough to show the existence of h, σ > 0 such that

rt-1 i=1 W ( ) i -ht √ t law → N (0, σ 2 ). Let τ := E[ τ 1 ] ∈ (0, ∞), W ( ) i := W ( ) i
-h w and τ i := τ i -τ . For all λ ∈ R, if we set

p λ := P r t -1 i=1 W ( ) i -hw t τ √ t
≤ λ , we have

p λ = P rt-1 i=1 W ( ) i ≤ h w t τ + λ √ t =P h w r t + rt-1 i=1 W ( ) i τ r t + rt-1 i=1 τ i × rt-1 i=1 τ i t ≤ h w τ + λ √ t =P h w τ 1 + rt-1 i=1 W ( ) i /(h w r t ) 1 + rt-1 i=1 τ i /(τ r t ) rt-1 i=1 τ i t ≤ h w τ + λ √ t .
The sequence (W

( ) i ) i≥1 is i.i.d.
, and E[W ( ) 1 ] = 0. Since r t ≤ t a.s., we have by Doob's maximal inequality:

P(| rt-1 i=1 W ( ) i | ≥ t 3/5 ) ≤ P(sup 1≤j≤t | j i=1 W ( ) i | ≥ t 3/5 ) = o(1). Therefore, rt-1 i=1 W ( ) i = o(t 2/3
) with high probability as t → +∞, and rt-1 i=1 τ i = o(t 2/3 ) for the same reason. In addition, the strong law of large numbers implies that r t /t → 1/τ a.s.

Hence,

1 + rt-1 i=1 W ( ) i /(h w r t ) 1 + rt-1 i=1 τ i /(τ r t ) = 1 + 1 r t rt-1 i=1 W ( ) i h w - τ i τ + R t , (4.16)
where R t = o(t -2/3 ) in probability.

Furthermore, r t -1

i=1 τ i t = 1 + (τr t -t)-τ 1 t = 1 + o(t -2/3
) in probability, according to Proposition 4.1.19. Hence

h w τ 1 + rt-1 i=1 W ( ) i /(h w r t ) 1 + rt-1 i=1 τ i /(τ r t ) rt-1 i=1 τ i t = h w τ + h w τ r t rt-1 i=1 Z i + R t ,
where Z i := W ( ) i hw -τ i τ and R t = o(t -2/3 ) in probability, and

P rt-1 i=1 W ( ) i -hwt τ √ t ≤ λ =P 1 r t rt-1 i=1 Z i ≤ τ λ h w √ t -R t =P 1 √ r t rt-1 i=1 Z i ≤ τ λ h w r t t -R t √ r t =P 1 √ r t rt-1 i=1 Z i ≤ √ τ λ h w + R t ,
where R t = o(1) in probability.

The Z i 's are i.i.d. variables, and we have E[Z 1 ] = 0 and σ 2 Z := Var(Z 1 ) > 0. Indeed by A.3, for a fixed trajectory of ε 1 (hence a given value of W ( ) 1 ), τ 1 can take different values with positive probability, so that Z 1 is not deterministic. And Z 1 has exponential moments by Proposition 4.1.19, so that σ 2 Z is finite. Let r t := t/τ . A consequence of the CLT applied to the series associated to the sequence

(τ i ) i≥1 is that |r t -r t | ≤ t 2/3 w.h.p., so that rt-1 i=1 Z i - r t -1 i=1 Z i ≤ (t 2/3 ) 3/5 ≤ t 2/5 and rt-1 i=1 Z i ≤ t 3/5
w.h.p. by the same argument using Doob's maximal inequality as above (4.16). Moreover, we have w.h.p.

1 √ rt -1 √ r t ≤ 1 √ r t r t rt -1 ≤ t -2/3 and √ r t ≥ t 4/9 . Therefore, 1 r t r t -1 i=1 Z i - 1 √ r t rt-1 i=1 Z i ≤ 1 √ r t - 1 r t × r t -1 i=1 Z i + 1 √ r t rt-1 i=1 Z i - r t -1 i=1 Z i ≤ t -2/3 t 3/5 + t -4/9 t 2/5 = o(1)
w.h.p. so that

P rt-1 i=1 W ( ) i -hwt τ √ t ≤ λ = P   1 r t r t -1 i=1 Z i ≤ √ τ λ h w + R (3) t   where R (3) t
= o(1) with high probability. Applying the CLT to the series associated to the sequence (Z i ) i≥1 concludes the proof, and we have

h T G = h w τ and σ 2 T G = h 2 w σ 2 Z τ .
Note that we could not have applied directly the CLT to Fix x ∈ V n and • ∈ V T G such that the label of • is the type of x. Let (X t ) t≥0 be a RW on (T G , •) starting at the root. We couple (X t ) with a RW (X t ) t≥0 on G n in the following manner: let X 0 = x a.s. For all t ≥ 0, X t+1 is the unique vertex of V n such that there is an oriented edge from X t to X t+1 whose type is the label of (X t , X t+1 ). Clearly, (X t ) is well defined and is indeed a RW on G n .

We define a map φ from the set of oriented paths starting at x in G n to that of oriented paths starting at • in T G : for all m ≥ 1 and p := ( -→ e 1 , .

P({W tn ≤ h T G t n -λσ T G √ t n } ∩ {|he(X tn ) -st n | > t 2/3 n )}) ≥ Φ(λ). (4.21)
For n ≥ 1, define the sets

A n := {W tn ≤ h T G t n -λσ T G √ t n } ∩ {|he(X tn ) -st n | > t 2/3 n } and U n := {y ∈ V T G | A n ∩ {X tn = y} = ∅}. Note that for all R > 0, y∈V T G ,he(y)=R exp(-W (y)) = 1.
Hence for n large enough,

|U n | ≤(2t 2/3 n + 1) exp(h T G t n -λσ T G √ t n ) ≤ log n h T G exp log n + (λ -λ) σ T G 2 h T G log n ≤n exp (λ -λ) σ T G 4 h T G log n . Let B n = {x ∈ V n | A n ∩ {X tn = x } = ∅}. By Remark 4.1.22, |B n | ≤ |U n |, hence d x (t n ) ≥ x ∈Bn (P tn n (x, x ) -π n (x )) ≥P(A n ) -π max n -1 |B n | ≥P(A n ) -π max exp (λ -λ) σ T G 4 h T G log n , t ≤ t 1 .
For j ≥ 1, if the exploration has not been stopped before time t j , set x j := X t j and let α j be the R-ancestor of x j . Reveal the pairings of the yet unmatched half-edges of the non-marked vertices of L R+j . Erase the edges closing a cycle, and place a mark at their endpoints. Extend φ to L R+j+1 , and place a mark at vertices y such that φ(y) ∈ N (β). Let then O j be the (R + 1)-offspring of α j in T . Stop the exploration at time t j if there is a marked vertex in O j .

Else, for t j < t ≤ t j+1 , let X t := φ -1 (X t ) and stop the exploration at time t if X t = α j .

Stop the exploration at time t n if it was still running.

For k ≥ 1, we say that the exploration is k-successful if it does not stop before time k. If the exploration is k-successful, then (X t ) 0≤t≤k is a RW on G n .

Remark that by construction, T is indeed a tree. Informally, we take at each level all possible vertices v such that W (φ(v)) is above a certain threshold, without creating a cycle. In fact, it will turn out in the sequel that by Theorem 4.1.3, (X t ) has a negligible probability to visit before time t n a vertex whose counterpart in T G has a small weight (provided a is small enough), and keeping these vertices in T could create cycles, which we want to avoid. This construction of T depends on β (but it will turn out that the choice of β is not important for the results of this Section).

Remark also that the purpose of stopping the exploration if (X t ) visits α j after x j is to ensure that (X t ) stays at distance ≤ R of its ray to infinity.

We insist that there are two sources of randomness: the matchings of the half-edges in G n and the trajectory of (X t ). Denote P ann the annealed probability on G n and (X t ), and P Gn the quenched probability on (X t ) conditionally on the realization of G n .

Proposition 4.1.23. Fix ε > 0. There exists a < 0 such that for n large enough, P ann (the exploration is t n -successful) ≥ 1 -ε.

Proof. Remark that for every j ≥ 1, if the exploration is t j -successful, then it is not t j+1successful only if:

• (X t ) visits α j between its respective first visits at x j and x j+1 , or

• a cycle appears while matching the (R + 1)-offspring of α j+1 , or

• W (φ(α j )) ≤ n -1 exp(β √ log n)w -(R+1) min , or • t j+1 > t n .
Note in particular that if the third point does not hold, φ(O j ) ⊆ N (β) and no vertex in O j is marked because its counterpart would have a too small weight. For all j ≥ 1, let E j := {the exploration stops at a time t ∈ {t j-1 + 1, . . . , t j }} ∩ {t j ≤ t n }.

It is enough to prove that for a < 0 small enough, for n large enough, there exists J 0 > 1 such that

P ann ({t J 0 ≤ t n } ∪ {∪ J 0 j=1 E j }) ≤ ε.
For n ≥ 1, let J 0 := h -1 s log n + γ √ log n where the constant γ ≤ 0 is such that we have

P(W (ξ J 0 ) > n -1 exp(β √ log n)) ≥ 1 -ε/
P ann t J 0 ≤ t n ≤ ε/4.
Thus, it remains to prove that

P ann (∪ J 0 j=1 E j ) ≤ 3ε/4. (4.23)
The probability that a cycle arises while revealing B(x, R + 3) is at most

∆ R+4 ∆ R+4 | - → E G |n -2∆ R+4 = O (log n) 2C log ∆ /n .
Indeed, the set L k contains at most ∆ k vertices for all k ≥ 0, so that we proceed to at most unmatched half-edges belonging to vertices in T .

∆ + ∆ 2 + . . . + ∆ R+3 ≤ ∆ R+4
If B(x, R + 3) has no cycle, for n large enough, for every y ∈ φ(B(x, R + 2)),

W (y) ≥ w R min > n -1 exp(β √ log n),
so that there is no marked vertex in B(x, R + 2). Thus,

P ann (E 1 ) = o(1).
Now, for j ≥ 1, let E j ⊆ E c j be the event that (X t ) visits α j between its respective first visits at x j and x j+1 , let E j ⊆ E c j ∩ (E j ) c be the event that a cycle arises while revealing O j+1 , and let

E (3) j ⊆ E c j ∩ (E j ) c ∩ (E j ) c be the event that W (φ(α j )) ≤ n -1 exp(β √ log n)w -R min . Remark that we have E j+1 ⊆ E j ∪ E j ∪ E (3) j .
On E j , (X t ) t≥t j visits φ(α j ), a vertex at distance R of its initial position. Thus, by the second part of Proposition 4.1.13 and our choice for R in (4. [START_REF] Amit | Random lifts of graphs[END_REF]

, P ann (E j ) ≤ C 1 exp(-C 2 R) = o (log n) -1
, and this bound is uniform in j.

O(n -1 ∆ R+1 (|L R+j | + |O j+1 |)) = O(∆ R exp(-β √ log n)) = o (log n) -1 .
Hence P ann (E j ) = o (log n) -1 . Again, this is uniform in j.

We have E

(3)

j ⊆ E * , with E * := {∃k ≤ t n , W (X k ) ≤ n -1 exp(β √ log n)w -R min }. Note that E * ⊆ {∃k ≤ t n , d(X k , ξ) ≥ R} ∪ {W (ξ J 0 ) ≤ n -1 exp(β log n)} ∪ {t J 0 < t n }.
By Proposition 4.1.13 and (4.22), a union bound on all 0 ≤ k ≤ t n yields

P ann (∃k ≤ t n , d(X k , ξ) ≥ R) = o(1).
Moreover, by our choice of γ and a, we know that

P ann ({W (ξ J 0 ) ≤ n -1 exp(β √ log n)} ∪ {t J 0 < t n }) ≤ 3ε/8.
Hence, for n large enough, P ann (E * ) ≤ ε/2.

All in all, we have

P ann (∪ J 0 j=1 E j ) ≤ P ann (E 1 ) + J 0 -1 j=1 P ann (E j ∪ E j ∪ E (3) j )
≤ P ann (E 1 ) +

-ν n (V n ) = x ∈Vn P t n n (x, x ) -ν n (x ). Remark that for all a, b, m ∈ R, such that a ≥ b, a -a ∧ m ≤ (a -b) + (b -b ∧ m), so that 1 -ν n (V n ) ≤ x ∈Vn\T P t n n (x, x ) -ν n (x ) + x ∈T P t n n (x, x ) -P t n n (x, x , T ) + P t n n (x, x , T ) -ν n (x )
where P k n (x, x , T ) is the probability that a RW on G n started at x reaches x in k steps without leaving the exploration tree T , and ν n (x ) := P

t n n (x, x , T ) ∧ exp(K √ log n) n
. Note that if the exploration is t n -successful, then (X t ) 1≤t≤t n stays in T . Applying Markov's inequality to Proposition 4.1.23, we get that with probability at least 1

- √ ε, G n is such that P Gn (X t ) 1≤t≤t n leaves T ≤ √ ε. For such G n , x ∈Vn\T P t n n (x, x ) -ν n (x ) + x ∈T P t n n (x, x ) -P t n n (x, x , T ) ≤ √ ε, thus 1 -ν n (V n ) ≤ √ ε + S n with S n := x ∈T P t n n (x, x , T ) -ν n (x )
. We now prove that for n large enough and for any realization of G n that satisfies P Gn (X

t ) 1≤t≤t n leaves T ≤ √ ε, S n ≤ √ ε. (4.24)
Note that for all x ∈ T , we have P

t n n (x, x , T ) = P t n T G (•, φ(x ), T) where P k T G (•, φ(x ), T)
is the probability that a RW on (T G , •) started at • reaches φ(x ) in k steps without leaving T, the subtree of (T G , •) corresponding to T . Thus,

S n ≤ y ∈T G P t n T G (•, y , T) -P t n T G (•, y , T) ∧ exp(K √ log n) n ≤ y ∈T G P t n T G (•, y ) -P t n T G (•, y ) ∧ exp(K √ log n) n ≤ 1 - y ∈T G P t n T G (•, y ) ∧ exp(K √ log n) n ≤ 1 -1 -P Gn P t n T G (X 0 , X t n ) ≥ exp(K log n)/n ≤ P Gn P t n T G (X 0 , X t n ) ≥ exp(K log n)/n .
Since (X t ) t≥0 is independent of the realization of G n , we have that

P Gn P t n T G (X 0 , X t n )≥ exp(K √ log n) n = P ann P t n T G (X 0 , X t n )≥ exp(K √ log n) n = P P t n T G (X 0 , X t n ) ≥ exp(K √ log n) n
where P is the probability associated to (X t ). Let A n be the R-ancestor of X t n . Clearly,

W (A n ) ≤ w -R min W (X t n ) ≤ exp( √ log n)W (X t n ) for n large enough. Hence if K is large enough,
then by Theorem 4.1.3, for large enough n:

P W (A n ) ≤ exp(K √ log n/2)/n ≥ 1 - √ ε.
And for every y, y ∈ T G such that y is in the R-offspring of y, we have that

{A n = y, A n ∈ ξ} ⊇ {X t n = y } ∩ {(X t ) t≥t n does not visit y}.
Since d(y, y ) = R, by Proposition 4.1.13 and the Markov property applied to (X t ) at time t n ,

P(A n = y, A n ∈ ξ) ≥ P t n T G (X 0 , y ) × (1 -C 1 exp(-C 2 R)), so that W (A n ) ≥ (1 -C 1 exp(-C 2 R)) P t n
T G (X 0 , X tn ) P-a.s. Hence, with P-probability at least 1 -√ ε, X t n is such that

P t n T G (X 0 , X t n ) ≤ exp(K √ log n/2)/n 1 -C 1 exp(-C 2 R) ≤ exp(K √ log n) n .
Therefore,

P Gn P t n T G (X 0 , X t n ) ≥ exp(K √ log n)/n ≤ √
- √ ε, G n is such that ν n (V n ) ≥ 1 -2 √ ε.
Since ε > 0 was chosen arbitrarily, this concludes the proof.

b) The last jump for mixing

In this section, we complete the mixing initiated in Corollary 4.1.5, proving a weak version of Theorem 4.1.1, for most starting points of the RW on G n : for all ε > 0, there exists K(ε) > 0 such that if S ⊆ V n is the set of vertices s satisfying |t

(n) s (ε) -h -1 log n| ≤ K(ε) √ log n, then π n (S) → 1 as n → +∞.
We use the fact that w.h.p., G n is an expander: There exists L > 0 such that w.h.p. as n → +∞,

L(G n ) ≥ L. (4.25)
We postpone the proof to the end of the section. In the literature, L(G n ) is usually called the conductance of G n . The largest L such that (4.25) holds is the Cheeger constant of (G n ) n≥0 .

An interesting application of this property is the contraction of L 2 norms: Proposition 4.1.26. There exists κ > 0, only depending G, such that for all n, t ≥ 1, and all

x ∈ V n , V ar πn (k (n) t,x -1) ≤ V ar πn (k (n) t-1,x -1)(1 -κL(G n ) 2 ),
where k

(n) t,x (x ) := P t n (x,x ) πn(x ) for x, x ∈ V n and t ≥ 0, and V ar πn (f ) :

= x∈Vn f (x) 2 π n (x) for f : V n → R.
This is a classical property. Arguments for the proof can be found in Section 2.3 (see in particular (2.11) and (2.12)) and in Theorem 6.8 of [START_REF] Montenegro | Mathematical aspects of mixing times in Markov chains[END_REF]. We stress that it is not necessary for P n to be reversible, as long as it is lazy.

It remains to link the total variation distance and the L 2 distance. By Corollary 4.1.5, for n large enough and for all p ≥ 0, with probability at least 1 -ε, G n is such that:

d x (t n + p) = P t n +p n (x, •) -π n (•) T V ≤ ε + ν n P p n (, •) -π n (•) T V ≤ ε + D ν n P p n (, •) -π n (•) L 2 (πn) ,
for some positive constant D (where ν n P p n (, y) = x ∈Vn ν n (x, x )P p n (x , y) for every vertex y, and recall the definition of d y (t n +p) from ( 4

q ≥ D √ log n, ν n P q n (, •) -π n L 2 (πn) ≤ ε, (4.26)
so that d x (t n + q) ≤ 2ε. This concludes the proof of this weak version of Theorem 4.1.1.

Proof of Proposition 4.1.25. The proof is a corollary from that of Theorem 1 in [START_REF] Amit | Random lifts of graphs: edge expansion[END_REF], which states that there exists δ > 0 such that w. This section is organized as follows: first, we prove that with high probability on G n , the random walk has a high probability to reach a root in O(log log n) steps, uniformly in the starting point (Proposition 4.1.28). Second, we prove that with high probability on G n , for every root x of G n , a RW starting at x has a probability at least 1 -3ε to leave B(x, r) at a vertex y from which the exploration described in Section 4.1.4 a) is t n -successful. This relies on the fact that conditionally on {x ∈ R n }:

To prove this statement, notice first that a cut-exploration from a good context has a large probability to be good.

Lemma 4.1.30. For n ∈ N, let x ∈ R n and y ∈ ∂B(x, r). Suppose that G n has been partially revealed and is a good context for y. Then we have P (the cut exploration from y is good) ≥ 1 -ε.

Proof. One checks that all the arguments in the proof of Proposition 4.1.23 remain valid. In particular, the probability of creating a cycle while matching the (R + 1)-offspring of the Rancestor of the RW in the first t n steps is o(1) and on the event that the exploration is not stopped, the RW has a probability o(1) to visit α(y, x).

Proof of Proposition 4.1.29. Let x be a root. Let α 1 , . . . , α q be the vertices of ∂B(x, r -R),

with q = |∂B(x, r -R)|. For all i ∈ {1, q}, let A i be the set of vertices y ∈ ∂B(x, r) such that α i is on the shortest path from x to y. We say that A i is intact whenever for all y ∈ A i , for all j < i and y ∈ A j , no vertex in the cut exploration from y is matched to y. On the contrary, if such a matching exists, say that the relevant edge is a mismatch. Denote V α the set of vertices α i such that A i is not intact. Denote V mis (resp. B) the set of y ∈ ∂B(x, r) which are not intact (resp. which are intact and whose cut exploration is not good). It is enough to prove that

P x (λ x (B ∪ V mis ) ≥ 3ε) = o(1/n),
where the o(1/n) is uniform in x ∈ R n .

Let I n be the cardinality of V α , and J n the number of mismatches. Clearly, I n ≤ J n a.s. and there exists K > 0, independent of x, such that for large enough n, |∂B(x, r)| ≤ log K n. Hence by Remark 4.1.24, while performing the cut explorations of all y ∈ A i for all i ≤ q, less than n exp(-β √ log n/2) edges are created for n large enough. Edges arise from independent matchings, so that for n large enough, J n is stochastically dominated by a sum of n exp(-β √ log n/2) independent Bernoulli random variables of parameter 2 log K n/n. This entails for all integers U ≥ 1:

P x (I n ≥ U ) ≤ P x (J n ≥ U ) ≤ n exp(-β √ log n/2) U 2 log K n n U ≤ 2 log K n exp(β √ log n/2) U since M N ≤ M N for M, N ∈ N. Letting U = 3 √ log n/β , we obtain P x (I n ≥ U ) = o(1/n),
and this is uniform in

x ∈ R n . But if I n ≤ 3 √ log n/β , by Proposition 4.1.13, λ x (V mis ) ≤ P((X t ) hits a non-intact α i before leaving B(x, r) ) = O log ne -C 2 (r-R) .
For C large enough in the definition of r,

√ log ne -C 2 (r-R) = o(1).
Therefore,

P x (λ x (V mis ) ≥ ε) = o(1/n) uniformly in x ∈ R n . Hence, it remains to prove that P x (λ x (B) > 2ε) = o(1/n).
For i ≤ q, let F i be the σ-field generated by the cut explorations of the vertices in ∪ i j=1 A j , and define

Y i := λ x (B ∩ A i ), Z i := Y i -E x (Y i | F i-1 ) and W i := i j=1 Z j . According to Lemma 4.1.30, for all i ≤ q, E x (Y i | F i-1 ) = 1 {A i is intact} y∈A i λ x (y)P x (B y |F i-1 ) ≤ ελ x (A i ),
where B y = {the cut-exploration from y is not good}. In particular, λ x (B) ≤ ε + W q . And for all i ≥ 0,

(W i+1 -W i ) 2 = Z 2 i+1 ≤ 2(Y 2 i+1 + E x (Y i+1 | F i ) 2 ) ≤ 2λ x (A i+1 ) 2 + 2ε 2 λ x (A i+1 ) 2 ≤ 4λ x (A i+1 ) 2 , so that q-1 i=0 (W i+1 -W i ) 2 ≤ 4 q j=1 λ x (A j ) 2 ≤ 4 max 1≤i≤q λ x (A i ).
By Proposition 4.1.13, max 1≤i≤q λ x (A i ) ≤ C 1 e -C 2 (r-R) . We apply Azuma-Hoeffding's inequality to the martingale (W i ) 1≤i≤q to get

P x (W q ≥ ε) ≤ exp - ε 2 2C 1 e -C 2 (r-R) = o(1/n), so that P x (λ x (B) > 2ε) = o(1/n), the o(1/n) being uniform in all x ∈ R n . This concludes the proof.
From this and the conclusions of 4.1.4 b), we deduce the following: Corollary 4.1.31. For every ε > 0, with high probability on G n , for all x ∈ R n ,

λ x {y | d y ( t n + D log n ) ≤ ε} ≥ 1 -ε,
x ∈ V n , d x (T n ) ≤ ε + sup x ∈Rn d x (T n -cr). (4.28)
Fix a realization of G n . For all x ∈ R n , again by the triangle inequality,

d x (T n -cr) ≤ y∈∂B(x ,r)   2s -1 r =0 P n (x , y)d z (T n -cr -)   + P Gn (τ e (x ) > 2s -1 r)
where τ e (x ) is the hitting time of ∂B Gn (x , r) by a RW started at x . But t → d y (t) is a non-increasing function for all y ∈ V n , so that

d x (T n -cr) ≤ P Gn (τ e (x ) > 2s -1 r) + y∈∂B(x ,r) λ x (z)d y (t n + D log n).

Relaxing the assumptions

We now establish that assumptions A. Remark that we can suppose that C 1 and C 2 don't lie on the same set of non-oriented edges, so that E 1 and E 2 are independent, and the probability that E 1 happens is the probability that a uniform permutation of {1, . . . , n} sends at least k(1 -ε) elements of {1, . . . , k} in {1, . . . , k}.

An estimation of this quantity via a union bound on the choice of the k(1 -ε) elements and Stirling's formula, and a union bound over all possible subsets V ⊆ V n of cardinality at most In general, computing exact values for h T G and s is a difficult problem. Nagnibeda and Woess [START_REF] Nagnibeda | Random walks on trees with finitely many cone types[END_REF] give two formulas for s depending on Green functions. The first one (equation (5.8)) is for all -→ e ∈ -→ E G (Proposition 2.5 in [START_REF] Nagnibeda | Random walks on trees with finitely many cone types[END_REF]), where -→ e ← -→ e means that the end vertex of -→ e is the initial vertex of e . One can establish this by decomposing the trajectory of a RW (X t ) t≥0 as follows: if x, y are such that (x, y) has label -→ e , and if X t = y, then either X t+1 = x, or X t+1 = y for some y ∈ T G \ {x}, which happens with probability w( -→ e ) where -→ e is the label of (y, y ). Now, (X t ) can come back to x in k steps only by reaching y in k ≤ k -1 steps and then reaching x in k -k steps.

s -1 = -→ e ∈ -→ E G π( - → e ) F-→ e -1 (
Letting q(x, y) := F (x,y) (1) the probability that (X t ) reaches y at least once if it starts at x, the transition matrix Q of the NBRW on T G satisfies Q( -→ e , -→ e ) = w( -→ e )(1 -q( -→ e -1 ))

-→ e ← -→ e , -→ e = -→ e w( -→ e )(1 -q( -→ e -1 )) . (4.29) Indeed, if (x, y) and (y, y ) have respective labels -→ e and -→ e , if (X t ) starts at y, in order to leave to infinity through y :

• either it goes through (y, y ) and never returns back from w to v: this has probability w( -→ e )(1 -q( -→ e -1 )),

• or it goes through any (y, y ), comes back to y, and will leave to infinity through y : this has probability -→ e ← -→ e w( -→ e )q( -→ e -1 ) ŵ( -→ e ).

Recall that ŵ( -→ e ) is the probability that (X t ) leaves to infinity through y. Hence ŵ( -→ e ) = w( -→ e )(1 -q( -→ e -1 )) + -→ e ← -→ e w( -→ e )q( -→ e -1 ) ŵ( -→ e ), thus ŵ( -→ e ) = w( -→ e )(1 -q( -→ e -1 ))

1 --→ e ← -→ e w( -→ e )q( -→ e -1 ) .

Recall that Q( -→ e , -→ e ) = ŵ( -→ e )

1-ŵ( -→ e -1 ) , from which we derive the formula (4.29). Note that -→ e ← -→ e w( -→ e )q( -→ e -1 ) < 1 since q( -→ g ) < 1 for all -→ g ∈ -→ E G . Then, it remains to compute the unique invariant probability measure of Q.

Knowing (F-→ e (1)) -→ e ∈ -→ E G and π, one can compute s. However, those quantities are the solutions of non-linear systems of equations, for which no explicit general solutions have been found.

Even for the seemingly simple case where G has only one oriented cycle (and the inverse cycle), continuous fractions are involved to derive non simple expressions for the F-→ e 's in [START_REF] Woess | Random walks and periodic continued fractions[END_REF]. The second formula for the speed in [START_REF] Nagnibeda | Random walks on trees with finitely many cone types[END_REF] is derived from a powerful theorem in [START_REF] Sawyer | The rate of escape for anisotropic random walks in a tree[END_REF], and involves again those series. For the SRW on periodic trees, Takacs obtains similar equations and gives explicit solutions for three particular cases (examples 4.7, 4.8 and 4.10 in [START_REF] Takacs | Random walk on periodic trees[END_REF]). hence once one manages to compute s, computing h T G is straightforward (recall that h T G = sh W ).

Gilch [START_REF] Lorenz | Asymptotic entropy of random walks on regular languages over a finite alphabet[END_REF] studies transient random walks in the more general context of regular languages, and obtains a law of large numbers:

-log P t T G (X 0 , X t )/t a.s.

→ h (4.31)

for some positive h if the random walk is transient. He proves that h is an analytic function of the weights in G and discusses the possibilities to compute the entropy. He obtains a formula that reduces to (4.19) in our particular context: h is given as the product of three factors, h = λ -1 h(Y ) (Theorem 2.5), where in our particular setting, λ = 1 is the expected distance between X θ i and X θ i+1 , h(Y ) = h W and = s is the rate of escape, whose computation in [START_REF] Lorenz | Rate of escape of random walks on regular languages and free products by amalgamation of finite groups[END_REF] is the equivalent of that of [START_REF] Nagnibeda | Random walks on trees with finitely many cone types[END_REF] for regular languages.

Note that in our setting, (4.31) can be deduced from Kingman's subadditive ergodic theorem.

We have P t T G (X 0 , X t ) ≥ P s T G (X 0 , X s )P t-s T G (X s , X t ) a.s. for all 0 ≤ s ≤ t. Moreover, if the label of X 0 is distributed according to π (hence, so is the label of X r for all r ∈ N), the random variables

P t-s
T G (X s , X t ) and P t-s T G (X 0 , X t-s ) have the same distribution that only depends on t -s, and we can apply Kingman's theorem to obtain (4.31). Since π is strictly positive on all labels and the convergence is a.s., the result holds if the label of X 0 is chosen arbitrarily. Note that this does not imply that h > 0.

Remark 4.1.32. Comparing the formula in [START_REF] Lorenz | Asymptotic entropy of random walks on regular languages over a finite alphabet[END_REF] with (4.19), we get h = h T G . However, one can prove this equality without using the results of [START_REF] Lorenz | Asymptotic entropy of random walks on regular languages over a finite alphabet[END_REF]. The fact that h T G ≤ h is a consequence from the ray localization of Proposition 4.1.4. Indeed, for all ε > 0 w.h.p. as t goes to infinity, P t T G (X 0 , X t ) ≥ exp(-(h + )t), and (X n ) n≥t has a probability o(1) to visit the (log log t)-ancestor of X t (the o(1) is uniform conditionally on X t , by Proposition 4.1.4), so that w.h.p., this ancestor and X t itself have a probability at least exp(-(h + 2ε)t) to be in the ray to infinity, so that

h T G ≤ h + 2ε.
Conversely, by (4.31), there exists a set S t of at least exp((h -ε)t) vertices, such that X t ∈ S t w.h.p. as t → +∞ and such that for all x ∈ S t , exp(-(h + ε)t) ≤ P t T G (X 0 , x) ≤ exp(-(h -ε)t). By (4.18), we can impose that st -εt 2/3 ≤ he(x) ≤ st + εt 2/3 . The fact that the vertices of S t are localized in less than 3εt 2/3 consecutive levels implies that we can decompose S t = m i=1 S m for some m ≥ |S t |/∆ 3εt 2/3 ≥ exp((h -2ε)t) for t large enough, where S i is a subtree of (T G , •) with some root x i , and of height less than 3εt 2/3 . x i maximizes W (x) for x ∈ S i , which implies that W (x i ) ≥ exp(-(h T G -ε)t) for all i ≤ m, and that 1≤i≤m W (x i ) ≥ m exp(-(h T G -ε)t).

Moreover, we can impose that x i is in the offspring of no other x j , so that 1≤i≤m W (x i ) ≤ 1.

From this, we deduce that h T G ≥ h -3ε. Further details are left to the reader.

Appendix 2: is laziness necessary?

It is easily checked that the fact that α > 0 is not necessary except for the proof of Proposition A sufficient condition to guarantee that the results of [START_REF] Montenegro | Mathematical aspects of mixing times in Markov chains[END_REF] required for the proof of Proposition 4.1.26 hold would be that there exists c > 0 such that for all n large enough, inf f,V arπ n (f )=1

E P * n Pn (f, f ) ≥ c inf f,V arπ n (f )=1 E Pn (f, f ) (4.32)
where E Pn (f, f ) := 1 2

x,y∈Vn (f (x) -f (y)) 2 P n (x, y)π n (x) and P * n (x, y) := πn(y) πn(x) P n (y, x), and E P * n Pn is defined analogously (note that π n is invariant for P n and P * n P n ). This is clearly true for α > 0: for all x, y ∈ V n , P * n P n (x, y) ≥ P * n (x, y)P n (x, x) ≥ αP * n (x, y), hence E P * n Pn (f, f ) ≥ αE Pn (f, f ) for all f (note that E Pn (f, f ) = E P * n (f, f ) for all f ). -→ E n ), defined as in (4.9).

It is irreducible by (4.8). We suppose that it is also aperiodic. One checks easily that P is bistochastic, so that its invariant measure is the uniform distribution. The same holds for P n .

We We denote ρ the pairing of the half-edges of -→ E n . We will abusively identify each half-edge u with the oriented edge starting at u. We define t n,u (ε), t min n (ε) and t max Note that σ 2 = 0 iff τ -1 u τ-→ e -1 t=0 log(deg(Z t )) is constant a.s., so that many graphs G satisfy A.5. For instance, it is enough that there are two cycles containing -→ e on which the average log-degree is not the same.

Without loss of generality, we impose σ > 0. Second, we prove that starting from a bulb, the probability that the NBRW is not on a root after r steps is o(1), where the speed of convergence to 0 only depends on G (in particular, it neither depends on the choice of the initial bulb, nor on the lift structure).

Let

p r ≤ 2 -r/| -→ E G | .
This yields (4.41) and the conclusion follows.

Proof of Proposition 4.2.6

Consider two distinct half-edges x, y. Recall the sequential construction of G n from Lemma 4.1.6. Pairing half-edges one after another, we grow a couple of trees T x and T y , starting from

x and y respectively. These trees will have final height t/2 (recall that t is even), and are made of non-backtracking paths. We then estimate the probability that at least one half-edge of T x is matched with one of T y , and this yields a lower bound of P t n (x, y). This procedure is called "exposure process". It was already used in [START_REF] Lubetzky | Cutoff phenomena for random walks on random regular graphs[END_REF] and then [START_REF] Ben | Cutoff for nonbacktracking random walks on sparse random graphs[END_REF], which we follow more closely.

A noticeable difference with [START_REF] Ben | Cutoff for nonbacktracking random walks on sparse random graphs[END_REF] is that one has to pay attention to the type of the leaves since one can match two half-edges if and only if their types correspond.

The construction

At the start, all half-edges are unpaired, and T x and T y are reduced to x and y respectively. A move consists of three steps:

Step 1: pick an unpaired half-edge u ∈ T , where T := T x ∪ T y , of height less than t/2 and of maximal weight. The height of u in T is the height of its vertex in T x or T y . The weight of u, denoted (u), is the weight of the path from x or y to u, depending on whether u ∈ T x or u ∈ T y . Choose any total order of the half-edges to break possible ties.

Step 2: match u with a random (uniform) unpaired half-edge v.

Step 3: add the vertex of v and all its half-edges in T , if none of them was already in T . This condition ensures that both T x and T y will not contain any cycle.

The construction ends when no half-edge remains available for Step 1.

After the construction, let F be the set of remaining unpaired half-edges in T , called leaves of the trees. A leaf u is said to be good whenever (u) > n -3/4 . Let L x and L y be the sets of good leaves of T x and T y . Denote H the set of the other leaves. where θ := 1 n log 2 n . We claim that it is enough to prove the following three Lemmas. Fix indeed ε > 0, and suppose that the conclusions of these Lemmas hold. Note that the o(1/n 2 ) is uniform over the pairs x, y ∈ R n in every Lemma (since the events considered might only depend on the types of x and y, for which we have no more than | -→ E G | choices). Hence, by a union bound, these conclusions hold w.h.p. for all pairs x, y ∈ R n . Then

P t n (x, y) ≥ n -1    -→ e ∈ -→ E G Υ - -→ e -ε    ≥ n -1    -→ e ∈ -→ E G Υ - -→ e + -→ e ∈ -→ E G Υ + -→ e - Φ(λ) | - → E G | + ε -ε    ≥ n -1    -→ e ∈ -→ E G Υ - -→ e + -→ e ∈ -→ E G Υ + -→ e    - Φ(λ) N - 2| - → E G |ε N
by Lemmas 4.2.7 and 4.2.9. Moreover,

-→ e ∈ -→ E G Υ + -→ e + Υ - -→ e = -→ e ∈ -→ E G    u∈L - → e x (u) v∈L - → e y (v)    ≥ | - → E G | 1 | - → E G | 2 -ε ≥ 1 | - → E G | -ε
by Lemma 4.2.8. This yields

P t n (x, y) ≥ 1 N - | - → E G |ε N - Φ(λ) N - 2| - → E G |ε N ≥ 1 -Φ(λ) N - 3| - → E G |ε N .
Since ε > 0 was arbitrary, the conclusion follows.

The remaining subsections are devoted to establishing these Lemmas.

Proof of Lemma 4.2.7

Fix ε > 0. We use Stein's method on concentration inequalities for exchangeable pairs, as presented in [START_REF] Chatterjee | Stein's method for concentration inequalities[END_REF]. We need the weights to be not too large, and this is the reason why we needed to introduce Υ - -→ e and Υ + -→ e . Since good leaves have height t/2, we have Therefore, it only remains to establish (4.48).

Proof of (4.48). Take δ = ε/4. We split the proof into two natural parts: first, we prove that

P u∈H (u) ≥ δ = o 1 n 2 .
Second, we show that for n large enough. We treat u∈Hy (u) the same way, and the conclusion follows.

It remains to prove (4.49), which we do by introducing a martingale of which we control the first and second moments.

Suppose that x, y ∈ R n . Let τ be the random number of moves in the exposure process, and (F j ) j≥0 be the natural filtration associated to this process: τ is a stopping time with respect to this filtration. Let U j be the set of unpaired half-edges in T after j moves for j ≥ 0, and K j := u∈U j (u). Observe that the sequence (K j ) j≥0 is non-increasing, and that we have K 0 = K r = 2 since x and y are roots. Then, when we pick an unpaired half-edge u ∈ T and match it with another half-edge v, it might happen that v is also in T . Thus, we lose (u) and (v). Formally, for j ≥ r + 1,

K j = K j-1 -1 j≤τ 1 v j ∈U j-1 ( (u j ) + (v j )) , (4.50)
where u j is the unpaired half-edge picked at the j-th move and v j is the half-edge to which it is matched. Remark that (u j ) is F j-1 -measurable. It holds:

E [K j -K j-1 |F j-1 ] = -1 j≤τ   (u j )P(v j ∈ U j-1 |F j-1 ) + v∈U j-1 (v j )P(v j = v|F j-1 )   ≥ -1 j≤τ (u j )|U j-1 | + K j-1 n -2j + 1 .
Indeed, 2j -2 half-edges have been paired, so that there are still at least n -2j + 1 unpaired half-edges (other than u j ) of each type. Hence, for every v ∈ U j-1 , P(v j = v|F j-1 ) ≤ We also have (K j -K j-1 ) 2 = 1 j≤τ 1 v j ∈U j-1 ( (u j ) + (v j )) 2 , hence E (K j -K j-1 ) 2 |F k-1 ≤ 1 j≤τ (u j ) 2 |U j-1 | + 2 (u j )K j-1 + v∈U j-1 (v) 2 n -2j + 1 .

Since half-edges are selected in decreasing order of weight, for every u ∈ U j-1 , its parent has weight at least (u j ), thus (u) ≥ (u j ) ∆ . But the total weight of the elements of U j-1 is K j-1 ≤ 2, so that |U j-1 | ≤ 2∆ (u j ) . Note also that (v) 2 ≤ (v) for any v. Hence

E [K j -K j-1 |F k-1 ] ≥ -1 j≤τ 2∆ + 2 n -2j + 1
, and

E (K j -K j-1 ) 2 |F k-1 ≤ 1 j≤τ 2∆ (u j ) + 6 n -2j + 1 .
We sum those inequalities for j = 1, . . . , τ . Observe that the sum of the weights of all the half-edges u 1 , . . . , u τ that were once selected in the exposure process is at most t + 2, since there are t/2 + 1 generations on each tree T x , T y , the total mass on each generation being at most 1. Together with (4.54), where we can take γ > 0 arbitrarily small, this yields the result.

The irreducibility of P and the existence of a half-edge with degree at least ∆ -1 ≥ 2 imply that every non-backtracking path on G meets an oriented edge with degree at least 2 every | -→ E G | steps at most. The same holds on the balls of radius r around x and y, which are included in T x and T y respectively since x, y ∈ R n . This divides the weight of each path in T by at least 2 after at most | -→ E G | steps. Hence, for n large enough, for every j ≥ r, max( (u j ), (v j )) < γ/2 almost surely, so that max j≤τ |K j-1 -K j | ≤ γ (recall that K 0 = . . . = K r = 2). This yields (4.55).

Proof of Lemma 4.2.9

We proceed as in the first part of the proof of (4.48). Fix ε > 0 and -→ e ∈ -→ E G . Let (Y j ) j≥1 and (Z j ) j≥1 be independent NBRWs starting at x and y respectively. Let E be the event that Y t/2 is of type -→ e , that Z t/2 is of type -→ e -1 , and that where we recall that r = log log n. The proof of Lemma 4.1.8 in [START_REF] Lai | Markov chains with stationary transition probabilities[END_REF], which relies on an excursion decomposition of the trajectory of the random walk, allows the following slight modification: in the random walk (X t ) 0≤t≤n , we can replace (X t ) n/2≤t≤n by a random walk (X t ) 0≤t≤n/2+1

starting at an arbitrary state in Ω. We apply this version of the Lemma by concatenating the trajectories of (Y j ) 1≤j≤t/2-r and (Z j ) 1≤j≤t/2-r .

Diameter of random lifts

In this section, we compute the oriented diameter D n of G n . For n ≥ 1, we set It remains to prove that this lower bound is optimal. We proceed in three steps. Fix ε > 0.

First, we show that w.h.p., for every x ∈ V n , the ball B(x, R n ) for R n := (1/2-ε) log -1 D log n has no more than 2/ε + 1 cycles, so that its successive generations grow exponentially at rate log D, as B(•, R) when R → +∞. Second, for y ∈ B(x, R n ), we build similarly the "adjoint" ball B * (y, R n ), made of the oriented paths reaching y. The corresponding adjacency matrix is the transpose of A G , so that it has the same Perron-Frobenius eigenvalue D. Third, we use the expansion to connect B(x, R n ) to B * (y, R n ) in K G ε log n steps, K G being a constant uniquely depending on G.

Step 1 Let R n := (1/2 -ε) log -1 D log n . We claim that w.h.p., G n is such that for every for n large enough, by Lemma 4.3.3. Hence, the probability that more than k ε cycles appear during those pairings is less than

x ∈ V n , n 1/2-2ε ≤ |∂ B(x, R n )| ≤ n 1/2-ε/2 . ( 4
n 1/2-ε/2 n -n 1/2-ε/2 kε n 1/2-ε/2 k ε ≤ 2 kε n (1/2-ε/2)2kε n -kε ≤ n -3/2 .
Note indeed that at each step of the construction, at most n 1/2-ε/2 unpaired half-edges have been discovered and at least n -n 1/2-ε/2 half-edges of each type have not been paired. This yields (4.58).

Then, let T x be a spanning tree of B(x, R n ): when building B(x, R n ), do not include in T x the edges that close a cycle. Step 2 Let y ∈ V n \ B(x, R n ) and v * (y, R n ) be the ball centred at y made of the oriented paths of length R n arriving at y. Define similarly B * (•, R n ). We can readily adapt Lemma 4.3.3 and thus (4.57), A G being replaced by its transpose A * G , the latter having the same Perron-Frobenius eigenvalue.

Step 3 If B * (y, R n ) intersects B(x, R n ), then there exists an oriented path from x to y of length at most 2R n ≤ log n log D . Suppose that this is not the case. We expand B(x, R n ) until we hit B * (y, R n ).

Let K G := Lπ min /2, where we recall that π min is the smallest weight of the invariant probability distribution π of the SRW on G, and where L was defined in (4.25). By Denote respectively A y and A x these sets of vertices. Now, we build a path of length at most | -→ E G | from A x to A y via a binomial argument. Let z 1 , z 2 , . . . be the vertices of A x , listed in an arbitrary order. For i = 1, 2, . . . successively, if no oriented path has been discovered from some z j , j < i to A y , reveal one path from z i to some oriented edge of type -→ e (such a path exists by (4.8) applied to -→ e a = -→ e and -→ e b = -→ e ). The probability that this path ends on A y is at least n 1/2-3ε /n. Therefore, the probability that for some i ≥ 1, one of these oriented paths lands on

A y is at least 1 -P(Z = 0), where Z ∼ Bin( n 1/2+5ε , n -1/2-3ε ).

But for n large enough, P(Z = 0) = (1 -n -1/2-3ε ) n 1/2+5ε ≤ exp( n 1/2+5ε log(1 -n -1/2-3ε )) ≤ exp(-n ε ) ≤ n -3 .

By a union bound on x, y ∈ V n , the probability that such a connection does not exist for some

x, y ∈ V n is o(1).

All in all, we have shown that w.h.p., G n is such that for every x, y ∈ V n , there is an oriented path of length at most 2R n + R n ≤ log -1 D log n + 10ε log -1 (1 + K G ) log n from x to y. Since ε > 0 can be taken arbitrarily small, this concludes the proof of the upper bound and hence of Theorem 4.3.1.

3. 3

 3 The Gaussian Free Field on T d . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 C h • as a branching process . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Exponential growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Approximate recursive construction of ψ Gn . . . . . . . . . . . . . . . . . . . . . . 3.5 Exploration of ψ Gn around a vertex . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.1 Successful exploration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.2 Aborted exploration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Existence of a giant component . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.1 Connecting two successful explorations . . . . . . . . . . . . . . . . . . . . 3.6.2 Average number of connections in E ≥h ψ Gn . . . . . . . . . . . . . . . . . . . 3.7 Uniqueness of the giant component . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7.1 Lower bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7.2 First phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7.3 Second phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7.4 Third phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7.5 Revealing ψ Gn on the three phases . . . . . . . . . . . . . . . . . . . . . . 3.8 Properties of C (n) 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8.1 The local limit: proof of Theorem 3.1.3 . . . . . . . . . . . . . . . . . . . Chapter 1

Theorem 2 . 1 . 1 .

 211 Fix α ∈[START_REF] Abächerli | Local picture and level-set percolation of the gaussian free field on a large discrete torus[END_REF][START_REF] Abächerli | Level-set percolation for the gaussian free field on a transient tree[END_REF]. Then under the conditions above, there exists a sequence of random compact measured metric spaces (G 1 , d 1 , µ 1 ), (G 2 , d 2 , µ 2 ), . . . (whose law depends on α) such that, as n → ∞,

  [START_REF] Addario-Berry | The continuum limit of critical random graphs[END_REF].)Conditionally on the ordered lengths ζ 1 , ζ 2 , . . . of the excursions and numbers M 1 , M 2 , . . . of Poisson points, we may give an attractive alternative description of the excursions encoding the spanning forests of the α-stable and Brownian graphs. These are closely related to the canonical family of random R-trees encompassing the Brownian continuum random tree[START_REF] Aldous | The continuum random tree[END_REF][START_REF] Aldous | The continuum random tree. II. An overview[END_REF][START_REF] Aldous | The continuum random tree[END_REF] and α-stable trees[START_REF] Duquesne | Random trees, Lévy processes and spatial branching processes[END_REF][START_REF] Duquesne | A limit theorem for the contour process of conditioned Galton-Watson trees[END_REF], which are the scaling limits of critical Galton-Watson trees conditioned to have size n with offspring distribution in the domain of attraction of a Normal or α-stable distribution respectively.

Theorem 2 . 1 . 2 .

 212 Conditionally on the lengths ζ 1 , ζ 2 , . . . of the excursions and the numbers

1 , G ER,n 2 ,

 12 for the Erdős-Rényi random graph, G(n, p) at its critical point. More precisely, consider the graph G ER n obtained by taking n vertices and connecting any pair of them by an edge independently with probability p = 1/n. Write G ER,n . . . for the components listed in decreasing order of size. Define L and R as in (2.4) with β = µ = 1, let ζ 1 , ζ 2 , . . . be the lengths of the excursions of R and let M 1 , M 2 , . . . be the numbers of points of a Poisson process of intensity 1 falling in each excursion.

Aldous' work in

  order to prove convergence to the β = µ = 1 Brownian graph, in the sense of an 4 version of the Gromov-Hausdorff distance. (It is straightforward to improve this to a convergence in an 4 version of the Gromov-Hausdorff-Prokhorov distance, which appears as Theorem 4.1 of Addario-Berry, Broutin, Goldschmidt & Miermont [7].)

  ii) D n d -→ D as n → ∞, along with the convergence of its first two moments, for some random variable D with P (D = 1) > 0, E [D] = µ and E [D(D -1)] /E [D] = θ > 1

E

  * = E ∪ {∂}, where the extra state, ∂, represents the empty excursion, with ζ(∂) = 0.

  (t) as n → ∞. Moreover, the sequence of random variables (Φ(n, tn α α+1 )) n≥1 is uniformly integrable.

Lemma 2 . 4 . 5 .

 245 Let L(λ) := E [exp(-λD 1 )]. Then as λ → 0,

Figure 2 . 1 :

 21 Figure 2.1: For simplicity, the labels given are those corresponding to the depth-first order.

Figure 2 . 2 :

 22 Figure 2.2: Top: three components of F n (ν). Left: X n (k) drawn with the vertices on the stack indicated. Empty squares represent half-edges which are available to be connected to as back-edges. Filled squares are marked vertices. Right: the corresponding height process (with the extra vertices in red).

Lemma 2 . 6 .

 26 [START_REF] Berestycki | Random walks on the random graph[END_REF]) is equal to (1 + o(1)) times 11 gives us the requisite uniform integrability in order to now deduce the result from Theorem 2.2.5 and the continuous mapping theorem.Let us briefly sketch how this result gives a scaling limit for a single component of M n (ν) or G n (ν) conditioned to have size xn α/(α+1) . First note that for any > 0 there exists a time T > 0 such that any component of size xn α/(α+1) is discovered before time T n α/(α+1) with probability exceeding 1 -, uniformly in n sufficiently large. Any such component discovered before time T n α/(α+1) corresponds to a tree of size ≈ xn α/(α+1) in the forest encoded by S n and G n and, indeed, this tree is asymptotically indistinguishable in the Gromov-Hausdorff-Prokhorov sense from a spanning tree of the graph component. The locations of the back-edges can then be handled in exactly the same way as in the unconditioned setting.

  3.2). On the other hand, the probability to create a cycle is o(1) as long as we reveal o( √ n) vertices (since we perform o( √ n) pairings of half-edges having each a probability o( √ n)/n to involve two already discovered vertices). Thus, with P ann -probability η(h) + o(1), T x and ∂T x will reach a size Θ( √ n log -κ n) for some constant κ > 0 (Proposition 3.5.1).

x|

  = o( √ n) with P ann -probability 1 -η(h) + o(1), we deduce in the same way that at least (1 -η(h) + o(1))n vertices are in connected components of size o( together force the existence of a connected component of size (η(h) + o(1))n.

  R). If x and y are neighbours, we denote B G (x, y, R) the subgraph of G obtained by taking all paths of length R starting at x and not going through y. The tree excess of a finite graph G is tx(G) = e -v + 1, where v := |V | and e is the number of edges in G.An important remark (in particular for Proposition 3.4.1) is that for any subgraph A of G and R ∈ N, tx(B G (A, R)) ≥ tx(A), with equality if and only if B G (A, R) has the same number of cycles and the same number of connected components as A. Note also that if G is connected, tx(G) = 0 if and only if G is a tree, i.e. has no cycle.

Lemma 3 . 2 . 2 (

 322 Binomial number of connections). Let m ∈ N, let W 0 , W 1 be disjoint subsets of V n . Write m 0 := |W 0 | and m 1 := |W 1 |. Suppose the only information we have on G n is a set E of its edges that has been revealed. Let m E := |E| and denote P E the law of G n conditionally on this information. Repeat the following operation m times: pick an arbitrary vertex v ∈ W 0 having at least one unmatched half-edge, and pair it with an other half-edge.

  .14) b) for k = k(n) → +∞ and n large enough, if we have m E + m < n/2 and kn > 6(m 1 + m 0 + m E )m, then P E (s ≥ k) ≤ 0.99 k . (3.15)

  99 k , and (3.15) follows.

m 1

 1 is replaced by m 0 + m in (3.13) and (3.14), and (3.15) does not change.

  2.1 and (3.11), w.h.p. G n satisfies (3.21).

Proposition 3 . 3 . 1 (

 331 Recursive construction of the GFF,[START_REF] Abächerli | Level-set percolation of the Gaussian free field on regular graphs I: regular trees[END_REF]). Define a Gaussian field ϕ on T d as follows: let (ξ y ) y∈T d be a family of i.i.d. N (0, 1) random variables. Let ϕ(•):= d-1 d-2 ξ • . For every y ∈ T d \ {•}, define recursively ϕ(y) := d d-1 ξ y + 1 d-1 ϕ(y), where y is the parent of y. Thenϕ d. = ϕ T d .Proposition 3.3.1 is the corollary of a more general domain Markov property (see for instance Lemma 1.2 of[START_REF] Rodriguez | Phase transition and level-set percolation for the gaussian free field[END_REF] where it is stated for Z d , but the proof can readily be adapted to any transient graph). Let G be a transient graph, G G the Green function on it and ϕ G the associated GFF.

  d and U = T y the subtree from y in T d , for every y ∈ T d \ {•}, to get Proposition 3.3.1. See [3], (1.4)-(1.9) for details. Write P T d for the law of ϕ T d , and P T d a for P T d ( • |ϕ T d (•) = a), a ∈ R (such conditioning is well-defined, (ϕ T d (x)) x∈T d being a Gaussian process). This construction gives a monotonicity property for ϕ T d . A set S ⊆ R T d is said to be increasing if for any (Φ (1)

T

  d as in Proposition 3.3.1. Then for every z ∈ T d of height k ≥ 0, ϕ

Figure 1 .

 1 Figure 1. The unicity of y comes from the fact that when building B Gn (A, a n ) from A, no cycle appears and no connected components of A join, since tx(B Gn (A, a n )) = tx(A).

  3 in [140] and of the following observation. If (X j ) j≥0 is a SRW on T d , then the trajectory of its height (h T d (X j )) j≥0 is distributed as a random walk on N ∪ {0} with transition probabilities 1/d towards the left neighbour and (d -1)/d towards the right neighbour, and reflected at 0. Lemma 3.4.2 (Geometric repulsion). Let s ∈ N, and A ⊆ V n such that tx(B Gn (A, s)) = tx(A). Let x ∈ V n \ B Gn (A, s), let (X j ) j≥0 be a SRW started at x and τ its first hitting time of A. Then τ dominates stochastically a geometric random variable of parameter (d -1) -s . Proof of Proposition 3.4.1. Let us first prove (3.34). By Proposition 3.2.3,

  and• the conditions of Proposition 3.4.1 hold for each vertex of T x , until a generation at which|∂T x | ≥ n 1/2log -κ-6 n (which happens with probability 1 -o(1), mainly because we have a probability o(1) to create a cycle when pairing o( √ n) half-edges), we can apply Proposition 3.4.1 to bound the difference between ϕ T d and ψ Gn by log -1 n, ensur-ing that T x ⊆ C Gn,h x , the connected component of x in E ≥h ψ Gn . The exploration. Fix x ∈ V n . Let b n := (d -1) -an log -6 n,(3.46)where we recall the definition of a n from(3.33). Let (ξ y ) y∈T d be a family of independent variables, each of law N (0, 1), independent of the pairing of the half-edges in G n . Define the GFF ϕ T d as in Proposition 3.3.1.

'Figure 2 .

 2 Figure 2. Illustration of the exploration with k = 3, a n = 2, h = 0.3 and n = 148 (so that log -1 n 0.2). Thick vertices and edges represent T x after two steps. Red vertices have not been included in T x because ϕ T d at their counterparts in T d is below h + log -1 n. Any number

. 48 ) 3 . 5 . 2 (

 48352 Remark Exploration size). Denote R x the set of vertices seen during the exploration (i.e. such at least one of their half-edges has been paired). Note that for n large enough, for every x ∈ V n , by C2, C4 and (3.46), T x contains less than

Proposition 3 . 5 . 4 .

 354 P ann (A(x)) -→ n→+∞ 1 -η(h).

  Write J := {j ≤ |∂T x |, B * (z j , a n ) is a joining ball} and S (x, y) := S(x, y)∩{|J| ≥ log γ-3κ-18 n} (S(x, y) was define above (3.58)).

Figure 3 .

 3 Figure 3. Illustration of a joining ball B * (z j , a n ). Here a n = 2 and a n = 9. Dashed lines represent subtrees that have not been fully pictured. The blue tree has total height a n , the red trees a n -3a n , and the green trees 2a n .

  for some constant a > 0. More precisely, we give a mark to each vertex y such that | ψ Gn (y) -h| ≤ n -a , and explore each connected component C of C \ M, where M is the set of marked vertices, until -(i) C is fully explored and has no more than O(log n) vertices, or -(ii) K log n vertices of C have been seen but not yet explored, for some constant K fixed in the second phase.

1 Figure 4 .

 14 Figure 4. Case 1. Marked vertices are in red. C consists of the thick edges. T y and T y are delimited by the purple contours. Remark that we could have y = y (it is not the case here).

{y 1 , 1 Figure 5 .

 115 Figure 5. Case 2. Marked vertices are in red.

  T d ) on less than d vertices. And for any vertex y ∈ T d \ {•}, by Proposition 3.3.1, max a ≥h

4 Figure 6 .

 46 Figure 6. Marked vertices are in red. There are four fertile subexploration trees (T 1 rooted at y 1 , T 2 rooted at y 2 , T 3 rooted at x and T 4 rooted at y 4 ), and two infertile ones. Lightgray

6 . 1 .

 61 Since there are less than log 2 n w i 's (and |R 1,2 | is controlled by (3.94)), we can replace (3.61) by less than n 1/2 log γ+2 n vertices are seen during the three phases. (3.98) Let B * := ∪ m q=1 ∪ z∈Bq B * (z, a n ). (3.63) becomes

  0≤k≤log γ-3κ-18 n P(Z = k) by (3.99), (3.100) and (3.101) ≤ 4n -5 .

4 . 1

 41 corresponds to Case 1 and Case 2 of the first phase of the exploration. Let us introduce some notations before stating the Lemma. Recall that for m ≥ 3, G m is the connected (and infinite) d-regular graph with a unique cycle C m of length m. Recall the definitions of α k , β k , γ k , α k and γ k from (3.86) and (3.87).

. 121 )

 121 Then(3.4) follows by a second moment argument as in Lemma 3.6.3. Hence, it is enough to prove (3.120) and(3.121).

For simplicity of the

  notations, write V n , E n and -→ E n for the vertex set, edge set and oriented edge set of G n . Define as previously the type of an element of -→ E n as the corresponding oriented edge of -→ E G , and give to each -→ e ∈ -→ E n the weight of its type. Denote π the invariant probability measure of the RW on G (if it exists). Note that the RW on G n has an invariant measure π n such that π n ((x, i)) = π(x)/n for all x ∈ V G , i ∈ [n]. Write P n for the transition matrix of the RW on G n . Let π min and π max be the smallest and largest values taken by π on V G . Finally, we denote w min > 0 the smallest positive weight in G and ∆ the largest degree in G (the degree of a vertex being the number of edges attached to it).

Figure 1 :

 1 Figure 1: a weighted graph G and a 3-lift of G (not all weights are written on the picture).

σ

  {u,u} = (2 3 1), σ {u,v} = (2 1) (3), σ {u,x} = (1 3 2), σ {v,x,red} = (1) (2) (3), σ {v,x,black} = (2 1) (3).

  d is even: in the very particular case when G consists of a single vertex and d/2 loops having weight 1/d on both orientations, a random n-lift of G is a random d-regular multigraph (but its distribution is neither that of a uniform d-regular multigraph, nor that of G n (d)). Our results allow us to conjecture the cutoff for the SRW (which is a RW with holding probability α = 0): Proposition 4.1.2 holds for α = 0 and gives the lower bound,. The upper bound would come from Theorem 4.1.1 with α > 0 arbitrarily small, but it is not clear that the arguments would work for α = 0 (see Appendix 4.1.7). One gets h 0 = (d-2) log(d-1) d (tools for its computation are in Section 4.1.

G in Section 4 . 1 . 3 ,

 413 and prove Proposition 4.1.2 and Theorem 4.1.1 in Section 4.1.4, under some additional assumptions on G introduced in Section 4.1.2. We show in Section 4.1.5 that those assumptions on G are not necessary. We discuss the computation of the constant h in Section 4.1.6, and the case α = 0 in Section 4.1.7.

4. 1 . 3 a

 13 simple transience criterion for the RW on the universal cover in terms of the base graph (Proposition 4.1.10), which we have not found in the literature. In particular, under assumptions A.1, A.2, A.3 and A.4 on G, the RW on (T G , •) is transient (Corollary 4.1.12).

  3 and Proposition 4.1.20). Transience of the RW on T G In this section, we give necessary and sufficient conditions on G for the RW on (T G , •) to be transient, and we prove Proposition 4.1.4. Proposition 4.1.10. Suppose that A.1 holds for G. Then the RW on (T G , •) is transient if and only if:• either G satisfies Assumption A.2,• or G has one oriented cycle ( -→ e 1 , . . . , -→ e m ) for some m ≥ 1 such thatw( -→ e 1 ) × . . . × w( -→ e m ) = w( -→ e -1 m ) × . . . × w( -→ e -1 1 ).Proof. If G has no oriented cycle, then (T G , •) is finite and isomorphic to G, and the RW is recurrent.From now on, assume that G has at least one oriented cycle. Lemma 4.1.11 below deals with the case where A.4 does not hold for G. Hence, suppose now the contrary. If A.2 does not hold, then by A.4, G is reduced to a cycle. Denote C = ( -→ e 1 , . . . , -→ e m ) one orientation of this cycle. (T G , •) is a line, and the transition probabilities along this line are periodic and are given by the w( -→ e i )'s and w( -→ e -1 i )'s. It is recurrent if and only if w( -→ e 1 ) × . . . × w( -→ e m ) = w( -→ e -1 m ) × . . . × w( -→ e -1 1

1 )

 1 × . . . × w( -→ e m ) w(e 1 ) × . . . × w( -→ e m ) + w( -→ e -1 m ) × . . . × w( -→ e -1 1 )

  w( -→ e 1 ) × . . . × w( -→ e m ) = w( -→ e -1 m ) × . . . × w( -→ e -11).

  , R) is the set of oriented edges with initial vertex in ∂B(•, R) and end vertex in ∂B(•, R + 1). Hence to check (iii), it is enough to prove that the sequence (s R ) R≥1 is summablewith s R := max (x,y)∈∂ -→ B (•,R) ν x,y . One checks easily that the irreducibility of the RW on G, A.2 and A.4 together imply that every non-backtracking path on G, after at most | -→ E G | steps, meets an oriented edge -→ e leading to at least two other oriented edges. Hence for all x ∈ (T G , •), if R = he(x) and x 1 is the parent of x, the shortest path from • to x 1 contains at least R -1/| -→ E G | such edges -→ e , so that

2

 2 and A.4 together imply that: for any -→ e a , -→ e b ∈ -→ E G with positive weight, there exists a non-backtracking path ( -→ e 1 , . . . , -→ e m ) in G with m ≤ | -→ E G | such that -→ e 1 = -→ e a and -→ e m = -→ e b . (4.8) We apply this with the label of (•, x) as -→ e a and -→ e b = -→ e . Together with the projection property of (T G , •) on G (Lemma 4.1.9), it implies that there is an upward path ( -→ e 1 , . . . , -→ e m ) in T G such that m ≤ | -→ E G |, -→ e 1 = (•, x) and -→ e m has label -→ e (recall the definition of upward path from the end of Section 4.1.

- 1 b(

 1 and we can take C 0 = w | -→ E G | min p. It remains to prove (4.8). Let -→ e a , -→ e b ∈ -→ E G . The irreducibility of the RW (A.1) implies that there is a non-backtracking path starting from -→ e a and leading to an oriented edge whose end vertex is the initial vertex x b of -→ e b . Hence, the NBRW can reach -→ e b or -→ e note that if the NBRW reaches x b through -→ e b -1, its next step cannot be -→ e b ). If w( -→ e -1 b ) = 0, then the NBRW might arrive at x b through another edge than -→ e -1 b , hence its next step can be -→ e b . It remains to show that if w( -→ e b )w( -→ e -1 b ) = 0, there is a non-backtracking path from -→ e -1 b to -→ e b . By A.4, there is a cycle C through -→ e b and a cycle C through -→ e -1 b , and we can impose by A.2 that none of them is reduced to { -→ e b , -→ e -1 b }. If C is not the inverse of C, let -→ e c be the first oriented edge of C (starting from -→ e b ) such that -→ e -1 c is not in C , and x c its initial vertex. Then x c is the end vertex of an oriented edge -→ e d of C . Therefore, a NBRW can go from -→ e -1 b to -→ e d , then to -→ e c and finally to -→ e b since -→ e c , -→ e b ∈ C. If C is the inverse of C, by A.2, there is another oriented cycle in G. Since G is connected, for some vertex x c of C, there is an oriented edge -→ e c ∈ -→ E G \ {C ∪ C }, with initial vertex x c . By A.1, there exists a non-backtracking path starting at -→ e c and ending at some oriented edge -→ e d ∈ -→ E G \ {C ∪ C } whose end vertex x d is on C. Note that there is a non-backtracking path from -→ e d to any oriented edge of C. Hence, there is a non-backtracking path from -→ e -1 b to -→ e c , then to -→ e d and finally to -→ e b . Moreover, one might impose that this path from -→ e a to -→ e b is injective (by deleting its cycles), so that m ≤ | -→ E G |. Proof of Proposition 4.1.13. Let R ≥ 0 and x ∈ V T G . By the same reasoning as in the paragraph above (4.7), on the shortest path p from • to x, there are at least m := (R -2)/| -→ E G | vertices (excluding • and x) that are the initial vertex of two upward edges in (T G , •). Denote x 1 , . . . x m

1 W

 1 The R-offspring of α j+1 contains at most ∆ R vertices. Hence, at most ∆ R+1 pairings of halfedges are performed to reveal O j+1 . We claim that L R+j contains at most ∆n exp(-β√ log n) vertices. If y ∈ L R+j and y is its parent, then φ(y ) ∈ N (β) ∩ L R+j-1 . Since y ∈N (β)∩L R+j-(y ) ≤ y ∈∂B(•,R+j-1) W (y ) =1, and since W (φ(y )) ≥ n -1 exp(β √ log n), there are at most n exp(-β √ log n) such vertices y (recall that φ is a bijection). Each has at most ∆ children, and this proves the claim. Thus, the probability to create a cycle while revealing O j+1 , i.e. to connect a vertex of O j+1 with a vertex of O j+1 ∪ L R+j is

Proposition 4 . 1 . 25 ( 2 W

 41252 Expansion). Let L(G n ) := min S⊆Vn, πn(S)≤1/(S,S c ) πn(S) for n ≥ 1, where for all A, B ⊆ V n , W (A, B) := x∈A,y∈B, -→ e :x→y π n (x)w( -→ e ) is the total weight from A to B.

  .20)). The last inequality is due to Cauchy-Schwarz and the fact that π n,min := inf u∈Vn πn (u) = π min /n. By definition of ν n , ν n -π n L 2 (πn) ≤ exp(2K √ log n). Hence by Propositions 4.1.25 and 4.1.26, there exists a constant D such that for

  h.p., min S⊆Vn, |S|≤|Vn|/2 |E(S, S c )| |S| ≥ δ, where E(S, S c ) is the set of non-oriented edges with one endpoint in S and one in its complement. Noticing that |E(S, S c )| = |E(S c , S)|, one might extend this property in the following way: for all θ ∈ (0, 1), there exists δ(θ) > 0 such that w.h.p., min S⊆Vn, |S|≤θ|Vn| |E(S,S c )| |S| ≥ δ(θ).Let now S ⊆ V n , such that π n (S) ≤ 1/2. Recall that the invariant distribution of the RW associated to G n is π n (x) = π(u)/n for all x ∈ V n of type u. Then there exists θ 0 ∈ (0, 1) such that for anyS ⊆ V n , if π n (S) ≤ 1/2, then |S| ≤ θ 0 |V n |. But W (S,S c ) πn(S) ≥ w min π min πmax |E(S,S c )| |S| .This implies (4.25), with L ≥ δ(θ 0 )w min π min /π max .c) Extending the starting pointFix ε ∈ (0, 1). Let r := C log log n for some constant C > 0 such that r > 2R, where R was defined in(4.22). Say that x ∈ V n is a root if B(x, r) contains no cycle, and denote R n the set of roots. Denote λ x the hitting measure on ∂B(x, r) of a RW started at x.

  where D is defined as in(4.26) and d y is defined in (4.20). Proof of Theorem 4.1.1. Let ε > 0. Define T n := t n +D √ log n+cr +2s -1 r. For any realization of G n and all vertices x, y ∈ V n , P Tn n (x, y) = x ∈Vn P cr n (x, x )P Tn-cr n (x , y), so that the distance to equilibrium starting from x satisfies 2d x (T n ) = y∈Vn x ∈Vn P cr n (x, x )P Tn-cr n (x , y) -π n (y) ≤ x ∈Vn P cr n (x, x ) P Tn-cr n (x , •) -π n 1 by the triangle inequality. Thus by Proposition 4.1.28, w.h.p. as n → +∞, G n is such that for all

1 1

 1 This implies straightforwardly Proposition 4.1.2, possibly with σ = 0. One checks readily that the reasoning of Sections 4.1.4 a), 4.1.4 b) and 4.1.4 c) is still true, so that Theorem 4.1.1 holds. For Proposition 4.1.25 in particular, the original proof of Amit and Linial for simple non-directed graphs is based on an argument that only requires A.1 and A.2: pick V ⊆ V n , that contains say k vertices of a given type u. Those vertices lead to k vertices of type v by irreducibility of the RW associated to G. Hence if V has less than k(1 -ε) vertices of type v, W (V , V c )/π n (V ) ε, and we can suppose that every type is represented in V 'almost' in the same proportion as the others. By A.2, there exist two cycles C 1 and C 2 = C -of G such that from each of those k vertices of type u, we can go along two trajectories featuring the types of C 1 and C 2 respectively. If we want W (V , V c )/π n (V ) ε, then at least (1 -ε)k of the k C 1 -like (resp. C 2 -like) trajectories should end in those k vertices of type u. Denote E 1 (resp. E 2 ) this event.

V n / 2

 2 finishes the proof.The fact that B(x, R) contains more vertices than B(x, R) := {y| there is an oriented path of length at most R from x to y} does not change anything to our reasoning (we only need that | B(x, R)| grows exponentially with R, which is the case by A.2).Getting rid of A.4As in Lemma 4.1.11, one can decompose G into a core c(G) satisfying A.4 and "branches" planted on this core. A similar decomposition holds for G n , as a n-lift c(G) n of c(G) and branches isomorphic to those of c(G). By Lemmas 4.1.7 and 4.1.8, after t 0 steps of a RW(X t ) t≥0 in G n , the number n(t 0 ) of steps in c(G) n follows a CLT: (n(t 0 ) -at 0 )/ √ t 0 → N (0, b 2 )for some constants a > 0, b ≥ 0 depending on G only. Moreover, the trajectory of (X t ) on c(G) n is a RW associated to c(G) n by the strong Markov property, so that Theorem 4.1.1 holds (h is replaced by ha). As for Proposition 4.1.2, the excursion theory presented in Section 4.1.3 is still true, and so is Proposition 4.1.2.

4. 1 . 6

 16 Appendix 1: computing h = h T G and s

  [START_REF] Abächerli | Local picture and level-set percolation of the gaussian free field on a large discrete torus[END_REF] w( -→ e -1 )(1 -F-→ e -1[START_REF] Abächerli | Local picture and level-set percolation of the gaussian free field on a large discrete torus[END_REF]),where π is the invariant distribution of the NBRW associated to G (defined in Section 4.1.3), andF-→ e (z) = F x,y (z) := n≥0 P(X n = y and ∃k < n, X k = y|X 0 = x)z n ,for all z ∈ C and x, y ∈ V T G such that the oriented edge (x, y) exists and has label -→ e , is the "first passage" Green function. Note that the series F-→ e has a positive radius of convergence.The functions (F-→ e ) -→ e ∈ -→ E G satisfy a non-linear system of equations given by F-→ e -1 (z) = zw( -→ e -1 ) + z-→ e ← -→ e w( -→ e )F-→ e -1 (z)F-→ e -1 (z)

h

  W has a simple expression in terms of Q and π, namelyh W = -→ e ∈ -→ E G π( -→ e )(log(1 -ŵ( -→ e )) -log ŵ( -→ e ),(4.30)

4. 1 .

 1 [START_REF] Ben-Hamou | Comparing mixing times on sparse random graphs[END_REF], so that the rest of our reasoning still holds for α = 0 with minor changes (for instance, one might have σ = 0 in Proposition 4.1.2 if G does not satisfy A.3* and if T G has a cylindrical symmetry, and the RW on G might have a period d > 1, in which case one should look at the RW (X t ) t≥0 at times {t = kd + r, k ∈ N}, for each residue r modulo d, details are left to the reader).

4. 2

 2 An example with a precise cutoff window: the NBRW on non-weighted graphs4.2.1 SettingLet G be a finite multigraph such that A.1, A.2 and A.4 hold. Suppose that for every u ∈ V G , all oriented edges going out of u have the same weight (this uniformity on the weights is also called the "non-weighted" case). In particular, A.3 is satisfied. Denote N = n| -→ E G | the cardinality of -→ E n .Let P (resp. P n ) be the transition matrix for the NBRW on -→ E G (resp.

n

  (ε) as the ε-mixing times from u ∈ -→ E n , in the worst and in the best case, as in (4log-degree of an element of -→ E G (and thus of -→ E n ), where we set, for -→ e ∈ -→ E G : deg(x) := |{y ∈ -→ E G , P (x, y) > 0}|. Define similarly deg(x) for x ∈ t ) t≥0 is a NBRW on -→ E G with Z 0 = -→ e a.s. and τ-→ e is the first hitting time of -→ e after 0. Note that deg(x) ≥ 1 for all x ∈ -→ E G by A.4 and that the maximal degree of an oriented edge is ∆ -1 ≥ 2 by A.2 (recall that ∆ is the maximal degree of a vertex in G). Thus µ and σ are well-defined, and µ > 0. By Lemma 4.1.8, σ 2 does not depend on the choice of -→ e . We make the additional assumption A.5 σ 2 > 0.

  For n ≥ 1, let t * n := log n µ and w * n := σ log n µ 3 . (4.33) for some constant C > 0 since K = o(n) and N/n = | -→ E G |. Note that K 4 = o(N ), so that there is a probability o(1/N ) that u is not a bulb. A union bound on u ∈ -→ E n yields the result.

Fory

  be the subsets of good leaves of type -→ e in T x and type -→ e -1 in T y respectively.

Lemma 4 . 2 . 7 .Lemma 4 . 2 . 8 .

 427428 For all ε > 0, For all ε > 0, and -

  P t n (x, ρ(y)) ≥ -→ e ∈ -→ E G W-→ e with W-→ e := u∈L - → e x ,v∈L -→ e y (u) (v)1 u=ρ(v) 1 (u) (v)≤θ . Now, fix -→ e ∈ -→ E G . The sets of half-edges of type -→ e (resp.-→ e -1 ) that are still unpaired at the end of the construction have the same cardinality m. A uniform random matching between those sets shall take place in order to complete the construction of the random lift (due to Lemma 4.1.6). We apply Proposition 1.1 in[START_REF] Chatterjee | Stein's method for concentration inequalities[END_REF] to the random variable W-→ e := W-→ e θ : for any a ≥ 0,P(| W-→ e -E[ W-→ e ]| ≥ a) ≤ 2 exp -a 2 4E[ W-→ e ] + 2a, where the expectation is taken with respect to the pairing. Note that E[ W-→ e ] = 1 mθ Υ - -→ e and that m = n + o(n) uniformly over all possibilities for the exposure process. Therefore, taking a = ε 2N θ in the previous inequality for all -→ e , we getnP t n (x, ρ(y)) ≥ -→ e ∈ -→ E G Υ - -→ e -εwith probability at least 1 -exp(-ε 2 Cnθ ) for some constant C (note that Υ - -→ e is bounded above by 1, so that E[ W-→ e ] ≤ 1 mθ ). Since ρ = 1/(n log 2 n), this probability is 1 -o(1/n 2 ) and we are done.Proof of Lemma 4.2.8By symmetry, it is enough to prove the Lemma forL -→ e x . Fix ε > 0. Take Y 0 = x a.s. Then u∈L -→ e x (u) = P EP (Y t/2 is of type -→ e and is on F \ H) ≥ P EP (Y t/2 is of type -→ e ) -P(Y t/2 ∈ F \ H),where P EP is the law of (Y k ) conditionally on the exposure process. Suppose thatP({ u∈F \H (u) ≤ 2 -ε/2} ∩ {x, y ∈ R n }) = o 1 n 2 . (4.48) This entails that the event {P EP (Y t/2 is not located on F \ H) > ε/2} ∩ {x, y ∈ R n } has probability o(1/n 2 ). Since P is bistochastic and irreducible, (Y k ) has a unique stationary distribution, which is uniform. For n (and thus t) large enough, P EP (Y t/2 is of type -→ e ) = P EP (Y t/2 is of type -→ e ) > 1/| -→ E G | -ε/2, for every realization of the exposure process. Hence, -ε} ∩ {x, y ∈ R n }   = o(1/n 2 ).

≤ 2 - 2 . ( 4 . 49 )

 22449 δ} ∩ {x, y ∈ R n }) = o 1 nAs for the first part, set Y 0 = x a.s. Then u∈Hx H x := T x ∩ H. By Lemma 4.1.8,

1 n

 1 -2j+1 , which impliesP(v j ∈ U j-1 |F j-1 ) ≤ |U j-1 | n-2j+1 and v∈U j-1 (v j )P(v j = v|F j-1 ) ≤ v∈U j-1 (v) n-2j+1 = K j-1 n-2j+1 .

(u j ) ≤ t + 2 .E(K j- 1 - 9 β β + 9γ 2 9+β/γ 2 ≤

 21922 Since t = o(n 1/8 ), we have τ = o(n) and we get for n large enough:τ k=1 E [K j -K j-1 |F k-1 ] ≥ -3∆tn -1(K j -K j-1 ) 2 |F k-1 ≤ 3∆tn -1/4 . (4.52)Denote α and β the RHS of (4.51) and (4.52) respectively. Fix γ > 0 and define the martingaleM γ by setting M γ (0) = 0 and M γ (k) := k j=1 K j ) ∧ γ -E [(K j-1 -K j ) ∧ γ|F j-1 ] (4.53) for k ∈ N. It is straightforward that P(|M γ (k) -M γ (k -1)| ≤ γ) = 1, and we may apply Proposition 2.1 (p.5) in[START_REF] Freedman | On tail probabilities for martingales[END_REF] with a = 9γ, b = β and K = γ to get:P(M γ (τ ) ≥ 9γ) ≤ e Cβ 9for some constant C > 0 uniquely depending on γ. And β = O(n -9/40 ) by definition of t in Proposition 4.2.6 and by (4.52), so thatP(M γ (τ ) ≥ 9γ) = o 1 n 2 . (4.54)To conclude, it is enough to show that for n large enough,max j≤τ |K j-1 -K j | ≤ γ almost surely. (4.55)Indeed, on {max j≤τ |K j-1 -K j | ≤ γ}, K τ ≥ K 0 -M γ (τ ) + α. Moreover, K 0 = 2 and α = o(1).

  j ) deg(Z j ) ≥ θ.Clearly, it holds that Υ +-→ e ≤ P(E).Denote (Z k ) k≥0 the projection of (Z k ) k≥0 on -→ E G , which is a NBRW by Lemma ??. Note that Y k and Y k are of the same type for all k, and similarly for (Z k ), so that P

  (Y j )) + log(deg(Z j )) ≤ -log θ (Y j )) + log(deg(Z j )) ≤ µ(t -2r) + λ √ t -2r + o( √ t -2r)   =Φ(λ) + o(1).

56 ) 4 . 3 . 1 .Remark 4 . 3 . 2 . 4 . 3 . 1

 56431432431 We make the assumptions A.1, A.2 and A.4 on G. Note that the distribution of D n only depends on n and on which oriented edges of G have a positive weight. To compute the nonoriented diameter, it is enough to give a positive weight to both orientations of every edge, hence our setting already includes this case.Let A G be the oriented adjacency matrix of G: it is a matrix indexed by the set of oriented edges with a positive weight, and for every (-→ e , -→ e ) ∈ ( -→ E + G ) 2 , A G ( -→ e , -→ e ) = 1 { -→ e -1 and -→ e start at the same vertex, and -→ e = -→ e -1 } . Note that A G is irreducible by (4.8). Let D be the Perron-Frobenius eigenvalue of A G , i.e. its largest eigenvalue, which is positive and simple by the Perron-Frobenius theorem. Then Theorem As n → +∞, D n log n P → log -1 D. Let us give a brief intuition on this result. Since G n is an expander (Proposition 4.1.25) with bounded degrees, its diameter should be of order log n. The constant log -1 D comes from the fact that D is the growth rate of the universal cover, and that there are too few cycles in G n to reduce this rate. Using the same core/branches decomposition than in Section 4.1.5, we see easily that the result holds without A.4, and that A G and A c(G) have the same Perron-Frobenius eigenvalue. The growth rate of the universal cover As in Section 4.1.3, root T G at some vertex •. Denote B(•, R) the oriented ball centred at • of radius R, i.e. it consists of all oriented paths of length R started at •.

1 G- → a 1 .D

 11 result is independent of the choice of •. Proof. For R ≥ 1, let a R := |∂ B(•, R)|, and for every -→ e ∈ -→ E G , let a R ( -→ e ) be the number of vertices x in |∂ B(•, R)| such that if y is the parent of x, then (y, x) has type -→ e . Denote -→ a R the vector (a R ( -→ e )) -→ e ∈ -→ E G . Then for every R ≥ 1, -→ a R+1 = A G -→ a R so that by an immediate induction: -→ a R = A R-is the largest eigenvalue of A G and has multiplicity 1. Moreover, by A.2, | B(•, R)| grows exponentially with R so that D > 1. Therefore, there exist constants c, c > 0, uniquely depending on G, such that for every R ≥ 1,a R = -→ e ∈ -→ E G a R ( -→ e ) ∈ [cD R , c D R ].The conclusion follows.

4. 3 . 2 1 Lemma 4 . 3 .R ≥ 1

 321431 Proof of Theorem 4.3.3 and the fact that | B(x, R)| ≤ | B(•, R)| for every x of the type of • and every

. 57 )

 57 Proof. Let k ε = 2/ε + 1. We first show that w.h.p., for every x ∈ V n , B(x, R n ) contains at most k ε (non-oriented) cycles. (4.58) Let x ∈ V n . Reveal B(x, R n ) in a breadth-first way, by successive pairings of half-edges. Since B(x, R n ) contains no more vertices of each type than B(•, R n ), the total number s of pairings satisfies s ≤ ∆| B(•, R n )| ≤ n 1/2-ε/2

  For k = 1 to R n and every -→ e ∈ -→ E , let a k,n ( -→ e ) be the number of oriented edges of T x from the k-th to the (k + 1)-th generation, and -→ a k,n the vector (a k,n ( -→ e )) -→ e ∈ -→ E G . It satisfies the same recursive equation as ( -→ a R ) R≥0 in Lemma 4.3.3, with at most k ε "-1" distributed arbitrarily to some a k,n ( -→ e ), for k ≥ 1 and -→ e ∈ -→ E G . Unless a k,n := -→ e ∈ -→ E G a k,n ( -→ e ) = 0 for some k ≥ 1 (w.h.p., this happens for no x ∈ V n , by Proposition 4.1.25), a bounded number of "-1"'s does not prevent the exponential growth of a k,n at rate log D. Indeed, in the first log log n generations, there are at least (2k ε ) -1 log log n consecutive generations without a -1, during which a k,n ( -→ e ) becomes larger than log log n for every -→ e (for n large enough). Then, each subsequent -1 divides the size of a k,ε ( -→ e ) by a factor less than 2. Therefore, as in Lemma 4.3.3, there exist constants c, c > 0 such that for n large enough, a Rn,n ∈ [cD Rn , c D Rn ]. The conclusion follows.

•

  Proposition 4.1.25, w.h.p. G n is such that for every x ∈ V n and fori ≥ 1, | B(x, R n + i)| ≥ (1 + K G ) i | B(x, R n )|,(4.59)as long as| B(x, R n + i)| ≤ nπ min /2. Let R n := 10ε log -1 (1 + K G ) log n. Starting from B(x, R n ), reveal B(x, R n + R n -| -→ E G |) in a breadth-first way. If we hit B * (y, R n ),then there is an oriented path from x to y of length at most 2R n + R n . Else,|∂ B(x, R n + R n -| -→ E G |)| ≥ n 1/2+6εby (4.59). Then, by Step 1, Step 2 and the pigeon-hole principle, there exist -→ e there are at least|∂ B * (y, R n )|/| -→ E G | ≥ n 1/2-3ε vertices of ∂ B * (y, R n ) thatare the initial vertex of an oriented edge of type -→ e (and that edge does not belong to B * (y, R n )),• there are at least|∂ B(x, R n +R n -| -→ E G |)|/| -→ E G | ≥ n 1/2+5ε vertices of ∂ B(x, R n +R n -| -→ E G |)that are the initial vertex of an oriented edge of type -→ e (and that edge does not belongto B(x, R n + R n -| -→ E G |)).

  Sparse random graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 The phase transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The configuration multigraph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 Exploration of the multigraph . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.2 Convergence of the depth-first walk and marks . . . . . . . . . . . . . . . 2.5.3 Back-edges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5.6 Height process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.7 The convergence of the metric structure . . . . . . . . . . . . . . . . . . . 2.6 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.1 A change of measure for spectrally positive Lévy processes . . . . . . . . 2.6.2 Size-biased reordering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.3 Convergence of the measure-change in the Brownian case . . . . . . . . . 2.6.4 Convergence of a single large component for α ∈ (1, 2) . . . . . . . . . . . 3 Gaussian Free Field level-set percolation on regular random graphs 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

	Contents
	1 Introduction
	1.1

1.3 The Gaussian Free Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 The mixing time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Scaling limit of a critical configuration model with power-law degrees 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Our results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 The configuration model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 The phase transition and critical behaviour of the component sizes . . . . 2.2.3 Branching processes and their metric space scaling limits . . . . . . . . . 2.2.4 Our method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.5 Related work on scaling limits of critical random graphs, universality, and open problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.6 Plan of the rest of the paper . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 The limit object: the stable graph . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 An absolute continuity relation for spectrally positive α-stable Lévy processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Excursion theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Convergence of a discrete forest . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 2.5.4 Components of the finite graph . . . . . . . . . . . . . . . . . . . . . . . . 2.5.5 Marked excursions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

23.1.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.4 Discussion and open questions . . . . . . . . . . . . . . . . . . . . . . . . 3.1.5 Proof outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.6 Plan of the rest of the paper . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.7 Further definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Basic properties of G n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Structure and Green function . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 GFF on G n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

  d n 1 , . . . , d n n be an ordered sample of i.i.d. random variables D 1 , . . . , D n with the same distribution as D.

	Then conditions (i) and (ii) above are satisfied almost surely for some sequence of random vari-
	ables ϑ 1 , ϑ 2 , . . . (see Section 2.2 of [62]). Perform percolation at parameter p = 1/θ to obtain
	new degrees D 1 , D 2 , . . . , D n which are mildly dependent but whose ordered version behaves very

  For suitable test functions f and g 1 , g 2 , g 3 , . . ., and any n ≥ 1, we have

		. Proposition 2.3.7 implies that we may use the
	Radon-Nikodym derivative Φ(t) to change measure at the random times σ . As earlier, we
	write (ε	( ) i , i ≥ 1) for the excursions of L occurring before time σ in decreasing order of length.
	For an excursion ε ∈ E * = E ∪ {∂}, write a(ε) =	ζ(ε) 0	ε(u)du for its area.
	Proposition 2.3.8.

  dt → ∞ and -I t → ∞ as t → ∞. The latter implies that σ < ∞ a.s., and since -I t < ∞ for each t > 0, we also have σ → ∞ as → ∞. Hence, sup{ζ(ε) : ε is an excursion of R starting after time σ }

	as t p → 0
	as → ∞. It follows that
	(ζ( ε	( ) i ), i ≥ 1) → (ζ( ε i ), i ≥ 1) a.s.
	in the product topology, as → ∞. The result then follows from Proposition 2.3.8 since the
	expressions there do not depend on the value of .
	This enables us to give the proof of Theorem 2.1.2 assuming the definition of ((G i , d i , µ i ), i ≥
	1) from L given following Theorem 2.1.1.
	Proof of Theorem 2.1.2. With Proposition 2.3.9 in hand, it remains to deal with the Poisson
	points which give rise to the vertex-identifications. We have straightforwardly that, given ε i ,
	the number M i of points falling under the excursion is conditionally independent of the other
	excursions and has a Poisson distribution with parameter 1 µ	∞ 0 ε i (u)du. Moreover, conditionally
	on the number of points, their locations are i.i.d. uniform random variables in the area under
	the excursion. For any suitable test function g,
			.
	Proof. The excursions of R occurring before local time has been accumulated are a strict
	subset of all the excursions that ever occur. By Lemma 2.3.5, we have that
	sup{ζ(ε) : ε is an excursion of R starting after time t}	p → 0

  Proof of Proposition 2.4.2. Denote the set of permutations of {1, 2, . . . , n} by S n . By definition,

1 , D2 n , . . . , Dn n ) of the degrees. First, we prove Proposition 2.4.2.

  .31) By (3.23), lim inf n→+∞ P T d (E n ) =: p > 0. Let a 1 be such that P(ϕ T d (•)≥a 1 )<p /4. For n large

	enough,
	a≥h P T d a (E n )ν(da) > p /2, hence	a 1 h P T d a (E n )ν(da) > p /4,
	By Lemma 3.3.3 again,

where we recall that ν is the density of ϕ T d (•). Since E n is an increasing event, by Lemma 3.3.3:

min a≥a 1 P T d a (E n ) ≥ p /4.

  e. no vertex was added to T x during the (k -1)-th step),C4 k > log λ h n. Else, denote x k,1 , x k,2 , . . . , x k,m the neighbours (in G n ) of vertices of O k-1 that are not

in O k-2 , for some m ∈ N (note that m = (d -1)|O k-1 |, each vertex of O k-1 having one neighbour in O k-2 , its parent, and d-1 other neighbours at distance k of •). Add to T x the vertices x k,i of O k-1 such that ϕ T d (Φ(x k,i )) ≥ h + log -1 n. Add to T x the corresponding vertices Φ(x k,i ).

  and log λ h+δ > 9 log λ h /10. The map h → η(h ) is continuous on R \ {h } by Theorem 3.1 of[START_REF] Abächerli | Level-set percolation of the Gaussian free field on regular graphs I: regular trees[END_REF] and the map h → λ h is an

	homeomorphism from (-∞, h ) to (1, d -1) by Proposition 3.3.4, hence such δ exists. It is clear
	that
	lim inf n→+∞

  hence by the first equation of Proposition 3.3.4 applied to h + δ,

	lim inf n→+∞

1/2 b n ≥ lim inf n→+∞ P ann |Z h+δ log λ h n | > n 9/10 / log 2 λ h n ,

  .56) By Remark 3.5.2, we have |T x | ≤ n 1/2 . By (3.55), (3.56) and a union bound on y ∈ T x , P Gn (∪ y∈Tx E (y)) ≤ n -5/2 with E (x) := E(x). And P Gn (∪ (k,i):x k,i ∈Tx E

	(n) k,i ) ≤ n -2 for large enough
	n, by Lemma 3.2.5.
	Thus for n large enough, if G n is a good graph,

  It remains to show that the typical distance in C -ε) log λ h n. Modify the lower exploration of Section 3.5.2: say that it is aborted if • C1 did not happen, and• it is stopped at some step k ≤ (1/2 -ε/2) log λ h n, or less than n 1/2-ε/10 vertices and half-edges have been seen at step (1/2 -ε/2) log λ h n . ∈ V n , perform the lower exploration from x, then that from y, and stop it if it meets a vertex of the exploration from x. This happens with P ann -probability o(1) by (3.14) with k = 1,P ann (E ab (x, y))=1 -o(1), with E ab (x, y):={the lower explorations from x and y are aborted}, Then, reveal ψ Gn on the exploration trees T x and T y . Applying Proposition 3.4.1 as below (3.54), we get that:

	(n) 1 is at least (1 Suppose that
	P ann (the lower exploration from x is aborted) = 1 -o(1).	(3.127)
	For x, y m 0 , m 1 , m, m E = o( √	n) (recall Remark 3.5.2).Hence by (3.127):

  10 (which is possible since h → λ h is decreasing and continuous, Proposition 3.3.4). By Proposition 3.3.8, there exists C > 0 (depending on ε) such that for n large enough, for every log log n ≤ k ≤ (1/2 -ε/2) log λ h n and a ≥ h,

  1 the RW on G is irreducible, A.2 [Two-cycles property] G has at least two oriented cycles which are not each other's inverse, where an oriented cycle of length m ≥ 1 is a cyclic order C = ( -→ e 1 , . . . , -→ e m ) of m oriented edges with positive weight such that the end vertex of -→ e i is the initial Suppose that G satisfies A.1 and A.2, and that G n is a uniform random lift of G. For any ε ∈ (0, 1), there exists K(ε) > 0 such that if t is the worst-case mixing time of the RW on G n , then w.h.p. on the realization of G n as n → +∞, R be the tail distribution of the standard normal. The following lower bound shows that the mixing time of Theorem 4.1.1 is almost the smallest possible among all n-lifts of G: Proposition 4.1.2. There exists σ > 0, uniquely depending on G, such that for any ε ∈ (0, 1),

	vertex of -→ e i+1 and -→ e i = -→ e -1 i+1 for all i (mod m), and the inverse of this cycle is the
	cycle C -1 = ( -→ e -1 m , . . . , -→ e -1 1 ).
	Theorem 4.1.1. (n) max (ε) |t (n) max (ε) -h -1 log n| ≤ K(ε) log n	(4.2)
	the constant h > 0 being the entropy of the universal cover of G (see Section 4.1.3).
	Let	Φ(λ) :=	1 √ 2π	λ +∞	e -u 2 2 du	(4.3)
	for λ ∈ for any arbitrary sequence

  Assume now that A.2 holds. If A.3 does not hold, then at least one edge of G has exactly one orientation -→ e with positive weight. In this case, the RW on (T G , •) is not irreducible. However, due to Lemma 4.1.9 and A.1, a RW (X t ) t≥0 started at • has probability at least w Assume now that A.3 holds, so that the RW on (T G , •) is irreducible (since all oriented edges have a positive weight). By Lemma 5.1 in [78], it is enough to prove Proposition 4.1.10 when the RW associated to G (and hence to (T G , •)) is reversible. Indeed, this result states that if a discrete irreducible Markov chain admits an invariant measure, then it is transient if the additive reversibilization of the chain is. The additive reversibilization of the RW on (T G , •) is the RW on (T * G , •), where (T * G , •) is obtained from (T G ,•) by modifying its weights as follows: for all x, y ∈ V T G , set G with initial vertex u 1 and end vertex u 2 . Let G * be the weighted graph with the same vertex and edge sets as G, and with edge weights w , the RW on G * is reversible. Therefore, the RW on (T G , •) is transient if the RW on the universal cover of a reversible base graph G * is transient. Now, we can apply a classical transience criterion for discrete reversible Markov chains. The earliest proof we found traces back to 1983 (see[START_REF] Lyons | A simple criterion for transience of a reversible Markov chain[END_REF]), and is derived from an analogous theorem of Royden for Riemannian surfaces. It states that if a discrete reversible Markov chain has state space Ω, transition matrix P and invariant measure Π, and if there exists a collection of weights ν = (ν i,j ) i,j∈Ω such that:

	| min to cross an oriented edge with label -→ E G | E G | steps. Hence, if T := inf{t ≥ 0, X t has label --→ e after at most | -→ → e }, then T is stochastically
	dominated by a geometrical random variable of parameter p ∈ (0, 1) depending on G. After
	this crossing, since T G is a tree, the RW can never come back to its starting point. Hence, it is
	transient.
		w * (x, y) =	1 2	w(x, y) +	π(y) π(x)	w(y, x) .
	(T * G , •) is the universal cover of the graph G * defined as follows. Define
		w * ( -→ e ) := 1 2 w( -→ e ) + π(u 2 ) π(u 1 ) w( -→ e -1 )
	for every -→ e ∈	-→ E ( -→ e ) for every -→ e ∈	-→ E G . By
	construction of G

* *

  Proposition 4.1.19. The random variables ( i ) i≥1 are i.i.d., and there exist C 6 , C 7 > 0 such that for all m, i ≥ 0, P( τ i ≥ m) ≤ C 6 exp(-C 7 m).

	(4.14)

  ≤ P( τ 1 ≥ -n/ log ŵmin ) = O(exp(C n/ log ŵmin )), so that W

				and define the
	quantity ŵmin := min e∈ -→ E G , ŵ( -→ e )>0 ŵ( -→ e ). Remark that for every i ≥ 1, W	( ) i	≤ -τ i log ŵmin .
	By Proposition 4.1.19, the (W	( ) i ) i≥1 's are i.i.d., and for some constant C > 0 and all n ≥ 1,
	P(W	( ) 1 ≥ n) ( ) 1	has moments of
	any order. Let h w

  Z i since r t is random and not a priori independent of the Z i 's (both depend on the τ i 's).For all n ≥ 1, let G n be an arbitrary n-lift of G. The proof goes as follows: we couple a RW (X t ) t≥0 on G n with a RW (X t ) t≥0 on T G . The estimate provided by Theorem 4.1.3 on W t implies that X t is concentrated on o(n) vertices with positive probability for t close to h -1 log n, and so is X t : cycles in G n can only reinforce this concentration. This implies a lower bound on d x (t),

		1 √ rt	rt-1
	Proposition 4.1.20. (Theorems D and E in [117]) There exist s, σ s > 0 such that
	he(X t ) t	a.s. → s	(4.17)
	and
	he(X t ) -st √ t
	The lower bound: Proof of Proposition 4.1.2
	where
	d x (t) := d T V (P t n (x, •), π n ),	(4.20)

i=1 law → N (0, σ 2 s ).

(4.18)

Proof. The proof is similar to that of Theorem 4.1.3. Again, the fact that σ s > 0 is due to

A.3.

Remark 4.1.21. The convergences in Theorem 4.1.3 and Proposition 4.1.20 do not depend on the choice of v * . Moreover, Theorem 4.1.3, Proposition 4.1.20 and Corollary 4.1.16 give

h T G = sh W . (4.19)

We discuss ways of computing h T G and s in the Appendix 1 (Section 4.1.6). 4.1.4 Proofs of Proposition 4.1.2 and Theorem 4.1.1 since π n is almost uniform on V n .

  . . , -→ e m ) an oriented path of length m in G n , Remark 4.1.22. For all y ∈ V T G , if p 1 (resp. p 2 ) is an oriented path of length t ≥ 1 from • to y, then φ -1 (p 1 ) and φ -1 (p 2 ) end at the same vertex of V n . The converse is not true as soon as G n has cycles: for every x ∈ V n , there are two distinct non-backtracking paths p 1 and p 2 from x to y , and the oriented paths φ(p 1 ) and φ(p 2 ) lead to two different vertices y 1 and y 2 of (T G , •).

	φ(p) = ( -→ e 1 , . . . , -→ e m ) where -→ e 1 is the unique oriented edge such that its initial vertex is •, and
	the label of -→ e 1 is the type of -→ e 1 , and for all i ≥ 2, -→ e i is the unique edge such that its initial
	vertex is the end vertex of -→ e i-1 and the label of -→ e i is the type of -→ e i .
	Now, let λ > λ ∈ R and define t n := log n h T G 4.1.20,	+ λ	σ T G 3/2 h T G	√	log n . By Theorem 4.1.3 and Proposition
	lim inf n→+∞

  8. By Corollary 4.1.16 and by (4.19), this is possible if one takes γ small enough. Then if a < 0 is small enough, by Proposition 4.1.20, we have that for n large enough,

  Proof of Corollary 4.1.5. Let ε, K > 0. Take a such that Proposition 4.1.23 holds. We have 1

	≤ 3ε/4
	for n large enough. This establishes (4.23) and concludes the proof.
	Remark 4.1.24. For large enough n, less than n exp(-2β	√	log n/3) vertices and edges of G n are
	discovered during the exploration. Indeed, as shown in the proof of Proposition 4.1.23, each level of T has at most ∆n exp(-β √ log n) vertices, and there are no more than h -1 log n levels, since we stop the exploration after t n ≤ h -1 log n steps. Hence, T has less than n exp(-3β √ log n/5)
	vertices. Each vertex seen in the exploration is either in T or a neighbour of a vertex of T , and
	every vertex of G

J 0 -1 j=1 (P ann (E j ) + P ann (E j )) + P ann (E * ) ≤ ε/2 + o(1) n has at most ∆ edges and ∆ neighbours. The result of Proposition 4.1.23 is annealed, and leads to a quenched result for "most" realizations of G n , Corollary 4.1.5.

  3 and A.4 are not necessary for Proposition 4.1.2 and Without loss of generality, we can assume that each edge admits at least one orientation with positive weight. Suppose that A.3 does not hold, hence, that at least one edge has exactly one orientation -→ e with positive weight. In this case, the RW on (T G , •) is not irreducible any more. so all constants in Section 4.1.3 can be made independent of v * and -→ e * . As a consequence, all results of Section 4.1.3 hold, with the following exceptions. Introduce the assumption (weaker than A.3): A.3* At least one edge has both orientations with positive weight. Then • if A.3* does not hold, (X t ) can not backtrack and Proposition 4.1.20 becomes he(X t ) = t a.s. for all t (hence s = 1), • if A.3* does not hold and σ W = 0 in Corollary 4.1.16, there exists K T G only depending on G such that Theorem 4.1.3 becomes sup t∈N |W t -h T G t| ≤ K T G . Indeed, X t = ξ t for all t ≥ 1 so that W t = W (ξ t ) and h T G = h W , and by Lemma 4.1.8, σ W = 0 implies that |W t -h T G t| is bounded.

	Theorem 4.1.1.
	Getting rid of A.3

It is still transient by Proposition 4.1.10. Moreover, assumptions A.1 and A.2 are enough to imply that the RW can reach every isomorphism class of subtrees, independently of the choice of •,

  will denote (Y k ) k≥0 a NBRW on -→ E n and (Y k ) k≥0 its projection on -→ E G . Note the following analogue of Lemma 4.1.7, with an identical proof: Lemma 4.2.1 (Lemma 4.1.7 for a NBRW). (Y k ) k≥0 is a NBRW on -→ E G .

  u ∈ -→ E n and take Y 0 = u a.s. If -→ B (u, 2r) contains no cycle, then the NBRW is in R n at each of the first r steps and we are done. Suppose that there is a unique cycle C in be the set of states from which the probability to be in R n after r -t steps is less than 1. For 1 ≤ t ≤ r, let p t be the probability that Y t ∈ C t . It is enough to prove that transition that the NBRW can take in order to keep a chance to attain C before leaving -→ B (u, 2r): else, there would be another cycle. There is one exception: when v is just outside C and points toward a vertex on it. Again, due to the uniqueness of C, the walk can meet such a v only once. Moreover, if (Y k ) leaves this path to C, then Y r will surely be in R n .

					-→ B (u, 2r).
	Let C t ⊆	-→ E n p r	P → 0	(4.41)
	uniformly in u and the structure of	-→ B (u, 2r). Remark that from any v ∈	-→ B (u, 2r), there is
	at most one

Therefore, for at least r -1 steps,

p t+1 = p t / deg(Y t ).

Note that by A.2 and A.4, there exists an oriented edge on -→ E G that the NBRW (Y k ) cannot avoid oriented edges of degree at least 2 for more than | -→ E G | -1 consecutive steps. Hence, the same is true for (Y k ), so that

  Then, uniformly on Y t/2-r , Y t is of type -→ e with probability 1 G is uniform, and the same holds for Z t (independently of Y t ).

	distribution of the NBRW on E Hence -→			|	-→ E G |	+ o(1) since the stationary
	Υ + -→ e ≤	Φ(λ) + ε | -→ E G | 2	,
	and we obtain the desired result by summing on -→ e ∈	-→ E G .

A rooted tree is a tree T with a distinguished vertex •, the root. The height h T (x) of a vertex x in T is d T (•, x). If T is finite, its boundary ∂T is the set of vertices of maximal height. The subtree from x is the subtree made of the vertices y such that x is on any path from • to y. The offspring of x is the set of vertices of its subtree. For r ≥ 0, the r-offspring of x is its offspring at distance r of x, and its offspring up to generation r is its offspring at distance at most r. If y is in the 1-offspring of x, then y is a child of x, and x is its parent.

In this case, write x = y.

If x, y are neighbours in T , the cone from x out of y is the rooted subtree of T with root x and vertex set {z ∈ T | y is not on the shortest path from x to z}.

An isomorphism between two rooted trees T and T is a bijection Φ : T → T preserving the root and the height, and such that for all vertices x, y ∈ T , there is an edge between x and y if and only if there is an edge between Φ(x) and Φ(y).

Unless mention of the contrary, all random walks are in discrete time. We will write T A (resp.

H A ) for the first exit (resp. hitting) time of A by a SRW.

For two probability distributions µ, µ on R, we write µ ≤ st.

µ (or µ ≥ st.

µ) if µ dominates stochastically µ, i.e. there exist two random variables X ∼ µ and X ∼ µ on the same probability space such that X ≤ X a.s.

Basic properties of G n

Structure and Green function

The graph G n can be generated sequentially as follows: attach d half-edges to each vertex of V n . Pick an arbitrary half-edge, and match it to another half-edge chosen uniformly at random. Choose a remaining half-edge and match it to another unpaired half-edge chosen uniformly and independently of the previous matching, and so on until all half-edges have been paired.

The resulting multi-graph M n is not necessary simple, i.e. it might have loops and multiple edges. The probability that M n is simple has a positive limit as n → +∞, and conditionally on {M n is simple}, M n is distributed as G n (see for instance Section 7 of [START_REF] Van Der Hofstad | Random graphs and complex networks: volume I[END_REF], in particular Proposition 7.13 for a reference).

In particular, an event true w.h.p. on M n is also true w.h.p. on G n , so that it is enough to prove all our results on M n . In the rest of the paper, we will even write G n for M n for the sake of simplicity. This Section is devoted to proving this result: Proposition 3.2.1. There exists K 3 > 0 such that w.h.p. as n → +∞, G n satisfies:

(I) G n is a K 3 -expander, i.e. the spectral gap λ Gn of G n is at least K 3 (the spectral gap is the smallest eigenvalue of I -P where I is the identity matrix and P the transition matrix of the SRW on G n ),

|D| ≥

|∂B Gn (Ty,an)| 2

≥ n 1/2 log -7 n by (3.46) and C2. Hence for n large enough, p 2 ≥ (d -1) log γ-2κ-1 n n 1/2 log -7 n dn 1 -n 1/2 log γ n dn

n -1/2 log γ-2κ-9 n.

With (3.67), this entails P ann (B * (z j , a n ) is a joining ball | S j ) ≥ p 1 p 2 p 3 ≥ n -1/2 log γ-2κ-10

for n large enough (uniformly on the realization of B * j ), and thus (3.66), so that the proof of the Lemma is complete.

The field ψ Gn on the joint exploration Suppose that we are on S (x, y). By Proposition 3.2.4, we can realize ψ Gn on T x as in (3.47) with the (ξ x,v ) v∈T d . Then we can realize it in a similar way on T y with the (ξ y,v ) v∈T d , letting recursively

where A k,i := T x ∪ {y ,j |( , j) ≺ (k, i)}, ≺ being the lexicographical order on N 2 , and Φ is the isomorphism between T y and T y .

Recall that J = {j ≥ 1, B * (z j , a n ) is a joining ball} and that for j ∈ J, we denote v j (0) the unique vertex of B * (z j , a n -2a n ) ∩ B Gn (T y , a n ). Since no cycle is discovered when revealing B * (z j , a n ) \ B * (z j , a n -2a n ), the intersection of B * (z j , a n -a n ) and T y is a unique vertex v j (a n ), which is in the a n -offspring of v j (0) in the tree B * (z j , a n ) rooted at z j . Then we realize ψ Gn on B * (z j , a n -2a n ) and on the shortest path P j from v(0) to v(a n ) as in (3.47), via a family of i.i.d. N (0, 1) random variables (ξ j,k,i ) k,i≥0 . In the tree T j := B * (z j , a n -2a n ) ∪ P j with root z j , denoting z j,k,i the i-th vertex at generation k and

the set of vertices where ψ Gn has already been revealed before z j,k,i , we let

(3.68)

Write S * (x, y) ⊆ S (x, y) the event that there exists j 0 ≥ 1 and a path from z j 0 to v j 0 (a n ) such that ψ Gn (v) ≥ h for every vertex v of that path. In particular, on S * (x, y), x and y are in the same connected component of E ≥h ψ Gn . Recall the definitions of κ (3.33) and γ (Lemma 3.6.1). Proposition 3.6.2. If κ and γ/κ are large enough, then

Proof of Proposition 3.6.2. Let γ > 3κ + 18. By Lemma 3.6.1, lim sup n→+∞ P ann (S * (x, y)) ≤ lim n→+∞ P ann (S (x, y)) = η(h) 2 .

Let E n := {G n is not a good graph} ∪ {max z∈Vn |ψ Gn (z)| ≥ log 2/3 n}. By Proposition 3.2.1 and Lemma 3.2.5, P ann (E n ) → 0. Therefore, it is enough to show that lim sup n→+∞ P ann (E c n ∩ (S (x, y) \ S * (x, y)) ) = 0.

By a straightforward adaptation of the reasoning below (3.54),

where S (x, y) We use again a binomial argument. For j ∈ J in increasing order, generate the GFF on T j as in (3.68). Denote E j the event that z j and v j (a n ) are in the same connected component of

Suppose that for every j ∈ J,

where K 8 := log d-1 ((1 + λ h )/2). Then, letting Z ∼ Bin( log γ-3κ-18 n , log γ(K 8 /3-1) n), we have

But if κ and γ/κ are large enough so that

we have lim n→+∞ P(Z = 0) = 0 and this yields (3.69). Thus, we are left with showing (3.70).

We split the proof of (3.70) in two parts. First, we prove that

where

Second, we show that for some constant K 9 > 0 (uniquely depending on d and h),

We prove that both hold for n large enough, uniformly in v ∈ T j and on realization of the ψ Gn on T 1 ∪ . . . ∪ T j-1 , as long as we are in E c n (so that we can apply Proposition 3.4.1). (3.71) and (3.72) imply indeed that Cov ann (1 S * (x,y) , 1 S * (w,t) ) = P ann (S * (x, y) ∩ S * (w, t)) -P ann (S * (x, y))P ann (S * (w, t)).

By Proposition 3.6.2, for n large enough, |P ann (S * (x, y))P ann (S * (w, t)) -η(h) 4 | ≤ ε.

(3.79)

Now, perform successively the exploration of Section 3.5.1 from x, then from y, then from z and finally from t (via an array of i.i.d. standard normal variables (ξ u,v ) u∈{x,y,w,t},v∈T d ). We add the following condition: for any u ∈ {x, y, w, t}, the exploration from v is stopped as soon as it meets a vertex seen in a previous exploration.

Then do the same from ∂T w to ∂T t , this time with

Finally, reveal ψ Gn on T x , T y , T w , T t and on the joining balls from T x to T y and from T w to T t , in that order.

One can adapt readily the proof of Lemma 3.6.1 to show that with P ann -probability 1-o(1), if the four explorations are successful then there are at least log γ-3κ-18 n joining balls from ∂T x (resp.

∂T w ) to ∂T y (resp. ∂T t ). Note in particular that the estimations of (3.61), (3.63) and (3.64) still hold. It is also straightforward to carry the arguments of the proof of Proposition 3.6.2, and we finally have |P ann (S * (x, y) ∩ S * (w, t)) -P ann (the explorations from x, y, w, t are all successful)| ≤ ε/2.

Together with (3.79) and (3.80), this yields (3.78).

Uniqueness of the giant component

In this Section, we prove (3.3). We start by the lower bound in Section 3.7.1, showing the existence P ann -w.h.p. of a component (different from

, we perform an exploration of a new kind, starting from some x ∈ V n . It consists of three phases (Sections 3.7.2 to 3.7.4), during which we assign a pseudo-GFF ψ Gn to the vertices that we visit. ψ Gn is defined via a recursive construction that mimics Proposition 3.3.1, as long as there are no cycles (in Section 3.5, the analogous of the pseudo-GFF was ϕ T d on T x ). Finally, in Section 3.7.5, we reveal the true values of ψ Gn one by one on the set of vertices we have explored via Proposition 3.2.4, and show that either |C Gn,h

by Remark 3.6.5.

Contrary to Section 3.5, we need this alternative to hold for every x ∈ V n , P ann -w.h.p. By a P ann (E i,n,meet ) + P ann (C1 happens) ≤ n -1/2 . Thus to establish (3.83), it only remains to show that for K small enough and n large enough,

Let v 1 , . . . , v d be the children of • in T d . Remark that for n large enough,

Using Proposition 3.3.1, one checks easily that p, p , p > 0. Taking K < -(12 log p ) -1 yields the result.

First phase

In this section, we define the first phase of the exploration, and show that it is successful with P ann -probability 1 -n -5/4 (Proposition 3.7.2). We will need a variant of Lemma 3.3.9, namely Lemma 3.7.3. We postpone its statement and proof to the end of this section.

Let a, K, K > 0. For every n ∈ N, define

Let δ ∈ (0, h -h) and ∈ N be such that the conclusion of Lemma 3.7.3 holds.

The exploration. Let x ∈ V n .

I -We first assume that we do not meet any cycle throughout the exploration. Let M be the set of marked vertices. Initially,

, stop the exploration. Else, give a mark to x (hence add it to M).

While M = ∅, pick y ∈ M in an arbitrary way and proceed to its subexploration.

The subexploration of y. Let T y be the subexploration tree, that we will build by adding subtrees of depth in a breadth-first way. Initially T y = {y}.

While 1 ≤ |∂T y | ≤ K log n, perform a step: take y 1 ∈ ∂T y of minimal height and if y 1 = x, let y 1 be its only neighbour where ψ Gn has already been defined. Note that if y 1 = y, y 1 is the parent of y 1 in T y . Reveal all the edges of B Gn (y 1 , y 1 , r n + ), where we recall that B Gn (y 1 , y 1 , r n + ) is the graph obtained by taking all paths of length r n + starting at y 1 and not going through y 1 .

Third phase

Suppose now that we are on S 2 (x). For 1 ≤ q ≤ m, denote y q one vertex of ∂T q whose exploration was successful and not back-spoiled in the second phase, T yq its exploration tree, and B q the boundary of T yq . In this section, we connect each B q to Θ(n) vertices in a fashion similar to Section 3.6.1. However, revealing the GFF on a positive proportion of the vertices would prevent us to use an approximation of the GFF as in Proposition 3.4.1.

To circumvent this difficulty, denote R 1,2 the set of vertices seen in the first two phases, and

for some constant K 11 > 0. We will connect each B q to a positive proportion of the vertices of D 1 only, with P ann -probability 1 -n -3 (Proposition 3.7.5), before revealing the GFF on the vertices of the first two phases, and on the connection from D 1 to B q in Section 3.7.5. The result will follow by symmetry of the D i 's and a union bound on 1 ≤ i ≤ r -1.

The exploration.

1) The w-explorations. Let w 1 , . . . , w K 11 log n be the vertices of D 1 . We proceed successively to the w-explorations of w 1 , w 2 , . . ., i.e. for i ≥ 1, we perform the exploration from w i as in Section 3.5.1, but stop it if we reach a vertex seen in the first two phases or in the w-exploration of some w j , j ≤ i -1. In particular, if w i was discovered during the exploration of some w j , j ≤ i -1, say then that w i is w-spoiled and do not proceed to its w-exploration. Denote R w the set of vertices seen during all the w-explorations.

For i ≥ 1, if we explore w i and C2 happens, say that the w-exploration from w i is w-successful.

Let s 0 be the number of w-successful vertices. Let w i 1 , . . . , w is 0 be the w-successful vertices with i 1 < . . . < i s 0 . Let T w i j be the exploration tree of w i j , for j ∈ {1, . . . , s 0 }. Take away

• from each ∂T w i j : the vertices that are seen in the w-exploration of some w i , for i > i j ,

• from each B q : the vertices z such that B Gn (z, z, a n ) intersects R w .

Say that those vertices are w-back-spoiled.

2) The joining balls. For q = 1, . . . , m successively, we develop balls from B q to the ∂T w i j 's, 1 ≤ j ≤ s 0 , with a few modifications w.r.t the construction of Section 3.6.1: let z 1,q , z 2,q , . . . be the vertices of B q . For z i,q ∈ B q , let B * i,q := ∪ (i ,q ):q <q or q =q,i <i B * (z i ,q , a n ), and let R i,q := R 1,2 ∪ R w ∪ B * i,q be the vertices seen before building B * (z i , a n ). Replace B * j , Q j and B Gn (T y , a n ) of Section 3.6.1 by B * i,q , R i,q and ∪ s 0 j=1 B Gn (T w i j , a n ) respectively. Say that B * (z i,q , a n ) is a J-joining ball if it hits B Gn (T w i J , a n ) at one vertex after a n -2a n steps, and no other intersection with vertices seen previously is created.

Since s 0 ≤ |D 1 | ≤ log 2 n by (3.95) and m ≤ K log n by D2, a union bound on (J, q) yields (3.96).

Proof of (3.97). We now estimate the probability that at most log n w-explorations are wsuccessful. Note that by Remark 3.5.2 and (3.94),

Moreover, for every i ≥ 1, conditionally on the fact that w i is not w-spoiled, the probability that the w-exploration from w i is stopped because it reaches a vertex of R 1,2 or a vertex seen in the exploration of some w j ,

Hence, a straightforward adaptation of the proof of (3.54) yields

One checks easily that if K 11 is large enough, then for large enough n,

and (3.97) follows.

Revealing ψ Gn on the three phases

Let R ψ 1 (resp. R ψ 2 ) be the set of vertices where ψ Gn has been defined during the first (resp. second) phase, and R ψ 3 be the set of vertices in the w-successful w-explorations and on the J-joining balls, for all 1 ≤ J ≤ m, on which we will realize ψ Gn on the third phase. By Proposition 3.2.4, we can realize ψ Gn jointly with G n by • proceeding to the three phases of the exploration from x,

• revealing the remaining pairings of half-edges of the G n ,

• defining ψ Gn on R ψ 1 ∪ R ψ 2 , in the same order as ψ Gn has been defined, using the same standard normal variables: we let

We deal with the second term of the RHS as (3.115) to show that it is O(n -3a ). As below (3.45), we have

if a is small enough, by (3.11). Now, by (3.17) applied to D := B Gn (y, y, r n ) (note that T D = τ ) and (3.41) (which still holds, as remarked in the proof of Lemma 3.7.7), we get

for any z ∈ ∂B Gm (E, 1) \ {z k }, by cylindrical symmetry of B Gm (E, 1). One checks easily that if a is small enough, then for n large enough,

One easily adapts the reasoning leading to (3.37), despite the presence of one cycle, to get

) for a small enough.

Note indeed that {T

starting at z has a probability decaying exponentially with r n to reach z k , since there are at most two injective paths from z to z k , and each contains at least r n -3 vertices where the SRW has a positive probability (only depending on d) to enter a subtree isomorphic T + d and to never leave it.

Since

All in all, we get that the first term of the RHS of (3.116) is O(n -3a ), and (3.114) follows.

Proof of Proposition 3.7.6. We proceed as below (3.54) in the proof of Proposition 3.5.1. Denote

On E n , ψ Gn is defined on at most K 0 log n vertices by D2 (and our choice of K = K 0 ), so that by the triangle inequality, either | ψ Gn (x) -ψ Gn (x)| ≥ n -a log -2 n, or there exists y such that

where R ψ y is the set of vertices where ψ Gn has been defined before ψ Gn (y). Let

exploration as in Section 3.5.1. Denote S T (x) the event that the exploration is successful, that

x and that B Gn (x, k) = T . We claim that

The proof goes as those of Lemma 3.5.3 and Proposition 3.5.1. In the proof of Lemma 3.5.3, replace

We also check easily that

in order to determine B Gn (x, k) P ann -w.h.p., as Proposition 3.5.1 only ensures that

Moreover, we get as for (3. (3.118)

Let ε > 0. Applying Bienaymé-Chebyshev's inequality as in Lemma 3.6.3, we get

where S T is the set of vertices x ∈ V n such that S T (x) holds. Let S ⊆ V n be the set of vertices such that their exploration is successful. By Theorem 3.1.1 and a reasoning as in the proof of Lemma 3.6.3, with P ann -probability 1 -o(1),

Suppose that these three assumptions and (3.119) hold. By (i),

1 ∩ S c |, so that by (ii), (iii) and (3.119),

, so that for ε small enough,

And since we can take ε arbitrarily small, the conclusion follows.

Chapter 4

Cutoff for random walks on random lifts Section 4.1 stems from the preprint [START_REF] Conchon-Kerjan | Cutoff for random lifts of weighted graphs[END_REF], that has been accepted for publication by Annals of Probability. Sections 4.2 and 4.3 are original material of this thesis.

Generic cutoff for random lifts of weighted graphs

Abstract. We prove a cutoff for the random walk on random n-lifts of finite weighted graphs, even when the random walk on the base graph G of the lift is not reversible. The mixing time is w.h.p. t mix = h -1 log n, where h is a constant associated to G, namely the entropy of its universal cover. Moreover, this mixing time is the smallest possible among all n-lifts of G. In the particular case where the base graph is a vertex with d/2 loops, d even, we obtain a cutoff for a d-regular random graph, as did Lubetzky and Sly in [START_REF] Lubetzky | Cutoff phenomena for random walks on random regular graphs[END_REF] (with a slightly different distribution on d-regular graphs, but the mixing time is the same).

Introduction

The cutoff phenomenon

The way random walks converge to equilibrium on a graph is closely related to essential geometrical properties of the latter (such as the typical distance between vertices, its diameter, its expansion, the presence of traps or bottlenecks, etc.), giving an important motivation for studying mixing times.

For a Markov chain on a discrete state space Ω, with transition matrix P , that has an invariant distribution π, the ε-mixing time from x is Proof. One checks easily that the obtained structure is a n-lift of G, and that all permutations along edges of G are uniform and together independent, as in the definition.

Random walks on lifts admit a natural projection property (whose proof is straightforward):

Lemma 4.1.7 (Projection of the lift). Fix n ∈ N. Let G n be a n-lift of G and let (X t ) t≥0 be a RW on G n . Let (X t ) t≥0 be the projection of (X t ) t≥0 on G, obtained by mapping X t to its type for all t ≥ 0.

Then (X t ) t≥0 is a RW on G.

Hence, walks on lifts inherit much of the structure of walks on the base graph.

It is well known that for a Markov chain (X t ) t≥0 on a finite set Ω with invariant measure π, for any u ∈ Ω, lim t→+∞ P(X t = u) = π(u) provided that the chain is aperiodic. We finally state a CLT refining this ergodic property (and not requiring aperiodicity):

Lemma 4.1.8 (CLT for Markov chains). Let f be a function from Ω to R. For n ∈ N, let S n := n-1 t=0 f (X t ). Let m := u∈Ω f (u)π(u). Fix u ∈ Ω, X 0 = u a.s. and let τ u be the first hitting time of u after 0. Let

Then v does not depend on the choice of u and

where N (0, 0) is the Dirac mass in 0.

This is a direct application of Theorem 16.1 and Corollary 16.1 (p.94) in [START_REF] Lai | Markov chains with stationary transition probabilities[END_REF].

Additional assumptions on G

Let us make the following additional assumptions on G (we will prove in that they are in fact not necessary, in Section 4.1.5):

A.3 All oriented edges have a positive weight,

A.4 every oriented edge lies on an oriented cycle.

For all u ∈ G and R ≥ 1, let ∂B(u, R) := B(u, R) \ B(u, R -1) be the boundary of B(u, R).

When A.3 holds on G, every non-oriented path gives rise to two oriented paths of positive weight (opposite to each other). We can define the distance d(x, y) between two vertices x, y as the length of the shortest path with positive weight from x to y (and d is indeed a metric).

We state those definitions for G but might use them for other graphs.

Study of the universal cover

The goal of this Section is to prove Theorem 4.1.3. Fix an arbitrary vertex v * ∈ V G and an arbitrary oriented edge -→ e * going out of v * (this choice has no importance for the sequel), and root the universal cover of G at a vertex • with label v * .

Definitions for labelled rooted trees

Label each vertex x ∈ V T G by the vertex v ∈ V G such that the non-backtracking path in G starting at v * , and corresponding to the path from • to x in (T G , •), terminates at v (see Figure 2). Give similarly a label in E G (resp.

-→ E G ) to each edge (resp. oriented edge) of (T G , •). In the literature, (T G , •) is sometimes called the directed cover of G, or periodic tree arising from G. Remark that the universal cover of an irreducible n-lift of G is also (T G , •).

Due to A.3, the RW on (T G , •) is an irreducible Markov chain with invariant measure π defined as follows: for all x ∈ V T G with label u, π(x) = π(u). Denote P T G its transition matrix.

An isomorphism φ between two rooted trees (T, •) and (T , • ) is a bijection between the vertices of (T, •) and those of (T , • ) such that φ(•) = • and such that there is an edge between x 1 and x 2 in (T, •) if and only if there is an edge between φ(x 1 ) and φ(x 2 ) in (T , • ).

Lemma 4.1.9 (Projection of the cover). If one projects (T G , •) on G by mapping each vertex, edge and oriented edge to its label, then the projection of a RW on (T G , •) is a RW on G.

Remark that this result is analogous to Lemma 4.1.7. Its proof is straightforward. Note that there exists at most |V | distinct rooted trees (T G , •) up to isomorphism. Indeed, two vertices in V T G with the same label induce isomorphic rooted trees.

and the same edges, weights and labels as (T G , •) on V x . Note as previously that there exist finitely many such subtrees up to isomorphism. V x is the offspring of x. If y ∈ V x , it is a descendant of x at generation he(y) -he(x), and x is the (he(y) -he(x))-ancestor of y. If in addition he(y) = he(x) + 1, y is a child of x and x is its parent. The height-R level of T x denotes ∂B(x, R), and B(x, R) ∩ V x (resp. ∂B(x, R) ∩ V x ) is also called the offspring of x up to generation R (resp. offspring of generation R, or R-offspring).

where x is the initial vertex of -→ e and w (x) is the total weight of the edges in G starting at x: this modified graph is c(G). Clearly, A.4 holds on c(G). Moreover, the RW associated to c(G) is also irreducible (erasing a leaf from G does not affect the irreducibility). Proof. Remark that c(T G ) = T c(G) , the erased vertices in T G being exactly those whose labels are erased in G. Let (X t ) t≥0 be a RW on T G . Note that a.s., (X t ) t≥0 visits infinitely many distinct vertices and edges of T c(G) . The trace of (X n ) on T c(G) is defined as follows: for all p > m ≥ 1 In Section 4.1.1, we defined the ray to infinity ξ of a RW (X t ) t≥0 in (T G , •) as:

with ξ t := ξ ∩∂B(X 0 , t). The next Proposition expresses the following fact: due to the regularity of T G , the RW is "uniformly transient" and the probability to make R steps in a given direction decreases exponentially w.r.t. R.

Proposition 4.1.13. For every base graph G, there exist two positive constants C 1 , C 2 so that if (X t ) t≥0 is a RW on (T G , •), then for all R ≥ 0 and x ∈ B(•, R),

Since those constants are independent of the choice of •, the Markov Property gives the following generalization: for all s > 0, y ∈ (T G , •) and x ∈ B(y, R),

The proof of this proposition requires the following intermediate result.

Lemma 4.1.14. There exists a positive constant C 0 , independent of the choice of v * , such that for every child x of •, if (X t ) t≥0 is a RW starting at •, then:

Proof. The assertion on the choice of the first oriented edge of (ρ t ) is clear by definition of ŵ.

It is then enough to prove that for all t 0 ≥ 1, for all x ∈ V T G of height t 0 + 1, and x i the vertex of height i in the shortest path from • to x, i ∈ {t 0 -1, t 0 }:

For simplicity, write p := P(ρ

and denote (ρ

t ) t≥0 the ray to infinity of (X τ +t ) t≥0 . We have

t ) t≥0 be the ray to infinity of a RW started at x t 0 . Then

by the Strong Markov Property applied to the stopping time τ .

We define the entropic weight of a vertex x ∈ V T G as the probability that x is in the ray to infinity of a RW started at •, and denote it W (x). Let (x 0 , . . . , x H ) be the vertices on the shortest path from • to x, where H = he(x), so that x 0 = • and x H = x. By Proposition 4.1.15,

Remark that the transition probabilities for the first step of the ray to infinity are different, since one does not condition on the event that the RW starting at x i will not go to infinity through x i-1 .

Corollary 4.1.16. There exist

Proof. From equation (4.11), we have

where ( -→ e ) is the label of -→ e for every oriented edge -→ e . By Proposition 4.1.15 and Lemma 4.1.9, ( (ρ i )) i≥0 is a NBRW on G. We conclude by Lemma 4.1.8, where the Markov chain is the

where we recall that π max := max v∈V G π(v). Thus, by (4.21),

Note that this result is uniform in x, due to Remark 4.1.21. This concludes the proof with

.

The upper bound: Proof of Theorem 4.1.1

From now on, we focus on the case where G n is a uniform random lift of G. The proof consists of three parts, as detailed in Section 4.1.1.

a) Almost mixing for a typical starting point

In this paragraph, we prove that with a large probability, G n is such that a RW (X t ) t≥0 started at a uniformly chosen vertex x ∈ V n has a large probability to stay on a subtree T of G n for

, where a is a constant given by Proposition 4.1.23. This allows us to couple (X t ) t≥0 with a RW (X t ) t≥0 on T G (keep all the notations of Section 4.1.3 for (X t )).

As underlined in the Introduction, by Proposition 4.1.4, (X t ) stays close to its ray to infinity (at distance O(log log n) during the first t n steps), and this reduces the number of vertices of G n that (X t ) could explore. This ray localization also ensures that W (X t ) and P t n (x, X t ) are close quantities, so that for some constant K, for most x ∈ V n , P tn n (x, x ) ≤ e K Let β > 0 be an arbitrary positive constant.

The exploration.

Let n ∈ N and N (β

with C large enough so that

at the root, independent of the realization of G n . For every j ≥ 1, let t j be the hitting time of B(•, R + j) by (X t ). We reveal the structure of G n edge by edge, starting from x and making systematic use of Lemma 4.1.6. At every moment of the exploration, let φ be an isomorphism between T and a subtree T of (T G , •), and denote L k and L k the respective height-k levels of T and T for all k ≥ 0.

At step 0, T and T are reduced to their respective roots x and •, so that φ(x) = •. Let

Reveal B(x, R + 1). If there is a cycle in B(x, R + 1), stop the exploration. Else, include

• the exploration from any given vertex y ∈ ∂B(x, r) has a probability at most ε not to be successful (Lemma 4.1.30), so that the mean value of λ x (y ∈ ∂B(x, r), the exploration from y is successful) is at least 1 -ε,

• and the explorations from two vertices y, y ∈ ∂B(x, r) whose common ancestor in B(x, r)

is at distance at least R (which is the case for most such couples (y, y ) ) are almost independent, so that for every x ∈ R n , λ x (y ∈ ∂B(x, r), the exploration from y is successful) is concentrated around its mean.

Third, we give a proof of Theorem 4.1.1 (under A.3 and A.4), using the results of Sections Proof. We proceed via a counting argument similar to the beginning of the proof of Proposition 4.1.23, while estimating the probability that the exploration stops at step 0: take x ∈ V n , proceed to the O(∆ 5r ) successive matchings of half-edges to generate B(x, 5r). Each matching has a probability O(∆ 5r /n) to close a cycle. Hence, the probability that at least two matchings close As a consequence,

Proof. Fix δ > 0 and a realization of G n such that Lemma 4.1.27 holds, for some n ≥ 0. Let x ∈ V n . If B(x, 5r) contains no cycle, then it is isomorphic to (T G , •) for some • ∈ V T G on its first 5r levels. For n large enough, for all

Else, there is only one cycle in B(x, 5r). Hence B(x, 5r) is a cycle C with trees T v attached to its vertices v of degree at least 3. Let L ∈ N be such that for all • ∈ V T G , for all y ∈ (T G , •) such that he(y) ≥ L, P(∃t ≥ 0, X t = y) ≤ δ where (X t ) t≥0 is a RW on (T G , •) started at the root (for any δ, such L exists, by Proposition 4.1.13). Let C(L) be the set of vertices at distance at most L of C, and let t 0 := inf{t ≥ 0, X t 0 ∈ C(L)}. We claim that for n large enough, P(t 0 > log r ) ≤ δ. By the triangle inequality, all trees in B(x, 5r) rooted on C that X t might visit for some t ≤ log r have a maximal height at least 5r -log r > L for n large enough. Hence for every 0 ≤ t ≤ log r ,

• either X t ∈ C, and is at distance less than L from B(x, 5r) \ C(L),

• or it is on C, and it is at distance at most |V | from a vertex v of C of degree at least 3, so that v is the root of a tree planted on C: a cycle in G cannot contain a path of |V + 1| consecutive vertices of degree less than 3, and this holds for any n-lift of G n . Suppose now that t 0 ≤ log r . Note that there exists some (random) y ∈ C such that X t 0 is on T y . As remarked in the proof of the claim, T y has height at least 5r -log r . Moreover, on its first 5r -log r levels, (T y , y) is isomorphic to a copy of (T G , • y ) where • y is the type of y, from which two branches starting at • y have been removed (corresponding to the two edges from y in C). Recall that d(X t 0 , C) ≥ L+1. By definition of L and Proposition 4.1.20, for n large enough, with probability at least 1 -2δ, (X t ) does not visit y (and hence C) for t 0 ≤ t ≤ t 0 + cr, and we have 4r/3 ≤ d(X t 0 +cr , C) ≤ 5r/3. On this event and on {t 0 ≤ log r }, B(X cr , r) contains no cycle so that X cr ∈ R n . Therefore, on this realization of G n , P Gn (X cr ∈ R n ) ≥ 1 -3δ uniformly in the starting point x.

Hence, we have shown that for all δ > 0, for n large enough and G n such that For x ∈ R n , let P x be the probability distribution of G n conditionally on the fact that x is a root, and for all y ∈ ∂B(x, r), denote α(y, x) the vertex at distance R of y on its shortest path to x.

In addition, if at most n exp(-β √ log n/2) edges of G n have been revealed (where β is the constant fixed at the beginning of Section 4.1.4), and if all revealed paths starting from y and leading to revealed cycles go through α(y, x), say that G n is a good context for y.

Define the cut exploration from y as the exploration performed in 4.1.4, except that some matchings may have already been revealed, and give a mark to α(y, x) (hence don't explore the offspring of φ(α) and stop the exploration if the RW hits α(y, x)). If the cut exploration from y is t n -successful, we say that it is good.

Proposition 4.1.29. With high probability on G n , for all x ∈ R n , λ x ({y| the cut exploration from y is good}) ≥ 1 -3ε.

Results

The results that we obtain are similar to those of Section 4.1.

Proposition 4.2.2. For any deterministic sequence (G n ) n≥1 of n-lifts of G, and for any ε ∈ (0, 1),

where Φ was defined in (4.3).

Note that this does not depend on the lift structure. For a typical lift (when n is large), this lower bound is attained:

Theorem 4.2.3. Suppose G n is a uniform random lift. Then for any ε ∈ (0, 1),

By comparison, in the case of a uniform regular graph, it has been shown in [START_REF] Lubetzky | Cutoff phenomena for random walks on random regular graphs[END_REF] that the window around the mixing time is of constant size.

In Section 4.2.

Fix ε ∈ (0, 1), take ε > ε and a < Φ -1 (ε ). If k = t * n + a w * n for some constant a < a, we have for n large enough by (4.33):

Thus (4.36) with b large enough such that Φ(a) -Φ(b) > ε yields

for n large enough, where τ n := exp(C √ log n)/n for some arbitrary constant C ∈ (0, (a-a )σ 2 √ µ ).

In other words, the NBRW has a probability at least ε to stay on paths of weight at least τ n during the first k steps. Since the probability that the walk took indeed a given path is equal to its weight, there are at most ε /τ n such paths. Thus, there exists

Thus for n large enough, the distance to equilibrium (defined as in (4.20)) satisfies