Raphaël Cerf

Nicolas Forien

Some toy models of self-organized criticality in percolation

Keywords: percolation, criticality, self-organized criticality Mathematics Subject Classification (2020): 82B43, 60K35, 82B20, 82B27

We consider the Bernoulli percolation model in a finite box and we introduce an automatic control of the percolation parameter, which is a function of the percolation configuration. For a suitable choice of this automatic control, the model is self-critical, i.e., the percolation parameter converges to the critical point pc when the size of the box tends to infinity. We study here three simple examples of such models, involving the size of the largest cluster, the number of vertices connected to the boundary of the box, or the distribution of the cluster sizes.

Introduction

Our goal is to present a simple model of self-organized criticality built upon the classical Bernoulli percolation model in Z d , which is amenable to a rigorous mathematical analysis. In the next subsection, we introduce a candidate model and we state our main theorem, which shows that the parameter of our model converges automatically towards the critical parameter of the Bernoulli percolation model.

Construction of the model and convergence result

Let Λ(n) be the box of side n centered at 0 in Z d with d 2, and let E n be the set of edges between nearest neighbours of Λ(n). Consider a sequence of increasing functions F n : {0, 1}

En → N and a parameter a > 0 and set, for ω : E n → {0, 1} a percolation configuration on the edges of the box,

p n (ω) = ϕ n F n (ω)
where

ϕ n (x) = exp - x n a .
This function p n will play the role of an automatic control of the percolation parameter, and in this paper we will study three examples of such a control, involving different functions F n (see theorem 1). The model we consider is given by the following probability distribution on the configurations, which is obtained by replacing the parameter p of Bernoulli percolation with our feedback function p n , with the appropriate normalization. Let

µ n : ω ∈ {0, 1} En -→ 1 Z n P pn(ω) (ω) (1)
where Z n = ω∈{0,1} En P pn(ω) (ω)

will be called the partition function, and P p is the Bernoulli percolation measure with parameter p, namely ∀ω ∈ {0, 1} En P p (ω) = e∈En p ω(e) (1 -p) 1-ω(e) .

For x ∈ Λ(n) and ω : E n → {0, 1}, we write

C(x, ω) = y ∈ Λ(n) : x ω ←→ y
for the open cluster of x in the configuration ω. We show the following convergence result, valid in any dimension d 2. The critical point of the Bernoulli percolation model is denoted by p c .

Theorem 1. If F n is one of the following sequences of functions:

(i) F n : ω -→ C max (ω) = max x∈Λ(n) |C(x, ω)| with 0 < a < d ; (ii) F n : ω -→ M n (ω) = x ∈ Λ(n) : x ω ←→ ∂Λ(n) with d -1 < a < d ; (iii) F n : ω -→ B b n (ω) = x ∈ Λ(n) : |C(x, ω)| n b with 0 < b < a < d ,
then the law of p n under µ n converges to δ pc when n → ∞, and we have the following control:

∀ε > 0 -∞ < lim inf n→∞ 1 (ln n)n v ln µ n |p n -p c | > ε lim sup n→∞ 1 n v ln µ n |p n -p c | > ε < 0 ,
where the (ln n) factor can be dropped in the case (i), and where the exponent v is given by

   v = a ∧ (d -a d ) in case (i) ; v = d -1 in case (ii) ; v = a ∧ d -b d in case (iii) .
We can see that, for a large interval of the parameter a, the mass of µ n concentrates on the configurations ω for which p n (ω) is very close to p c . Hence, our model presents a phenomenon of self-organized criticality: the percolation parameter concentrates around the critical point without the need to finely tune a parameter to a precise value (see section 1.3 about self-organized criticality).

An estimate on the convergence speed

In case (iii), an estimate on the convergence speed can be obtained, provided that we assume the existence of the critical exponents β and γ. Let us briefly recall the definition of these exponents (see [START_REF] Grimmett | of Grundlehren der Mathematischen Wissenschaften[END_REF]).

The exponent β is related to the percolation probability θ(p), which is the probability that the origin belongs to an infinite cluster in a percolation configuration on Z d , with percolation parameter p. It is believed (but unproven in general up to now) that θ(p c ) = 0 and that we have the power-law scaling θ(p) = (p -p c ) β+o (1) when p → p c with p > p c , for a certain exponent β > 0, which depends on the underlying graph Z d .

The exponent γ is related to the mean finite cluster size χ(p), which is defined as the mean size of the cluster of the origin, conditioned on the event that this cluster is finite. It is conjectured that we have a powerlaw χ(p) = |p -p c | -γ+o(1) when p → p c , for a certain exponent γ > 0. The following theorem indicates which scaling could be deduced from the unproven existence of these critical exponents. The existence of these exponents was proven in dimension 2 for the case of the triangular lattice [START_REF] Smirnov | Critical exponents for two-dimensional percolation[END_REF], with β = 5/36 and γ = 43/18, and our study could easily be adapted on the triangular lattice. We do not believe the condition on c to be optimal, since the term (d -bd -b)/β comes from a quite rough estimate (see lemma 19), and it does not allow to deal with b d/(d + 1). It may be possible to improve our technique to get rid of this limitation, and to obtain a similar estimate on the convergence speed for the two first models.

Self-organized criticality

Our model is intended as a toy model of self-organized criticality, a concept which was coined in by the physicists Bak, Tang and Wiesenfeld in their seminal paper [START_REF] Bak | Self-organized criticality: An explanation of the 1/f noise[END_REF]. Many physical models present a phenomenon called phase transition: there is a critical point or a critical curve in the parameter space separating two distinct regions characterized by very different macroscopic behaviours. In such systems, the behaviour of the model at criticality is of particular interest and presents some general features (e.g., fractal geometry or power-law temporal and spatial correlations) which are universal across a wide range of systems and do not depend much on the microscopic details of the system. Bak, Tang and Wiesenfeld pointed out that these "critical features" are very common in nature, which is rather surprising because it seems that the parameters need to be finely tuned for a system to be critical. To explain this paradox, they showed that some systems tend to be naturally attracted by critical points, without any fine tuning of the parameters. They call this phenomenon self-organized criticality.

To illustrate this idea, they defined a simple model inspired by the dynamics of a sandpile. The balance between avalanches and accumulation of sand leads to a state where the system looks critical, with a self-similar distribution of the sizes of the avalanches and the slope self-adjusting to the critical slope, which is the slope at which large-scale avalanches appear. But despite a very simple dynamics, their model turns out to be very difficult to analyze mathematically [START_REF] Dhar | Theoretical studies of self-organized criticality[END_REF][START_REF] Antal | Sandpile models[END_REF][START_REF] Hutchcroft | Universality of high-dimensional spanning forests and sandpiles[END_REF].

In [START_REF] Cerf | A Curie-Weiss model of self-organized criticality[END_REF], Cerf and Gorny constructed a self-critical model as a variant of the generalized Ising-Curie-Weiss model, by replacing the temperature with a function depending on the spin configuration. In this paper, we implement the same principle of a feedback from the configuration to the parameter, but within the framework of Bernoulli percolation. This technique to obtain self-organized criticality by "artificially" replacing the control parameter with a feedback function depending on the state of the model, which is explained in section 15.4.2 of [START_REF] Sornette | Critical phenomena in natural sciences[END_REF], was implemented by physicists to imagine self-critical variants of percolation in [Sor92, FSS93, SWdA + 00, CdML03]. However, the understanding of such models often relies on computer simulations and few models are amenable to rigorous mathematical analysis.

Self-critical models based on percolation

There have been several attempts to build mathematical models of self-organized criticality in the percolation setup.

Forest-fire models. One strategy to obtain a self-critical model consists in modifying a process of dynamical percolation in order to burn down the large or infinite clusters. In a model defined by Dürre [START_REF] Dürre | Existence of multi-dimensional infinite volume self-organized critical forest-fire models[END_REF][START_REF] Dürre | Uniqueness of multi-dimensional infinite volume self-organized critical forest-fire models[END_REF],

trees grow with rate 1 on each site of the square lattice Z d , and lightnings strike each site occupied by a tree with rate λ, which makes the cluster of this tree instantaneously become vacant. Thus, each cluster is burnt with a rate proportional to its size. If λ = 0, the state of this model at time t corresponds to Bernoulli site percolation with parameter p = 1 -e -t . The introduction of the lightning parameter λ > 0 is intended to prevent the appearance of too large clusters, and the most interesting behaviour is expected in a limit λ → 0, where finite clusters are almost never hit by lightnings, whereas no infinite cluster can survive without being immediately destroyed.

The study of this model proved quite challenging, but notable rigorous results have been derived for the one-dimensional case [START_REF] Van Den Berg | On the asymptotic density in a one-dimensional self-organized critical forest-fire model[END_REF][START_REF] Bressaud | On the invariant distribution of a one-dimensional avalanche process[END_REF]. In [START_REF] Ráth | Erdős-Rényi random graphs + forest fires = self-organized criticality[END_REF], Ráth and Tóth studied a similar model, but on the complete graph with n vertices, and where instead of trees growing on the sites, edges are added with a certain rate. Then, the most interesting regime is when each edge is added with rate 1/n and lightnings strike on each site with rate λ(n), with n -1 λ(n)

1. In this regime, the authors proved that, under certain conditions on the initial configuration, the stationary distribution of the cluster sizes converges when n tends to infinity to a power-law distribution, which shows that this mean-field model exhibits a phenomenon of self-organized criticality. Heuristically, this model behaves as if an infinite cluster was about to appear, but the lightnings prevent it from effectively forming.

In view of this, a natural idea is to try to build a model on an infinite graph where trees grow with rate 1 and any cluster of trees which becomes infinite is instantaneously destroyed. Such a model has been studied on non-amenable graphs [START_REF] Ahlberg | Bernoulli and self-destructive percolation on non-amenable graphs[END_REF] and in high dimension [START_REF] Ahlberg | Seven-dimensional forest fires[END_REF], but it turns out that such a model does not exist in dimension 2, as proved in [START_REF] Kiss | Planar lattices do not recover from forest fires[END_REF], confirming a conjecture of [START_REF] Van Den Berg | Self-destructive percolation[END_REF]. The argument is based on the instructive fact that there exists δ > 0 such that, if one takes a supercritical site percolation configuration on Z 2 , closes all the sites belonging to the infinite open cluster, and reopens each closed site with probability δ, then almost surely there is still no infinite cluster. Thus, after the destruction of an infinite cluster, it takes some incompressible time to reconstitute an infinite cluster. This stands in contradiction with the fact that, in a model where infinite clusters are instantaneously destroyed, there would be an accumulation of such destruction events just after having reached a critical density of trees.

These forest-fire models can be seen as continuous variants of the Drossel-Schwabl forest fire model [START_REF] Drossel | Self-organized critical forest-fire model[END_REF], where instead of instantaneously destroying the clusters, lightnings trigger fires which then spread progressively from one tree to its neighbours, and so on. This Drossel-Schwabl model has received much attention in the physics literature, with the hope to prove that it exhibits self-organized criticality, in the sense of power-law distributions for the cluster sizes and the duration and the sizes of the fires. But, despite its quite simple definition, this process has been mainly studied through computer simulations and heuristic reasoning, which gave contradictory predictions about its large-scale behaviour, and few mathematically rigorous results have been obtained (see [START_REF] Grassberger | Critical behaviour of the Drossel-Schwabl forest fire model[END_REF] and the references therein).

Frozen percolation. Instead of burning large or infinite clusters, another technique consists in freezing clusters when they reach a certain size. Once an open cluster is frozen, the closed sites on its boundary are forced to remain closed forever, preventing further growth of this cluster.

One may wish to freeze clusters when they become infinite. Aldous defined such a model on the infinite binary tree and showed that, as soon as half of the sites are open, the system gets blocked in a critical-like state, where finite clusters look like critical percolation clusters [START_REF] Aldous | The percolation process on a tree where infinite clusters are frozen[END_REF].

On the square grid Z 2 , such a process with freezing of the infinite clusters does not exist (see [START_REF] Van Den Berg | A signal-recovery system: asymptotic properties, and construction of an infinite-volume process[END_REF],

which explains an argument of Benjamini and Schramm). Instead, one may consider diameter-frozen percolation, where clusters are frozen when their diameter exceeds N [START_REF] Van Den | A percolation process on the square lattice where large finite clusters are frozen[END_REF] or volume-frozen percolation, where clusters freeze when they contain more than N vertices [START_REF] Van Den | Two-dimensional volume-frozen percolation: exceptional scales[END_REF][START_REF] Van Den | Two-dimensional volume-frozen percolation: deconcentration and prevalence of mesoscopic clusters[END_REF]. Then, some interesting properties arise in the N → ∞ limit. On the binary tree, one recovers the behaviour observed by Aldous when only infinite clusters were frozen [START_REF] Van Den | A percolation process on the binary tree where large finite clusters are frozen[END_REF]. In diameter-frozen percolation on Z 2 , when N → ∞ most frozen clusters freeze in a near-critical window around the critical time, and these clusters tend to look like critical percolation clusters [START_REF] Kiss | Frozen percolation in two dimensions[END_REF]. Surprisingly, in the diameter-frozen case, this phenomenon of self-organized criticality turns out to be quite sensible to the rule imposed on the boundary of the frozen clusters (namely, the behaviour changes when one does not close the sites on the boundary of a frozen cluster [START_REF] Van Den | Boundary rules and breaking of self-organized criticality in 2D frozen percolation[END_REF]).

In [START_REF] Ráth | Mean field frozen percolation[END_REF], a mean-field variant of frozen percolation is studied, where clusters are frozen when they are hit by lightnings. This model exhibits a similar behaviour to the mean-field forest-fire model described in [START_REF] Ráth | Erdős-Rényi random graphs + forest fires = self-organized criticality[END_REF], that we mentioned before. For a large regime of the lightning rate, the process gets stuck in a state which looks like a critical Erdős-Rényi random graph, where unfrozen clusters look like critical Galton-Watson trees.

Invasion percolation. Invasion percolation is another process constructed as a variant of percolation which exhibits a phenomenon of self-organized criticality. For each edge e of the lattice Z d , we draw a random variable τ e uniformly distributed on [0, 1], the variables (τ e ) e being independent. Invasion percolation can be defined as a random increasing sequence (G t ) t∈N of subgraphs of Z d . At time t = 0, we take G 0 to be the graph containing only the origin, and no edge. At each step t ∈ N, we look at the edges which connect a vertex in G t to a vertex outside of G t , and to obtain G t+1 we add to G t the edge among these edges for which τ e is minimal (and we also add the corresponding new vertex). Eventually, this exploration process gives an infinite tree G = ∪ n∈N G t , which turns out to look like the so-called incipient infinite cluster of critical percolation [START_REF] Wilkinson | Invasion percolation: a new form of percolation theory[END_REF].

Heuristically, this can be understood by considering the p-clusters of the underlying dynamical percolation, that is to say the clusters formed of all the edges for which τ e p. When the exploration process reaches an infinite p-cluster, then it stays inside this cluster forever and no more edge with τ e > p can be explored. Thus, progressively, the invasion percolation will reach infinite p-clusters for values p > p c more and more close to p c . On the contrary, for all p < p c , the (finite) p-cluster of the origin will eventually be entirely explored.

The above heuristics were made rigorous by [START_REF] Chayes | The stochastic geometry of invasion percolation[END_REF][START_REF] Häggström | Percolation on transitive graphs as a coalescent process: relentless merging followed by simultaneous uniqueness[END_REF], which confirmed that the invaded region asymptotically looks like the incipient infinite cluster. These results were later precised in the two-dimensional case [START_REF] Zhang | The fractal volume of the two-dimensional invasion percolation cluster[END_REF][START_REF] Antal | Invasion percolation and the incipient infinite cluster in 2[END_REF][START_REF] Damron | Limit theorems for 2D invasion percolation[END_REF], but this similarity between planar invasion percolation and critical percolation has some limits: in particular, both measures turn out to be mutually singular [START_REF] Damron | Relations between invasion percolation and critical percolation in two dimensions[END_REF], and the scaling limit of invasion percolation shows rotational and scaling invariance, but it is conjectured that it is not conformal invariant [START_REF] Garban | The scaling limits of the minimal spanning tree and invasion percolation in the plane[END_REF].

Our approach. The model presented in this article is defined in a different way, which may seem less natural but has some advantages. First, by defining a probability measure on the percolation configurations in a finite box, we avoid the risk to have an ill-defined process (as can be the case when one tries to burn or freeze the infinite clusters). Also, instead of dynamically adjusting the percolation parameter (like in invasion percolation or in the algorithmic models studied in the physics literature), we directly define this parameter as a function of the percolation configuration. This function encapsulates the feedback mechanism from the configuration onto the control parameter, which is a key ingredient of self-organized criticality. Thus, to investigate the self-critical behaviour of our model, we only need to study this feedback function, and in particular its behaviour in a nearcritical window (see paragraph 1.6.2). As we will see, this behaviour is related to challenging problems of finite-size scaling of the cluster sizes, some of which remain unsolved even for the square lattice Z 2 [GPS18a]. In the end, our toy model of self-organized criticality, which is intended to be as simple as possible in its definition, already requires some work and raises some interesting problems.

Heuristics for the construction of the model

Let us explain the heuristics which lead to the choice of the sequences F n which appear in the definition of our model. The role of the function p n is to introduce a negative feedback which assigns low values p n (ω) p c to percolation configurations which are "typical" of the supercritical phase p > p c , and high values p n (ω) p c to configurations which are "typical" of the subcritical phase p < p c . For example, if F n = |C max |, a configuration ω with a largest cluster containing a number of vertices of order n d will be assigned a very low value p n (ω) p c . Yet, for this value of the parameter p in Bernoulli percolation, it is very unlikely to have such a large cluster, which will give ω a very low weight in the measure µ n . Indeed, we will show that under µ n , configurations which are either "typically subcritical" or "typically supercritical" have a very low probability. Therefore, the mass of µ n concentrates on configurations ω with p n (ω) sufficiently close to p c , hence the self-critical behaviour of our model. In fact, the difficult point is to show that the weight of the "typically" supercritical or subcritical configurations is much smaller than the weight of the quasi-critical configurations (see paragraph 1.6.2).

Note that our parameter a does not need to be finely tuned for our result to hold, showing the robustness of the construction. Indeed, one could expect a different behaviour depending on whether a is smaller or larger than the so-called fractal dimension d f of the incipient infinite cluster (see for example [START_REF] Borgs | The birth of the infinite cluster: finite-size scaling in percolation[END_REF]), but p n tends to p c regardless of a. In fact, one can conjecture that, if a > d f , then our p n will tend to p c "from above", and the configurations in our model might look slightly supercritical, while they might look slightly subcritical when a < d f . This is plausible because the definition of our model more or less amounts to forcing the size of the largest cluster (or |M n |, or B b n ) to be of order n a . Our list of three models is of course not comprehensive, since many variants could be defined using the same approach. For example, the case of the largest cluster can be extended to the largest cluster in the torus, which means we can set periodic boundary conditions on the box Λ(n). In the model defined with B b n (point (iii) of theorem 1), one could consider the distribution of the cluster diameters instead of the cluster sizes, by setting

B b n (ω) = x ∈ Λ(n) : diam C(x, ω) n b , (2)
which gives exactly the same convergence result, under the same conditions for a and b, and with a similar estimate on the convergence speed.

Outline of the article

The proof of each case of theorem 1 requires two main steps. Recall the definition (1) of our model: for every percolation configuration ω, we have µ n (ω) = P pn(ω) (ω)/Z n . The first step is to prove that P pn(ω) (ω) tends to 0 exponentially fast and uniformly over all configurations ω for which p n (ω) / ∈ [p c -ε, p c + ε], for a fixed ε > 0. This step, described in paragraph 1.6.1, relies on classical large deviation estimates far from the critical point. But this step is not sufficient to prove our result, because of the normalization constant Z n . Therefore, the second step is to obtain an adequate lower bound on this partition function Z n . This step relies on a monotone coupling of percolation configurations and the search for a fixed point of a certain function (see paragraph 1.6.2). The crucial tool to construct this fixed point is a geometric surgery procedure, which allows to cut finite subgraphs of Z d in pieces of a given size, without closing too many edges. This geometric lemma is proved in section 3, after some standard definitions and notations are given in section 2. The last three sections 4, 5 and 6 are devoted to the proofs of the three items of theorem 1, each section containing the two steps described above (first the exponential decay estimates far from p c , and then the lower bound on Z n ). Eventually, theorem 2 is proved at the end of section 6.

Exponential decay estimates far from p c

Let ε be such that 0 < ε < min(p c , 1 -p c ). We start with an upper bound on the right tail of the law of p n . To this end, we define

t + n = n a -ln(p c + ε) . (3)
Grouping the configurations according to the value of F n , we can write

µ n p n > p c + ε = 1 Z n ω∈{0,1} En 1 {pn(ω)>pc+ε} P pn(ω) (ω) = 1 Z n ω∈{0,1} En 1 {Fn(ω)<t + n } P pn(ω) (ω) = 1 Z n t + n -1 t=0 ω∈{0,1} En 1 {Fn(ω)=t} P pn(ω) (ω) = 1 Z n t + n -1 t=0 P ϕn(t) F n = t (4) 1 Z n t + n -1 t=0 P ϕn(t) F n < t + n .
Yet, the variables F n are increasing, whence

µ n p n > p c + ε n d Z n P pc+ε F n < -ln(p c + ε) n a .
(5)

Similarly, we can show that

µ n p n < p c -ε n d Z n P pc-ε F n > -ln(p c -ε) n a . (6)
Therefore, the first step is to obtain exponential decay estimates for

P pc+ε F n < -ln(p c + ε) n a and P pc-ε F n > -ln(p c -ε) n a (7)
with ε > 0 fixed. This is done in subsections 4.1 and 4.2 in the case of F n = |C max | (case (i) of theorem 1), in subsections 5.1 and 5.2 for F n = |M n | (case (ii) of theorem 1) and 6.1 and 6.2 with F n = B b n (case (iii) of theorem 1). The estimates we obtain there are quite standard and follow from classical results in the literature about the behaviour of the cluster sizes in the subcritical and supercritical phases.

0 n d F n ω(t) n d t • • • • • • • F n ω ( t ) = t • F n t but "close" to t t = T Figure 1:
The partition function Z n may be expressed as the probability that the random function t → F n ω(t) admits a fixed point. This allows for the construction of a scenario where we can force such a fixed point to appear, with a reasonable probabilistic cost.

Lower bound on the partition function

The second step, which is the crucial and more interesting step, is to obtain a lower bound on the partition function Z n . Indeed, to show that (5) and ( 6) tend to 0 as n tends to infinity, one must not only show that the two terms in (7) are small enough, but also that Z n is not too small. To obtain this lower bound, we rewrite the partition function as

Z n = ω∈{0,1} En P pn(ω) (ω) = n d t=0 ω∈{0,1} En Fn(ω)=t P ϕn(t) (ω) = n d t=0 P ϕn(t) F n = t .
To make this expression more concrete, we construct a decreasing coupling ω(0) ω(1)

• • • ω(n d
) of percolation configurations, such that for every t ∈ 0, . . . , n d , the configuration ω(t) is distributed according to ∼ P ϕn(t) . Then Z n rewrites as

Z n = n d t=0 P F n ω(t) = t = P ∃t ∈ 0, . . . , n d F n ω(t) = t . (8)
Hence, the partition function Z n is equal to the probability that the random non-increasing function t → F n ω(t) admits a fixed point. This leads us to build the coupling step by step, and to consider a (random) stopping time T located just before this function goes under the first bisector (see figure 1). We then obtain a lower bound on the probability that the next steps of the coupling lead to a fixed point.

Because this instant T when we try to force a fixed point typically occurs for a percolation parameter close to p c , the classical estimates available in subcritical or supercritical percolation are of no use. Indeed, we need to study the behaviour of F n as p decreases towards p c , and to show that F n does not vary too abruptly close to the critical point. Our problem is therefore closely related to a question of finite-size scaling, i.e., the behaviour of the model when one takes n → ∞ and p → p c simultaneously (see [START_REF] Borgs | The birth of the infinite cluster: finite-size scaling in percolation[END_REF][START_REF] Garban | The scaling limits of near-critical and dynamical percolation[END_REF]).

Yet, we are able to bypass the use of (unproven) scaling laws thanks to the geometric argument of section 3, which is quite general and does not rely on the near-critical behaviour of F n . Roughly speaking, this geometric result indicates that to cut a piece of a precise size out of a subgraph of Z d of size N , one only needs to close O(N (d-1)/d ) edges. This geometric argument allows us to implement a surgery procedure on the configuration ω(T ) which leads to a fixed point by forcing a reasonable number of edges to be closed in the subsequent steps of the coupling. The surgery procedure is different for each of the three considered models, but the core ingredient is always this graph separation result.

Remark. An important goal is to build a similar model of self-organized criticality associated with the Ising model. A natural strategy consists in adapting the results presented here to the FK percolation model. However, a major complication arises with the FK model. Indeed, in a dynamical coupling of the FK processes, there is already a phenomenon of self-organized criticality in the way the edges become open when one approaches the critical point from below [START_REF] Duminil-Copin | The near-critical planar FK-Ising model[END_REF]. Whereas in dynamical Bernoulli percolation, the opening times of the edges are independent, in FK percolation this independence property is lost, and groups of edges tend to become open simultaneously when p becomes close to p c . As a consequence, our construction of the fixed point using the geometric surgery procedure does not work any more in FK percolation, because it would require to control this phenomenon of simultaneous openings of edges, which is not yet well understood. Yet, in the article [START_REF] Forien | A planar Ising model of self-organized criticality[END_REF], we have managed to bypass this problem in the particular setting of the planar FK-Ising model, using the estimates about the near-critical regime proved by [START_REF] Cerf | The 2D-Ising model near criticality: a FK-percolation analysis[END_REF] in this context.

Definitions and notations

The box

We fix an integer d 2 for the whole article. Let E d be the set of edges between nearest neighbours of Z d :

E d = {x, y} ⊂ Z d : x -y 1 = 1 .
Let n 1. Let us consider the box centered at 0 and containing n d vertices,

Λ(n) = - n 2 , n 2 d ∩ Z d = - n 2 , . . . , n -1 2 d .
For V ⊂ Z d a set of vertices, we write

E [V ] = {x, y} ⊂ V : x -y 1 = 1
for the set of edges in E d connecting two vertices of V , and we write in particular

E n = E [Λ(n)].
The inner boundary of the box Λ(n) will be denoted

∂Λ(n) = x ∈ Λ(n) : ∃ y ∈ Z d \Λ(n) x -y 1 = 1 .

Bernoulli percolation

For 0 p 1, on the space {0, 1}

E d
equipped with the σ-field generated by events depending on finitely many edges, let P p be the product measure such that the state of each edge follows a Bernoulli law of parameter p. An element ω :

E d → {0, 1}
ω e : f ∈ E d -→ 0 if f = e , ω(f ) otherwise
for the configuration obtained from ω by closing the edge e. Similarly, for any configuration ω : E d → {0, 1} and any set of edges H ⊂ E d , we will write

ω H : f ∈ E d -→ 1 if f ∈ H , ω(f ) otherwise and ω H : f ∈ E d -→ 0 if f ∈ H , ω(f ) otherwise
for the configurations obtained from ω by opening or closing all the edges of H. These notations naturally extend to configurations ω : E n → {0, 1} on the edges of the box Λ(n).

Clusters

Let ω : E d → {0, 1} be a percolation configuration on Z d . For x, y ∈ Z d , we write x ω ←→ y if there exists a path of open edges in the configuration ω joining x and y. For x ∈ Z d , we will write

C(x) = C(x, ω) = y ∈ Z d : x ω ←→ y for the connected component of x, which is called the cluster of x in ω. If x ∈ Z d and Y ⊂ Z d , we write x ω ←→ Y ⇐⇒ ∃ y ∈ Y x ω ←→ y .
All these notations naturally extend to percolation configurations restricted to the box Λ(n). Thus, for ω : E n → {0, 1} and x ∈ Λ(n), we will write C(x, ω) (or C(x)) for the set of the vertices in Λ(n) which are connected to x in Λ(n) by an open path in the configuration ω. When it is not clear whether we consider paths which stay in the box or not, for example if ω is defined on E d , we will specify C Λ(n) (x) to denote the set of the vertices which are connected to x by an open path with all its intermediate vertices belonging to Λ(n), i.e., the cluster of x in the configuration restricted to E n . For a percolation configuration ω : E n → {0, 1} in the box Λ(n), we will denote by C max (ω), or sometimes C max (Λ(n)), the largest cluster in ω, speaking in terms of the number of vertices. In case of equality between several maximal clusters, we choose one of them with an arbitrary order on subsets of Λ(n). For ω :

E n → {0, 1},
we also define

M n (ω) = x ∈ Λ(n) : x ω ←→ ∂Λ(n) and B b n (ω) = x ∈ Λ(n) : C Λ(n) (x, ω) n b , where b > 0 is a fixed parameter. Given p ∈ [0, 1], let θ(p) = P p |C(0)| = ∞
be the probability that the origin lies in an infinite open cluster in a percolation configuration drawn according to P p . We will write p c for the critical point of Bernoulli percolation in dimension d, defined by

p c = inf p ∈ [0, 1] : θ(p) > 0 .
3 Geometrical interlude

Main result

The purpose of this section is to show the following geometric inequality, which one could sum up as "separating a cluster of a given size in a graph (V, E) requires at most O(|V | (d-1)/d ) edges".

Lemma 1. There exists a constant

K = K(d) such that, for any finite connected subgraph G = (V, E) of (Z d , E d ),
for any vertex x ∈ V and for any integer m such that 1 m |V |, there exists a subset

E 0 ⊂ E of edges of G with cardinality |E 0 | K |V | d-1 d
such that the connected component of x in the graph (V, E\E 0 ) contains exactly m vertices.

We decompose the proof of this lemma in two steps. In section 3.2, we prove the "butcher's lemma", which allows to cut a graph into small components, which may be too small, in particular the component of x might have a cardinality strictly smaller than the goal size m. In section 3.3, we prove the "surgeon's lemma", which involves an adequate algorithm to reopen some of the edges closed by the butcher's lemma in order to reach the goal size m for the cluster of x.

The "butcher's lemma"

We start with an upper bound on the number of edges that one needs to remove from a connected graph to divide it into pieces which are all smaller than half of the initial graph.

Lemma 2 (The butcher's lemma). For every finite subgraph

G = (V, E) of (Z d , E d ), there exists a subset E 0 ⊂ E of edges of G with cardinality |E 0 | 4 d+1 d 2 |V | d-1 d
such that any connected component of the graph (V, E\E 0 ) contains at most |V | /2 vertices.

This separation lemma, which can be summarized by "cutting a graph in two requires O(|V | (d-1)/d ) edges", was proved in [START_REF] Benjamini | On the separation profile of infinite graphs[END_REF], corollary 3.3. For completeness, we present here a self-contained proof of this geometric result for the case of Z d . The more general technique of [START_REF] Benjamini | On the separation profile of infinite graphs[END_REF] would make it possible to extend our result to more general graphs, but we choose here to restrict our presentation to the d-dimensional square grid. For x ∈ Z d , we will write its coordinates x = (x 1 , . . . , x d ). For any finite non-empty subset V ⊂ Z d and any i ∈ {1, . . . , d}, we define

diam i V = max x∈V x i -min x∈V x i and diam V = max 1 i d diam i V .
If i ∈ {1, . . . , d} and m ∈ Z, then

T i,m = e = {x, y} ∈ E d : x i = m and y i = m + 1
will denote the slice of edges cutting Z d in two parts in the direction i between abscissa m and m + 1. We first prove an auxiliary lemma.

Lemma 3. For every k ∈ N and for any real number A 4, given a subgraph

G = (V, E) of (Z d , E d ) such that |V | A d and diam V 3 2 k (A -1) ,
there exists a subset E 0 ⊂ E of edges of G with cardinality

|E 0 | 2A d-1 + 36d 2 1 - 2 3 k A d-1
such that any connected component of the graph (V, E\E 0 ) contains at most A d /2 vertices.

Remark. In the sequel, this lemma will only be used with A = |V | 1/d but it will be helpful for the proof to keep this parameter A fixed rather than have it depending on the graph. Proof. Fix A 4. We will proceed by induction on k, and therefore we start with the case k = 0. Let G = (V, E) be a subgraph of (Z d , E d ) such that |V | A d and diam V A -1. Without loss of generality, we can assume that V ⊂ Λ(diam V + 1). Let us choose

E 0 = E ∩ T 1,-1 ∪ T 1,0 , whose cardinality satisfies |E 0 | 2 diam V + 1 d-1 2A d-1 . If C ⊂ V is a connected component of (V, E\E 0 ), then we have |C| max diam V 2 , diam V + 1 2 diam V + 1 d-1 diam V + 1 d 2 A d 2 .
We now perform the induction step. Take k 1 such that the result holds for k -1.

Let G = (V, E) be a subgraph of (Z d , E d ) such that |V | A d and diam V 3 2 k (A -1) .
We are going to trim the graph G to decrease its diameter by a factor 2/3. To this end, we will remove slices of edges in the directions i in which the diameter is "too big". Consider

I = i ∈ {1, . . . , d} : diam i V > 3 2 k-1 (A -1) ,
and take i ∈ I. Without loss of generality, one can assume that min x∈V x i = 0. By the pigeonhole principle, there exists an integer k i satisfying

diam i V 3 < k i 2 diam i V 3 and |E ∩ T i,ki | |E| diami V 3 .
We choose such a k i and we write, recalling that A 4,

diam i V 3 diam i V 3 - 2 3 1 3 3 2 k-1 (A -1) - 2 3 = 1 9 3 2 k-1 (A -1) + 2 9 3 2 k-1 (A -1) -3 1 9 3 2 k-1 (A -1) 1 9 3 2 k-1 3 4 A = 1 12 3 2 k-1 A . Noting that |E| d |V | dA d , we get |E ∩ T i,ki | 2 3 k-1 12 |E| A 2 3 k-1 12dA d A = 12d 2 3 k-1 A d-1 .
Consider now

E 1 = i∈I (E ∩ T i,ki ) ,
whose cardinality satisfies

|E 1 | 12d 2 2 3 k-1 A d-1 .
Let G = (V , E ) be a maximal connected component of the graph (V, E\E 1 ), in terms of number of vertices.

By construction, we have that, for i ∈ I,

diam i V max k i , diam i V -(k i + 1) 2 3 diam i V 3 2 k-1 (A -1) ,
while for i / ∈ I, the definition of I implies

diam i V diam i V 3 2 k-1 (A -1) .
Taking the maximum over i yields

diam V 3 2 k-1 (A -1) .
Besides, note that |V | |V | A d . Hence, by the induction hypothesis applied to G , there exists

E 2 ⊂ E such that |E 2 | 2A d-1 + 36d 2 1 - 2 3 k-1 A d-1 ,
and all connected components of the graph (V , E \E 2 ) contain at most A d /2 vertices. Now take

E 0 = E 1 ∪ E 2 .
We have

|E 0 | = |E 1 | + |E 2 | 12d 2 2 3 k-1 A d-1 + 2A d-1 + 36d 2 1 - 2 3 k-1 A d-1 = 2A d-1 + 36d 2 1 - 2 3 k A d-1 . If C is a connected component of the graph (V, E\E 0 ), then either C ⊂ V \V which, by maximality of V , entails |C| |V | /2 A d /2, or C ⊂ V in which case C turns out to be a connected component of the graph (V , E \E 2 ), which implies |C| A d /2 .
We can now prove the butcher's lemma, which is a mere rephrasing of lemma 3.

Proof of lemma 2. If |V | 4 d , this is a straightforward consequence of lemma 3 with

A = |V | 1/d and k = d ln A -ln(A -1) ln 3 -ln 2 because we then have diam V |V | = A d A -1 (A -1) 3 2 k (A -1)
and the lemma provides us with a subset E 0 ⊂ E with cardinality satisfying

|E 0 | 2 + 36d 2 A d-1 4 d+1 d 2 |V | d-1 d such that all connected components of (V, E\E 0 ) contain at most A d /2 = |V | /2 vertices. Otherwise, if we have |V | < 4 d , then E 0 = E answers the problem.

3.3

The "surgeon's lemma"

The application of the butcher's lemma allows us to separate a graph into connected components which are at least twice smaller than the original graph. If the connected component of x in the remaining graph still contains more vertices than the goal size m, one can apply again the butcher's lemma to this component of x, to obtain a connected component which contains at most a fourth of the initial number of vertices. This operation can be repeated until the connected component of x contains strictly less than m edges, which means that we have closed too many edges. The surgeon's lemma will fix this problem, by reopening some of the edges closed by the butcher's lemma.

Lemma 4 (The surgeon's lemma). Let k ∈ N and let G = (V, E) be a connected subgraph of (Z d , E d ) with |V | 2 k . Let x ∈ V and let m be an integer such that 1 m |V |. There exists a subset E 0 ⊂ E of edges of G with cardinality satisfying

|E 0 | 1 -a k 1 -a 4 d+1 d 2 |V | d-1 d , where a = 1 2 d-1 d
, such that, in the graph (V, E\E 0 ), the connected component of x contains exactly m vertices.

Proof. We proceed by induction on k. The result is trivial if k = 0, so we perform next the induction step.

Take k 1 such that the result holds for k -

1. Let G = (V, E) be a connected subgraph of (Z d , E d ) with 2 k-1 < |V | 2 k , let x ∈ V and let m be an integer such that 1 m |V |. According to lemma 2, we can choose a subset E 0 ⊂ E of cardinality |E 0 | 4 d+1 d 2 |V | d-1 d
such that any connected component of the graph (V, E\E 0 ) contains at most 2 k-1 vertices. The idea is to reopen the edges of E 0 one by one starting from the cluster of x, in order to make this cluster grow until it reaches or exceeds the size m. Then we will apply the induction hypothesis on the last piece added, which contains at most 2 k-1 vertices.

We are going to order the edges of E 0 by exploring them one by one starting from the cluster of x. We start by writing V 0 for the connected component of x in the graph (V, E\E 0 ). We have that

|V 0 | 2 k-1 < |V |, hence V 0
V . Yet the graph (V, E) is connected, therefore we can choose an edge e 1 ∈ E 0 incident to this cluster V 0 . Assume now that we have defined e 1 , . . . , e s ∈ E 0 for some s 1. Let V s be the connected component of x in the graph

V, E \ (E 0 \ {e 1 , . . . , e s }) .

If s < |E 0 |, then we can choose an edge e s+1 ∈ E 0 incident to V s . Such an edge exists because (V, E) is connected. We proceed with this construction until all the edges of E 0 are ordered in a sequence e 1 , . . . , e r where r = |E 0 |. We have then

x ∈ V 0 ⊂ V 1 ⊂ . . . ⊂ V r = V .
If we close all the edges of E 0 and then reopen these edges one by one in the order e 1 , . . . , e r , then after having reopened s edges, the cluster of x is V s . Therefore, we introduce

V 2 = V σ V 0 V 1 \V 0 V = V 2 \V 1 = V σ \V σ-1 V 4 \V 3 e 1 • x e 2 = e σ • x
Figure 5: Illustration of the proof of lemma 4: closing the edges of E 0 = {e 1 , e 2 , e 3 , e 4 } cuts the graph in pieces containing at most 2 k-1 vertices. We reopen the edges e i in this order until the number of vertices in the cluster of x reaches or exceeds m. In the case drawn here, σ = 2, and V 3 = V 2 because the edge e 3 connects two vertices which already belong to V 2 .

we have

|V σ-1 | < m |V σ |, hence V σ = V σ-1 . In that case, the edge e σ must connect a vertex of V σ-1 to a vertex x ∈ V σ \V σ-1 . Letting m = m -|V σ-1 |, we have that 1 m |V σ | -|V σ-1 | = |V σ \V σ-1 | .
Otherwise, if σ = 0, we set x = x and m = m, which entails 1 m |V 0 |.

Let us consider the graph G = (V , E ) of the connected component of x in (V, E\E 0 ). The choice of E 0 ensures that |V | 2 k-1 . What's more, we have that V = V σ \V σ-1 if σ 1 and V = V 0 otherwise, which in both cases leads to 1 m |V |. The induction hypothesis applied to the graph G = (V , E ) gives us a subset E 0 ⊂ E satisfying

|E 0 | 1 -a k-1 1 -a 4 d+1 d 2 |V | d-1 d 1 -a k-1 1 -a 4 d+1 d 2 a |V | d-1 d
and such that the connected component of x in (V , E \E 0 ), which will be denoted V x , contains exactly m vertices. Now, we consider the set

E 0 = {e σ+1 , . . . , e r } ∪ E 0 ,
which is such that

|E 0 | = (r -σ) + |E 0 | 4 d+1 d 2 |V | d-1 d + a -a k 1 -a 4 d+1 d 2 |V | d-1 d = 1 -a k 1 -a 4 d+1 d 2 |V | d-1 d . If σ = 0, then the connected component of x in the graph (V, E\E 0 ) is V x and thus it contains exactly m = m vertices. Otherwise, if σ 1, then this connected component is V σ-1 ∪ V x , which contains |V σ-1 | + m = m vertices.
4 Proof of case (i ) of theorem 1

This section is devoted to the proof of the item (i) of theorem 1. In this case, the function p n is defined by p n (ω) = exp(-|C max (ω)| /n a ), where C max (ω) denotes the largest cluster in the box Λ(n) in the configuration ω. As explained in the introduction, the first step is to show the exponential decay of the distribution of |C max | in the subcritical and supercritical phases.

Exponential decay in the subcritical phase

We first present a classical estimate about the size of the largest cluster below p c :

Lemma 5. For any a ∈ (0, d), for p < p c and A > 0, we have

-∞ < lim inf n→∞ 1 n a ln P p C max Λ(n) > An a lim sup n→∞ 1 n a ln P p C max Λ(n) > An a < 0 .
Proof. Let a > 0, p < p c and A > 0. For all n 1, we have that

P p C max Λ(n) > An a = P p max v∈Λ(n) C Λ(n) (v) > An a P p max v∈Λ(n) |C(v)| > An a n d P p |C(0)| > An a .
According to theorem 6.75 in [START_REF] Grimmett | of Grundlehren der Mathematischen Wissenschaften[END_REF], there exists a constant λ(p) > 0 such that, for all m 1,

P p |C(0)| m e -mλ(p) . (9)
It follows that, for all n 1,

P p C max Λ(n) > An a
n d e -Aλ(p)n a , which implies the desired upper bound. To create a cluster of size more than An a , one may simply open a self-avoiding path of An a edges and An a + 1 vertices, hence

P p C max Λ(n) > An a
p An a , which shows the lower bound.

Exponential decay in the supercritical phase

We establish a corresponding result in the supercritical regime: Lemma 6. For all a ∈ (0, d), for p > p c and A > 0, we have

-∞ < lim inf n→∞ 1 n d-a/d ln P p C max Λ(n) < An a lim sup n→∞ 1 n d-a/d ln P p C max Λ(n) < An a < 0 .
The upper bound is a consequence of the following result, which easily follows from the classical literature:

Lemma 7. For all p > p c , we have

lim sup n→∞ 1 n d-1 ln P p C max Λ(n) θ(p)n d 8 < 0 .
Proof. Assume first that d 3. From theorem 1.2 of [START_REF] Pisztora | Surface order large deviations for Ising, Potts and percolation models[END_REF], it follows that, for d 3, for all p > p c (where p c denotes the slab-percolation threshold),

lim sup n→∞ 1 n d-1 ln P p |C max (Λ(n))| θ(p)n d 2 < 0 .
In addition, Grimmett and Marstrand proved the identity p c = p c for d 3 in [START_REF] Grimmett | The supercritical phase of percolation is well behaved[END_REF]. The claim for d 3 thus follows immediately.

Consider now the case d = 2. Theorem 6.1 of [START_REF] Alexander | The Wulff construction and asymptotics of the finite cluster distribution for two-dimensional Bernoulli percolation[END_REF] implies that, for all p > p c , if we consider a percolation configuration on Z d and write C ∞ ⊂ Z d for the unique infinite cluster of the configuration, then

lim n→∞ 1 n ln P p |C ∞ ∩ Λ(n)| θ(p)n 2 2 < 0 .
Thereby, there exists L > 0 such that, for all n 1,

P p |C ∞ ∩ Λ(n)| θ(p)n 2 2 e -Ln .
Besides, if we set, for m k 1,

L k, m =
P p (L k, m ) 1 -C 2 me -C3k . ( 10
)
Define the rectangles

R 1 = Z 2 ∩ n 2 , n × [-n, n[ , R 2 = Z 2 ∩ [-n, n[ × n 2 , n , R 3 = Z 2 ∩ -n, - n 2 × [-n, n[ , R 4 = Z 2 ∩ [-n, n[ × -n, - n 2 ,
which are represented in figure 6. Following a classical argument (see the proof of theorem 7.61 in [START_REF] Grimmett | of Grundlehren der Mathematischen Wissenschaften[END_REF]), we As illustrated on figure 6, we have the inclusion F n ⊂ E n . In addition, by the FKG inequality, we have that

R 1 R 3 R 2 R 4 • x • y Λ(n) Λ(2n) 2n n
P p (F n ) P p L n/2 , 2n 4 .
In combination with (10), this yields

P p (E n ) P p (F n ) P p L n/2 , 2n 4 1 -2C 2 ne -C3 n/2 4 1 -8C 2 ne -C3 n/2 .
Yet if the event E n occurs, then all the vertices of Λ(n) which are connected by an open path to the boundary of Λ(2n) must be connected to each other inside Λ(2n), which implies that |C max (Λ(2n))| |C ∞ ∩ Λ(n)|. Therefore, we have the inclusion

E n ∩ |C ∞ ∩ Λ(n)| > θ(p)n 2 2 ⊂ C max (Λ(2n)) > θ(p)n 2 2 .
Considering complementary events leads to

P p |C max (Λ(2n))| θ(p)n 2 2 1 -P p (E n ) + P p |C ∞ ∩ Λ(n)| θ(p)n 2 2 8C 2 ne -C3 n/2 + e -Ln e -L n
for a certain constant L > 0, which concludes the proof.

We now briefly explain how to deduce lemma 6 from lemma 7:

Proof of lemma 6. We divide the box Λ(n) into smaller boxes of side

N n = 8An a θ(p) 1/d .
The box Λ(n) contains at least n/N n d disjoint boxes of side N n , so that we have

P p C max Λ(n) < An a P p C max Λ(N n ) < An a n/Nn d P p C max Λ(N n ) < θ(p)N d n 8 n/Nn d , which implies that lim sup n→∞ 1 n d-a/d ln P p C max Λ(n) < An a lim sup n→∞ n d N d n n d-a/d ln P p C max Λ(N n ) < θ(p)N d n 8 = θ(p) 8A 1/d lim sup N →∞ 1 N d-1 ln P p C max Λ(N ) < θ(p)N d 8 < 0 ,
where the last inequality comes from lemma 7. To obtain the lower bound, we divide the box Λ(n) into boxes of side N n = (An a ) 1/d -1, which all contain strictly less than An a vertices, and we consider the event that all the edges between two neighbouring boxes are closed. This leads to

ln P p C max Λ(n) < An a dN d-1 n n N n d ln(1 -p) n→∞ ∼ θ(p) 8A 1/d d ln(1 -p)n d-a/d ,
which shows that d -a/d is indeed the correct exponent.

Lower bound on the partition function

We show here the following inequality on the normalization constant Z n of our model:

Lemma 8. For any real number a such that 0 < a < d, we have

lim inf n→∞ ln Z n (ln n)n a(d-1)/d > -∞ .
Proof. As explained in the introduction, we define a monotone coupling of the probability distributions P ϕn(t) for t ∈ 0, . . . , n d . with Bernoulli law of parameter exp(-1/n a ). For t 0 ∈ 0, . . . , n d , define a random configuration

Construction

ω(t 0 ) : e ∈ E n -→ min 0 t<t0 X t,e .
Hence, for t 0 ∈ 0, . . . , n d and e ∈ E n , we see that

P ω(t 0 )(e) = 1 = t0-1 t=0 P X t,e = 1 = exp - t 0 n a = ϕ n (t 0 ) ,
therefore the configuration ω(t 0 ) has distribution P ϕn(t0) . What's more, configurations are coupled in such a way that

1 En = ω(0) ω(1) • • • ω(n d ) .
When going from the configuration ω(t) to the configuration ω(t + 1), a certain number or edges are closed (these are the edges e such that ω(t)(e) = 1 and X t,e = 0). In order to control the edge closures one by one, we define intermediate configurations. For t ∈ 0, . . . , n d -1 and s 0 ∈ {0, . . . , r}, we set ω(t, s 0 ) : e s ∈ E n -→ ω(t + 1)(e s ) if s s 0 , ω(t)(e s ) otherwise.

In this way, we have ω(t, 0) = ω(t) and for s 1, the configuration ω(t, s) is obtained from the configuration ω(t, s -1) by closing the edge e s if X t,es = 0, and by keeping everything unchanged if X t,es = 1. For s = r = |E n |, all edges have been updated, so ω(t, r) = ω(t + 1). The configurations are therefore coupled in such a way that (t, s) (t , s ) =⇒ ω(t, s) ω(t , s ) ,

where we use the lexicographic order on {0, . . . , n d -1} × {0, . . . , r}. As we have shown in (8), the partition function Z n is equal to the probability that the non-increasing function t → C max ω(t) admits a fixed point. Thus, we now look for an instant t = T situated before this function goes under the first bisector, and we will study what is needed on the variables X t,e for this function to actually cross the bisector at the instant t = T + 2.

Definition of the instant T : Still considering the lexicographic order, we define a pair of random variables

(T, S) = min (t, s) ∈ 0, . . . , n d -2 × {0, . . . , r} : ∃e ∈ E n C max ω(t, s) e t + 2 .
This minimum is well defined because one always has C max ω(n d -2, 0) n d . In addition, for every (t 0 , s 0 ), the event {(T, S) = (t 0 , s 0 )} only depends on the variables X t, es for (t, s) (t 0 , s 0 ), which means that (T, S) is a stopping time for the filtration generated by the variables X t,es . Also, closing one single edge cannot divide the size of the largest cluster by more than two, whence Let us prove that we also have

C max ω(T, S) 2(T + 2) . (11)
• (t, s) = (0, 0) • (T, S) • (T + 2, 0) ω = 1 En |C max (ω)| T + 2 ∃e ∈ E n |C max (ω e )| T + 2 C max |C max (ω)| = T + 2 close H H C max
C max ω(T, S) T + 2 . ( 12
)
We distinguish several cases.

• If S 1, then the minimality of (T, S) ensures that, for all e ∈ E n , C max ω(T, S -1) e > T + 2 .

Yet the configuration ω(T, S) is obtained from ω(T, S -1) by closing at most one edge, whence (12).

• If S = 0 and T 1, then, (T, S) being minimal, we have that

C max ω(T -1, r) > T -1 + 2 = T + 1 .
The configurations ω(T -1, r) and ω(T, 0) being identical, inequality (12) is also satisfied.

• The case (T, S) = (0, 0) does not happen because all edges are open in the configuration ω(0, 0). We build next a happy event, which implies the existence of the desired fixed point.

Construction of the happy event: Let (V, E) be the graph associated to the largest cluster in ω(T, S), that is to say V = C max ω(T, S) and E is the set of the edges between two vertices of V which are open in ω(T, S). Given (12), it follows from lemma 1 that there exists a (random) set of edges

H = H T, ω(T, S) ⊂ E , satisfying |H| K |V | d-1 d (13)
and such that the largest connected component of the graph (V, E\H) contains exactly T + 2 vertices. Note that we have defined H = H(T, ω(T, S)) as a deterministic function of the variables T and ω(T, S), this will be useful later. The existence of an edge e ∈ E n such that

C max ω(T, S) e T + 2
entails that, in ω(T, S), there is at most one cluster containing strictly more than T + 2 vertices. Thus, closing the edges of H is enough to ensure that the remaining largest cluster contains exactly T + 2 vertices, i.e.,

C max ω(T, S) H = T + 2 .
Hence, closing the edges of H and no other edge of E C max ω(T, S) between the instants (T, S) and (T + 2, 0)

ensures that C max ω(T + 2) = T + 2. However, the edges e s ∈ H are not necessarily labeled with numbers s > 20 S. It is therefore not generally possible to close all the edges of H between the instants (T, S) and (T + 1, 0).

For this reason, the event we consider is the one in which no edge of E C max ω(T, S) is closed between (T, S) and (T +1, 0), and the edges of E C max ω(T, S) which are closed between (T +1, 0) and (T +2, 0) are precisely the edges of H, that is to say

E =      ∀s > S e s ∈ E C max ω(T, S) ⇒ X T, es = 1 ∀e ∈ H X T +1, e = 0 ∀e ∈ E C max ω(T, S) \H X T +1, e = 1      .
If this event occurs, then in ω(T + 2), all the edges of H are closed, the other edges of E C max ω(T, S) which were open in the configuration ω(T, S) remain open, and all the other clusters contain at most T + 2 vertices, whence

E ⊂ C max ω(T + 2) = C max ω(T, S) H = T + 2 .
Conditional probability of the happy event: Coming back to the expression (8) of the partition function, we find that

Z n P C max ω(T + 2) = T + 2 P (E) . (14)
Let (t 0 , s 0 ) ∈ 0, . . . , n d -2 × {0, . . . , r} and ω 0 : E n → {0, 1} be such that P C t0, s0, ω0 > 0 where C t0, s0, ω0 = (T, S) = (t 0 , s 0 ) and ω(T, S) = ω 0 .

Having defined H as a deterministic function of T and ω(T, S), we can consider the event

E t0, s0, ω0 =      ∀s > s 0 e s ∈ E [C max (ω 0 )] ⇒ X t0, es = 1 ∀e ∈ H(t 0 , ω 0 ) X t0+1, e = 0 ∀e ∈ E [C max (ω 0 )] \H(t 0 , ω 0 ) X t0+1, e = 1     
, which satisfies

P E C t0, s0, ω0 = P E t0, s0, ω0 C t0, s0, ω0 . (15)
Now note that this event E t0, s0, ω0 depends only on the variables X t, es with (t, s) > (t 0 , s 0 ), whereas the event C t0, s0, ω0 depends only on the variables X t, es with (t, s) (t 0 , s 0 ). Thus, these two events are independent of each other, which allows us to write

P E t0, s0, ω0 C t0, s0, ω0 = P E t0, s0, ω0 = s>s0 es∈E[Cmax(ω0)] P X t0, es = 1 × e∈H(t0, ω0) P X t0+1, e = 0 × e∈E[Cmax(ω0)]\H(t0, ω0) P X t0+1, e = 1 e -1/n a 2|E[Cmax(ω0)]| 1 -e -1/n a |H(t0, ω0)| .
Combining this with (15) yields

P E (T, S, ω(T, S)) e -1/n a 2|E[Cmax(ω(T, S))]| 1 -e -1/n a |H(T, ω(T, S))| . ( 16
)
Yet, according to (11), we have

E [C max (ω(T, S))] d |C max (ω(T, S))| 2d(T + 2) .
Furthermore, by convexity of x → e -x , we get

1 -e -1/n a 1 n a 1 -e -1 1 2n a .
In addition, combining (13) and (11) leads to

|H| K |C max (ω(T, S))| d-1 d K 2(T + 2) d-1 d 2K T + 2 d-1 d .
Plugging the previous inequalities in equation ( 16), we obtain P E (T, S, ω(T, S))

exp - 4d (T + 2) n a 1 2n a 2K(T +2) d-1 d .
We take the conditional expectation with respect to T , and we deduce that

P E T exp - 4d (T + 2) n a 1 2n a 2K(T +2) d-1 d . ( 17
)
Upper bound on T : Next, we need a control on T in order to obtain a lower bound on P(E). Define

τ + n = n a -ln p c 2 . ( 18
)
Lemma 5 implies that

P pc/2 |C max | τ + n n→∞ -→ 1 .
This entails that, for n large enough,

P pc/2 |C max | τ + n 1 2 .
Given that

ϕ n τ + n ϕ n n a -ln p c 2 = p c 2 ,
we deduce that, for n large enough,

P T τ + n P C max ω(τ + n ) τ + n + 2 = P ϕn(τ + n ) |C max | τ + n + 2 P pc/2 |C max | τ + n + 2 1 2 .
Therefore, we can find κ 2 such that, for all n 1,

P T κn a 1 2 . ( 19
)
Conclusion: Combining ( 19) with (17) gives

P (E) P E ∩ {T κn a } = P T κn a P E T κn a 1 2 exp - 4d(κn a + 2) n a -2K(κn a + 2) d-1 d ln(2n a ) 1 2 exp -8dκ -4Kκ(ln 2)n a(d-1)/d -4Kκa(ln n)n a(d-1)/d ,
where we have used that 2 κn a . Now recall inequality (14) to deduce that

lim inf n→∞ ln Z n (ln n)n a(d-1)/d -4Kκa > -∞ ,
which is the required lower bound.

Proof of the convergence result

We are now in position to prove case (i) of theorem 1.

Proof of theorem 1, case (i). Let ε > 0 and a ∈ (0, d). Given that a(d -1)/d < a, the lower bound on Z n we have obtained in lemma 8 implies that lim inf n→∞ Z n /n a 0. Combining this with the result of lemma 5 and plugging it into the inequality (6) leads to

lim sup n→∞ 1 n a ln µ n p n < p c -ε lim sup n→∞ 1 n a ln P pc-ε |C max | > -ln(p c -ε) n a < 0 . (20)
Similarly, using lemma 6, inequality (5) and the fact that a(d -1)/d < d -a/d, we get

lim sup n→∞ 1 n d-a/d ln µ n p n > p c + ε lim sup n→∞ 1 n d-a/d ln P pc+ε |C max | < -ln(p c + ε) n a < 0 . ( 21
)
It remains to show that the exponent v = a ∧ (d -a/d) is optimal. To this end, we go back to our computation (4) and we recall that Z n was expressed as a probability in (8), whence Z n 1. Therefore, with t + n as defined in (3) we have

µ n p n > p c + ε = 1 Z n t + n -1 t=0 P ϕn(t) |C max | = t t + n -1 t=0 P ϕn(t) |C max | = t .
Using the notations of the last subsection, this becomes

µ n p n > p c + ε P ∃t ∈ 0, . . . , t + n -1 : C max ω(t) = t P E ∩ T t + n -3 , ( 22
)
since the occurrence of the event E implies that C max ω(T + 2) = T + 2. As we did in the proof of lemma 8, we can write

P T t + n -3 P C max ω(t + n -3) (t + n -3) + 2 = P ϕn(t + n -3) |C max | < t + n .
Now notice that ϕ n (t + n -3) → p c + ε, whence ϕ n (t + n -3) p c + 2ε for n large enough. Thus, for n large enough, we have

P T t + n -3 P pc+2ε |C max | < t + n .
Plugging this into (22) and using our lower bound (17) on the conditional probability of E with respect to T leads to

1 n d-a/d ln µ n p n > p c + ε - 4d t + n -1 n a+d-a/d - 2K t + n -1 d-1 d ln 2n a n d-a/d + 1 n d-a/d ln P pc+2ε |C max | < t + n = O 1 n d-a/d + O ln n n d-a + 1 n d-a/d ln P pc+2ε |C max | < t + n .
Taking the infinimum limit and using the lower bound given by lemma 6, we obtain

lim inf n→∞ 1 n d-a/d ln µ n p n > p c + ε > -∞ . ( 23
)
To handle the other tail, we choose a such that

a < a < d ∧ da d -1 ,
we define t - n = n a -ln(p c -ε) and we write

µ n p n < p c -ε P ∃t ∈ t - n + 1, . . . , n d : C max ω(t) = t P E ∩ T t - n -1 P E ∩ t - n -1 T n a . ( 24
)
It follows from (11) that

P T t - n -1 P C max ω(t - n -1) > 2 t - n + 1 P pc-2ε |C max | > 2 t - n + 1 .
Similarly, it follows from (12) that

P T > n a P C max ω n a n a + 2 P C max ω n a > n a P pc/2 |C max | > n a .
Therefore, we have

P t - n -1 T n a = P T t - n -1 -P T > n a P pc-2ε |C max | > 2 t - n + 1 -P pc/2 |C max | > n a e -Cn a -e -C n a e -Cn a 2 ,
with C, C > 0, using the exponential estimate of lemma 5. Plugging this into (24) and using again (17), we now obtain

1 n a ln µ n p n < p c -ε -C - ln 2 n a - 4d n a + 2 n 2a - 2K n a + 2 d-1 d n a ln 2n a = -C - ln 2 n a + O 1 n 2a-a + O ln n n a-a +a /d n→∞ -→ -C > -∞ . (25)
The first case of theorem 1 then follows from (20), ( 21), ( 23) and (25).

A variant on the torus

One can define a similar model on the torus of side n, which boils down to considering periodic boundary conditions on the box Λ(n). Clusters on the torus are at least as big as in the box, so the exponential decay in the supercritical phase for the model defined on the torus immediately follows from lemma 6. The analog of lemma 5 can be proved by noting that the size of the cluster of the origin in the torus is stochastically dominated by the size of the cluster of the origin in a configuration on all Z d . The same proof for the lower bound on the partition function applies in the case of the torus, by adapting our geometrical lemma to extend it to subgraphs of the torus. We therefore have the same convergence of p n to p c when n → ∞ for this alternative model.

Proof of case (ii ) of theorem 1

We prove here the point (ii) of theorem 1, namely the case of the model defined with F n = |M n |, where M n is the set of the vertices connected by an open path to the boundary ∂Λ(n) of the box Λ(n).

Exponential decay in the subcritical phase

Following the same method as for the first model, we start with an upper bound on the law of |M n | in the subcritical regime (the lower bound is straightforward, but we will not need it).

Lemma 9. For any a > d -1, for p < p c and A > 0, we have the upper bound

lim sup n→∞ 1 n a ln P p |M n | > An a < 0 . Proof. Take a > d -1, p < p c and A > 0. Write ∂Λ(n) = {x 1 , . . . , x t } with t = |∂Λ(n)|.
If A and T are two events, then A • T denotes the disjoint occurrence of these two events, which is defined in section 2.3 of [START_REF] Grimmett | of Grundlehren der Mathematischen Wissenschaften[END_REF].

Let ω : E n → {0, 1} be a configuration such that |M n (ω)| > An a . Define, for i ∈ {1, . . . , t},

n i = C Λ(n) (x i ) \ j<i C Λ(n) (x j ) = 0 if there exists j < i such that x i ω ←→ x j , C Λ(n) (x i ) otherwise.
We have that

t i=1 n i = t i=1 C Λ(n) (x i ) = |M n (ω)| > An a , and ω ∈ C Λ(n) (x 1 ) n 1 • . . . • C Λ(n) (x t ) n t .
Indeed, if n i = 0, then the event C Λ(n) (x i ) n i is trivial, whereas if we have n i > 0 and n j > 0 for some i = j, then the vertices x i and x j must belong to disjoint clusters. Whence the inclusion

|M n | > An a ⊂ 0 n1, ..., nt n d n1+•••+nt>An a C Λ(n) (x 1 ) n 1 • • • • • C Λ(n) (x t ) n t .
Note that, for all i ∈ {1, . . . , t}, the event C Λ(n) (x i ) n i is an increasing event, thus by the BK inequality,

P p |M n | > An a 0 n1,...,nt n d n1+•••+nt>An a t i=1 P p C Λ(n) (x i ) n i 0 n1,...,nt n d n1+•••+nt>An a t i=1 P p |C(0)| n i .
Furthermore, according to theorem 6.75 in [START_REF] Grimmett | of Grundlehren der Mathematischen Wissenschaften[END_REF], for p < p c , there exists a constant λ(p) > 0 such that, for all n 1,

P p |C(0)| n e -nλ(p) ,
which is also true if n = 0. It follows that

P p |M n | > An a 0 n1,...,nt n d n1+•••+nt>An a t i=1 exp -λ(p)n i 0 n1,...,nt n d exp -λ(p)An a n d + 1 t exp -λ(p)An a = exp |∂Λ(n)| ln(n d + 1) -λ(p)An a .
To conclude, note that

|∂Λ(n)| ln(n d + 1) = O (ln n)n d-1 = o(n a ) .
This completes the proof of the lemma.

Exponential decay in the supercritical phase

We now state a similar exponential decay property in the supercritical regime.

Lemma 10. For all a < d, for p > p c and A > 0, we have

lim sup n→∞ 1 n d-1 ln P p |M n | < An a < 0 .
Proof. Let p > p c and A > 0. As in the proof of lemma 6, we show that

lim sup n→∞ 1 n d-1 ln P p |M n | θ(p)n d 2 < 0 .
For d 3, the result follows from theorem 1.2 of [START_REF] Pisztora | Surface order large deviations for Ising, Potts and percolation models[END_REF], which proves it for p larger than p c , which was proved to be equal to p c in [START_REF] Grimmett | The supercritical phase of percolation is well behaved[END_REF]. In dimension d = 2, the claim follows from theorem 6.1 in [START_REF] Alexander | The Wulff construction and asymptotics of the finite cluster distribution for two-dimensional Bernoulli percolation[END_REF].

Lower bound on the partition function

We now establish a lower bound on the normalization constant Z n .

Lemma 11. For any real a such that d -1 < a < d, we have

lim inf n→∞ ln Z n (ln n)n a(d-1)/d > -∞ .
Heuristics of the proof: We wish to apply the same technique as in the proof for the case of the largest cluster (section 4.3), by constructing a decreasing coupling between the distributions P ϕn(t) for t varying from 0 (all edges open) to n d (almost all edges closed). We monitor the evolution of the variable |M n | until an instant t = T when |M n | is of order T . Then we find a set of edges H ⊂ E n whose closure would lead to

|M n | = T + 2 at the instant T + 2.
The hurdle is that, in order to find such a set H which is not too big (and thus whose closure is likely enough), we need a control on the size of the clusters which are connected to the boundary of the box at the instant T . To obtain such a control, a natural idea is to monitor first the evolution of the size of the clusters connected to the boundary, to wait for an instant T when these clusters have become small enough, and then to define the instant T in a way which ensures that it occurs later than T . However, unlike the size of the largest cluster C max , which can be at most divided by a factor 2 when closing an edge, the size of the largest cluster connected to the boundary can fall drastically with the closure of one edge. To avoid this, we choose to monitor the size of the largest cluster on the torus, that is to say in the box Λ(n) but with periodic boundary conditions. This variable has the advantage of being at most halved at each edge closure.

Proof. Sketch of the proof: We first define a decreasing coupling of configurations (ω(t, s)) t, s but on the edges of the torus. We then consider the first instant (T, S) when the largest cluster on the torus contains at most 2T +3 vertices. In what follows, we will reason conditionally on the fact that, at this instant, the largest cluster on the torus touches the boundary of the box. We will show that, at this instant, we have |M n (ω)| T + 2. Next we will construct a second instant (T , S ) (T, S) and a set of edges H such that, if the only edges of M n (ω) which are closed between (T , S ) and (T + 2, 0) are the edges of H, then we have |M n (ω(T + 2))| = T + 2. We will call this scenario the "happy event", and our aim is to obtain a lower bound on its probability. To this end, we will show that, with sufficiently high probability, we have T = O(n a ), which implies that, from the instant (T, S) onward, any of the clusters on the torus contains at most O(n a ) vertices. This control will allow us to show that it is possible to find H small enough to ensure that the happy event is likely enough. Construction of the coupling and definition of T : Take n 2. We use the same notations and definitions as in the proof of lemma 8, but we now consider configurations on the torus. To define the torus, write p : Z d → Λ(n) for the projection application, with is such that p(x) -x ∈ nZ d for every x ∈ Z d . The torus is the graph whose vertex set is Λ(n), and whose edge set is

• (0, 0) • (T, S) • (T , S ) • (T + 2, 0) ω = 1 En C T max C T max (ω) 2T + 3 |M n (ω)| C T max (ω) T + 2 M n |M n (ω)| T + 2 ∃e ∈ E n |M n (ω e )| T + 2 close H M n H |M n (ω)| = T + 2
E T n = p E d = p(x), p(y) : {x, y} ∈ E d ,
which amounts to adding edges between corresponding vertices on opposite faces of the box. We then write E T n = {e 1 , . . . , e r } with r = E T n , and we consider a collection of i.i.d. random variables For a configuration ω : E T n → {0, 1} and v ∈ Λ(n), we denote by C T Λ(n) (v, ω) ⊂ Λ(n) the cluster of v in the configuration ω on the torus, that is to say the connected component of the vertex v in the graph

(
Λ(n), e ∈ E T n : ω(e) = 1
.

For any ω : E T n → {0, 1}, we denote by C T max (ω) the largest cluster on the torus in the configuration ω. In case of equality between several clusters, we choose one with an arbitrary order on the subsets of Λ(n). We consider the pair of random variables (T, S) = min (t, s) ∈ 0, . . . , n d -2 × {0, . . . , r} : C T max ω(t, s) 2t + 3 , which is well-defined because C T max ω(n d -2, 0) n d . Let us show that, at this instant (T, S), we have

C T max ω(T, S) T + 2 . ( 26
)
We distinguish several cases :

• If S 1 then, (T, S) being minimal, we have C T max ω(T, S -1) 2T + 4. To obtain (26), note that closing a single edge can at most divide C T max by a factor two. • If T = 0 and S = 0 then, by minimality of (T, S), we have that

C T max ω(T -1, r) 2(T -1) + 4 = 2T + 2 T + 2 ,
which implies inequality (26), because the configurations ω(T -1, r) and ω(T, 0) are identical.

• The case (T, S) = (0, 0) never occurs because we have C T max ω(0, 0) = n d > 3. We have thus shown that (26) holds.

Definition of the reference vertex :

We now order the vertices of Λ(n) in a deterministic way (for instance the lexicographic order) and we denote by V the vertex of C T max ω(T, S) which is minimal for this order. Given that v∈Λ(n) Applying lemma 1 to the graph (C v , E v ) and the vertex v, we can choose a set

P(V = v) = 1 , we can find a vertex v 0 ∈ Λ(n) such that P(V = v 0 ) 1 |Λ(n)| = 1 n d . M n ω(T , S ) e • e • v C v close H H • e • v ω(T , S ) ω(T + 2, 0)
H ⊂ E v satisfying |H| K |C v | d-1 d (30)
and such that the cluster of v in the graph (C v , E v \H) contains exactly T + 2 -M n ω(T , S ) e vertices. We then have

M n ω(T , S ) H = T + 2 .
The edge e (and thus the vertex v) depends only on T , ω(T , S ) and V , thus we can choose such a set H which also depends only on T , ω(T , S ) and V . Besides, we have the following control over |C v |:

|C v | = C Λ(n) v, ω(T , S ) e C Λ(n) v, ω(T , S ) C T max ω(T , S ) . (31)
Note now that C T max (ω) is a decreasing function of ω and that, by definition, (T , S ) (T, S), whence

C T max ω(T , S ) C T max ω(T, S) 2T + 3 . (32)
Combining ( 31) and (32), we get |C v | 2T + 3, and therefore the upper bound (30) becomes

|H| K 2T + 3 d-1 d . ( 33
)
To sum up these two cases, we have defined a (random) set of edges H ⊂ E n whose size is controlled by (33) and which satisfies

M n ω(T , S ) H = T + 2 .
Therefore, conditionally on {V = v 0 }, if the edges belonging to H and no other edges of E M n ω(T , S ) are closed between the configurations ω(T , S ) and ω(T + 2, 0), then we have

M n ω(T + 2) = T + 2 . ( 34
)
This leads us to consider the event

E =      ∀s > S e s ∈ E M n ω(T , S ) ⇒ X T , es = 1 ∀e ∈ H X T +1, e = 0 ∀e ∈ E M n ω(T , S ) \H X T +1, e = 1      Yet, if V = v 0 and |M n (ω(τ + n ))| < τ + n ,
P E ∩ {V = v 0 } P V = v 0 and T κn a P E V = v 0 and T κn a 1 2n d exp - 2d(3κn a + 5) n a -2Ka(ln n) (2κn a + 3) d-1 d 1 2n d exp -6dκ - 10d n a -8Kκa(ln n)n a(d-1)/d .
Given (35), we obtain

lim inf n→∞ ln Z n (ln n)n a(d-1)/d lim inf n→∞ ln P {V = v 0 } ∩ E (ln n)n a(d-1)/d -8Kκa > -∞ ,
which is the required lower bound.

Proof of the convergence result

We now explain how the case (ii) of theorem 1 follows from the above lemmas.

Proof of theorem 1, case (ii). Let ε > 0 and a ∈ (d -1, d). Exactly as in section 4.4, the upper bounds lim sup n→∞ 1 n a ln µ n p n < p c -ε < 0 , lim sup n→∞ 1 n d-1 ln µ n p n > p c + ε < 0 , follow from our lower bound on Z n (lemma 11) and from the results of exponential decay in the subcritical (lemma 9) and supercritical phases (lemma 10). To obtain the lower bound, we go back to our computation (4) to write

µ n p n > p c + ε = 1 Z n t + n -1 t=0 P ϕn(t) |M n | = t P ϕn(t + n -1) |M n | = t + n -1 , (39)
where t + n is still given by (3). We implement now a simplified surgery procedure to force

|M n | = t + n -1 starting from a configuration ω such that |M n (ω)| > t + n -1.
According to lemma 10, we have

P ϕn(t + n -1) |M n | > t + n -1 P pc+ε |M n | > t + n -1 n→∞ -→ 1 . ( 40
)
Let ω ∈ {0, 1} En be a configuration such that |M n (ω)| > t + n -1. Consider the set E of the edges of E n which have exactly one endpoint in ∂Λ(n), which is such that

|M n (ω E )| = |∂Λ(n)| 2dn d-1 < t + n -1 (41)
for n large enough, because a > d -1. We write E = e 1 , .

P ϕn(t + n -1) |M n | = t + n -1 1 C n (2d+1)n d-1 × P ϕn(t + n -1) |M n | > t + n -1 ,
where

C n = 1 ∨ ϕ n (t + n -1) 1 -ϕ n (t + n -1) |E n | = O n d .
Plugging this into (39) and using (40) then yields

lim inf n→∞ 1 (ln n)n d-1 ln µ n |p n -p c | > ε lim inf n→∞ 1 (ln n)n d-1 ln µ n p n > p c + ε -(2d + 1)d > -∞ ,
as announced in theorem 1.

6 Proof of case (iii ) of theorem 1

The goal of this section is to prove the remaining part of theorem 1, namely the case (iii), where the function p n is defined by

p n (ω) = exp - B b n (ω) n a ,
where a and b are two fixed parameters such that 0 < b < a < d, and

B b n (ω) = x ∈ Λ(n) : |C(x, ω)| n b .
In subsection 6.5, we will also obtain the estimate on the convergence speed announced in theorem 2, which depends on the existence of the critical exponents β and γ. The rest of the section is organized as the two previous sections, with first the large deviation estimates far from p c and then the lower bound on Z n (with, this time, two different lower bounds).

Exponential decay in the subcritical phase

We now prove the following exponential decay in the subcritical regime :

Lemma 12. For every p < p c and any A > 0, we have

-∞ < lim inf n→∞ 1 n a ln P p B b n > An a lim sup n→∞ 1 n a ln P p B b n > An a < 0 .
Proof. Let p < p c and A > 0. Writing N n = 1 + An a-b and using the BK inequality as in the proof of lemma 9, we get

P p B b n > An a Nn k=1 x1, ..., x k ∈Λ(n) n b n1, ..., n k n d n1+•••+n k >An a P p C Λ(n) (x 1 ) n 1 • • • • • C Λ(n) (x k ) n k Nn k=1 x1, ..., x k ∈Λ(n) n b n1, ..., n k n d n1+•••+n k >An a k i=1 e -λ(p)ni
N n n d 2Nn e -λ(p)An a .

Therefore, we obtain

1 n a ln P p B b n > An a ln N n n a + 2N n d ln n n a -λ(p)A n→∞ -→ -λ(p)A < 0 ,
which proves the upper bound. The lower bound follows from the lower bound given by lemma 5, since An a n b for n large enough.

Exponential decay in the supercritical phase

We now deal with the deviations in the regime p > p c . We wish to thank an anonymous referee for having improved our proof, leading to a better (and in fact optimal) exponent.

Lemma 13. We have the upper bound

∀p > p c ∀A > 0 lim sup n→∞ 1 n d-b/d ln P p B b n < An a < 0 .
Proof. Let p > p c and A > 0. We shall partition the box Λ(n) into hypercubic boxes of side

N n = 8n b θ(p) 1/d .
We let

M n = min m ∈ N : Λ(n) ⊂ j∈Λ(m) N n j + Λ(N n ) , so that we have a partition Λ(n) = j∈Λ(Mn) N n j + Λ(N n ) ∩ Λ(n) .
By definition of B b n , we have

B b n = x ∈ Λ(n) : C Λ(n) (x) n b n b j ∈ Λ(M n ) : C max N n j + Λ(N n ) ∩ Λ(n) n b .
Lemma 14. For every a and b such that 0 < b < a < d, we have

lim inf n→∞ ln Z n n b > -∞ .
Proof. We use the same monotone coupling (ω(t, s)) 0 t n d , 0 s r as in the proof of lemma 8. Following a strategy similar to that of lemma 8, we define (T, S) = min (t, s) ∈ 0, . . . , n d -2 × {0, . . . , r} :

B b n ω(t, s) t + 1 + 2n b . (42)
When closing one single edge, B b n cannot decrease by more than 2n b (in the worst case, the edge cuts a cluster of 2 n b -2 vertices in two equal parts). Therefore, we always have B b n ω(T, S) T + 1. Thus, if we consider the instant T = B b n ω(T, S) , we have T + 1 T T + 1 + 2n b . In view of this, our strategy is to force all the edges of B b n ω(T, S) to remain open until the configuration ω(T , 0). This idea is much simpler than the strategy of the previous sections, because we do not perform any surgery step. Considering the event

E = ∀s > S e s ∈ E B b n ω(T, S) ⇒ X T, es = 1 ∀t ∈ {T + 1, . . . , T -1} ∀e ∈ E B b n ω(T, S) X t, e = 1 , equation (8) becomes Z n P B b n ω(T , 0) = B b n ω(T, S) = T P E . (43)
A lower bound on the probability of E is easily obtained by writing

P E (T, S, ω(T, S)) = s>S es∈E[B b n (ω(T, S))] P X T, es = 1 T -1 t=T +1 e∈E[B b n (ω(T, S))] P X t, e = 1 e -1/n a (T -T )|E[B b n (ω(T, S))]| e -1/n a (2n b +1)d(T +1+2n b ) . (44)
We then show an upper bound on T , using the same technique as in the proof of lemma 8. With τ + n defined as in (18), we can write

P T τ + n P B b n ω(τ + n ) τ + n + 1 + 2n b P pc/2 B b n τ + n + 1 + 2n b n→∞ -→ 1 ,
thanks to lemma 12. Combining this with (43) and (44) then leads to

ln Z n n b ln P T τ + n n b + ln P E T τ + n n b o(1) - (2n b + 1)d(τ + n + 1 + 2n b ) n a+b n→∞ -→ -2d -ln p c 2 , using that τ + n ∼ (-ln(p c /2))n a .
We now state the other lower bound we obtain using a more geometrical technique:

Lemma 15. For every a and b such that 0 < b < a < d, we have

lim inf n→∞ ln Z n (ln n)n c > -∞ where c = 1 - a d + b d ∨ a - a d .
Proof outline: We use again the same coupling (ω(t, s)) t,s as in the previous proof, and the same instant (T, S).

We then want to close edges to reach a fixed point, but the problem is that it is not always possible to do so by only closing edges. Imagine for example that, in the configuration ω(T, S), no cluster contains more than n b vertices, meaning that B b n is only made of clusters containing exactly n b vertices. Then, in this very unfavourable n only by closing edges. To circumvent this problem, we will change what happened before the instant (T, S) so as to ensure that, at this instant, we have at our disposal a cluster containing at least 2n b vertices. This will enable us to use our surgery procedure on this cluster, in order to reach the exact desired value for B b n . However, to do so, we need to intervene on the past of the instant (T, S), which will make notations more complicated. Namely, we will define a second coupling of configurations (ω (t, s)) t,s which is a copy of the first coupling, except that the closure times of a certain number of edges are drawn again, allowing us to close or to open these edges at different times. Before diving into the proof, we precise our surgery procedure in the following two lemmas. The first one is a lower bound on the number of edges we need to reopen to create a cluster of size at least 2n b .

• x • y H |C(x)| n b |C(y)| n b L n
L n = 2n 2n b |B b n (ω)| 1/d
, with boxes which may be smaller along the boundaries of Λ(n). The number of boxes is at most

n L n d     1 2 B b n (ω) 2n b 1/d     d <   1 2 B b n (ω) 2n b 1/d + 1   d B b n (ω) 2n b .
Hence, there are strictly less boxes than the number of clusters with size at least n b . Therefore, by the pigeonhole principle, at least one of these boxes must intersect two such clusters, which means that we can find x, y ∈ Λ

(n) such that |C(x, ω)| n b , |C(y, ω)| n b , x ω ←→ y and x -y ∞ L n .
We then have x -y 1 dL n , implying that there exists a path H ⊂ E n with at most dL n edges which connects x In all three cases, H 0 satisfies (47) and lemma 1 ensures that H 0 can be chosen with In both cases, obtain the claimed result, with K 1 = 2K + 11 × 4 d+1 d 2 .

|H 0 | K |C x |
We are now in a position to prove our second lower bound on Z n .

Proof of lemma 15. As explained above, we define two couplings, in order to be able not only to close edges, but also to reopen edges.

Definition of the two couplings:

The first coupling is defined as in the previous proofs. We write E n = {e 1 , . . . , e r } with r = |E n |, and we consider i.i.d. random variables (X t,e ) t∈{0, ..., n d -1}, e∈En with Bernoulli law of parameter exp(-1/n a ). For t 0 ∈ 0, . . . , n d , we define ω(t 0 ) : e ∈ E n -→ min 0 t<t0

X t,e .

In addition to this, we draw a uniform random M ∈ 0, . . . , n d , uniform independent edges ε 1 , . differs from the first one only on the edges ε 1 , . . . , ε M . This set of edges is chosen at random, but we will be interested in the event that {ε 1 , . . . , ε M } = H 1 ∪ H 2 , where H 1 is a set of edges which we want to leave open longer in the second coupling, and H 2 is a set of edges which we want to close sooner in the second coupling. Thus, this double coupling will allow us to perform the surgery procedure of lemmas 16 (which involves opening edges) and 17 (which involves closing edges), starting from a given configuration in the first coupling. Note that, whence

Z n P B b
n ω (T + 1, 0) = T + 1

P E ∩ τ - n T τ + n . (52)
As in the proof of lemma 8, we now take (t 0 , s 0 ) and ω 0 ∈ {0, 1} En such that τ - n t 0 τ + n and P C t0, s0, ω0 > 0 where C t0, s0, ω0 = (T, S) = (t 0 , s 0 ) and ω(T, S) = ω 0 .

Because H 1 and H 2 only depend on T and ω(T, S), we may consider the deterministic sets H 1 and H 2 associated with T = t 0 and ω(T, S) = ω 0 . Then we consider the event This bound being uniform with respect to t 0 , s 0 and ω 0 (as long as τ - n t 0 τ + n ), we obtain that

E t0, s0, ω0 =            M = |H 1 ∪ H 2 | ,
P E τ - n T τ + n 1 n d + 1 1+K n 1+b

Theorem 2 .

 2 Take F n = B b n (case (iii) of theorem 1). Assume that there exist real constants β, γ > 0 such that lim sup p→pc p>pc ln θ(p) ln(p -p c ) β and lim inf p→pc p<pc ln χ(p) ln(p c -p) -γ . Then, for any real parameters a, b and c, we have 0 < b < a < d and c < min b -b β =⇒ n c (p n -p c ) L -→ 0 .

  is called a percolation configuration. Edges e ∈ E d such that ω(e) = 1 are said open in ω, and the other edges are said closed in ω. Under the law P p , each edge is open with probability p and the states of different edges are independent of each other. For any configuration ω : E d → {0, 1} and any edge e ∈ E d , we will write

Figure 2 :

 2 Figure 2: Percolation in the box Λ(35) with, left, p = 0.48 and right, p = 0.52. Open edges belonging to the largest cluster are drawn in solid lines, while other open edges are in dotted lines.

Figure 3 :

 3 Figure 3: Percolation in the box Λ(35) with, left, p = 0.48 and right, p = 0.52. Open edges connected to the boundary of the box by an open path are drawn in solid lines, while other open edges are in dotted lines.

σ

  = min s ∈ {0, . . . , r} : |V s | m which is the number of reopened edges at which the size of the cluster of x reaches or exceeds the desired size m. This number σ is well-defined because |V r | = |V | m. Assume that σ 1. By minimality of σ, e 3 e 4

Figure 6 :

 6 Figure 6: If each of the four rectangles R 1 , R 2 , R 3 , R 4 is crossed by an open path in its long direction, then Λ(n) is surrounded by an open path in Λ(2n), and thus any two vertices x and y in the box Λ(n) cannot be connected to ∂Λ(2n) without being connected to each other by an open path inside Λ(2n).consider the events E n = There exists an open path in Λ(2n)\Λ(n) containing Λ(n) in its interior

  of the coupling: Write E n = {e 1 , . . . , e r } with r = |E n |, and consider a collection of i.i.d. random variables (X t,e ) t∈{0, ..., n d -1}, e∈En

Figure 7 :

 7 Figure7: Sketch of the proof: if E occurs, i.e., between the instants (T, S) and (T + 2, 0), the edges H are closed but no other edges of C max is closed, then the largest cluster in the configuration ω(T + 2, 0) contains T + 2 vertices.

Figure 8 :

 8 Figure 8: Illustration for the sketch of the proof of lemma 11.

Figure 9 :

 9 Figure 9: If (29) is a strict inequality, then closing the edge e in the configuration ω(T , S ) changes the number |M n | of vertices connected to the boundary of the box. This means that one end of the edge e, say v, happens to be disconnected from the boundary when e is closed. We then choose a subset H of the edges of the cluster C v which is disconnected by the closure of e, such that closing all the edges of H and no other edges of E [M n ] between (T , S ) and (T + 2, 0) implies M n ω(T + 2) = T + 2.

Figure 10 :

 10 Figure10: If there is no cluster with size 2n b , we reconstitute one by joining two clusters of size at least n b , with a path using less than dL n edges.

Lemma 16 .

 16 Let b ∈ (0, d). For every n 1 and for any configuration ω ∈ {0, 1} En such that B b n (ω) 2 d+1 n b , there exists a set of edges H ⊂ E n , such that C max ω H 2n b Let b ∈ (0, d) and ω ∈ {0, 1} En such that B b n (ω) 2 d+1 n b . If |C max (ω)| 2n b , then we choose H = ∅. Let us now assume that |C max (ω)| < 2n b . Then all the clusters in B b n (ω) contain between n b and 2n b vertices. Therefore, there are at least B b n (ω) /(2n b ) such clusters in the configuration ω. We divide the box Λ(n) into hypercubic boxes of side

and y . 2 H 1 C 2 H 1 C

 .2121 Figure 11: First case: when |C max | 7n b (picture on the left), we cut a piece of C max with the desired size m (by closing H 1 ) and we divide the remaining part of C max into pieces smaller than n b (by closing H 2 ). Second case: when |C max | < 7n b (picture on the right), we cut some intermediate clusters (by closing H 1 ) and we cut a piece of C max with the desired size m (by closing H 2 ). In both figures, the hatched region is B b n (ω)\B b n (ω H1∪H2 ), i.e., the vertices whose cluster is no longer larger than n b after the cutting procedure.

  (d-1)/d K |C max (ω)| (d-1)/d K 7n b (d-1)/d Ks b(d-1)/d . After closing this set of edges H 0 , we still have B b n s and, either B b n < s + n b or B b n has decreased by at least n b -1. Therefore, we can repeat this operation, and after at most 6 steps, we obtain s B b n < s + n b . Thus, we end up with H 1 ⊂ E n satisfying (46) and such that |H 1 | 42Kn b(d-1)/d . As we have not touched C max (ω) during this procedure, we still have 2n b C max ω H1 < 7n b . Letting now m = C max ω H1 -B b n ω H1 + s , it follows from (46) that n b m C max ω H1 . Hence, using again lemma 1, we can find H 2 ⊂ E n with |H 2 | K 7n b (d-1)/d Ks (d-1)/d and such that closing the edges of H 2 divides the cluster C max ω H1 into one connected component of size exactly m and one or several other connected components, whose total size is C max ω H1 -m = B b n ω H1 -s < n b . Therefore, writing H = H 1 ∪ H 2 , we have |H| 2Ks (d-1)/d and B b n ω H = B b n ω H1 -C max ω H1 + m = s .

d=

  P ϕn(t) , and the second coupling

P

  E C t0, s0, ω0 = P E t0, s0, ω0 C t0, s0, ω0 = P E t0, s0, ω0 n a |E[B b n (ω0)]|+|H1|(t0+1) 1 -e -1/n a |H2| .We now use the bounds (49) and (51) on |H 1 | and |H 2 |, the upper boundE B b n (ω 0 ) d B b n (ω 0 ) d t 0 + 1 + 2n b d τ + n + 1 + 2n band the fact that, for n large enough, 1 -e -1/n a 1/(n d + 1), to obtainP E C t0, s0, ω0 1 n d + 1 1+K n 1+b/d-a/d +2K n a(d-1)/d exp -d(τ + n + 1 + 2n b ) + (τ + n + 1)K n 1+b/d-a/d n a.

  /d-a/d +2K n a(d-1)/d exp -(d + K n 1+b/d-a/d )(τ + n + 1) + 2dn b n a .Plugging this into (52) and recalling (48), we getln Z n ln P τ - n T τ + n -1 + K n 1+b/d-a/d + 2K n a(d-1)/d ln n d + 1 -d + K n 1+b/d-a/d τ + n + 1 + 2dn b n a = o(1) + O n 1+b/d-a/d ln n + O n a(d-1)/d ln n = O n c ln n , with c = (1 + b/d -a/d) ∨ (a -a/d).

  The rectangle {0, . . . , k} × {0, . . . , m} is crossed by an open path in its long direction ,

	then it follows from equation (7.110) in [Gri99] that there exist positive constants C 2 (p) and C 3 (p) such that, for
	all m k 1,

  X t,e ) t∈{0, ..., n d -1}, e∈E T n all distributed with a Bernoulli law of parameter exp(-1/n a ). We set, for t 0 ∈ {0, . . . , n d },

	ω(t 0 ) : e ∈ E T n -→ min 0 t<t0	X t,e ,
	and for t ∈ {0, . . . , n d -1} and s 0 ∈ {0, . . . , r}, we define
	ω(t, s 0 ) : e s ∈ E T n -→	ω(t + 1)(e s ) ω(t)(e s )	if s s 0 , otherwise.

  then inequality (28) entails that T < τ + n . From this we can deduce that, for n large enough, P V = v 0 and T < τ + Therefore, we can find κ 2 such that, for n large enough,

		n	1 2n d .
	P V = v 0 and T	κn a	1 2n d .	(38)
	Conclusion: Combining (37) and (38) yields

  . . , e |E| with |E| 2dn d-1 , and we let B = max b ∈ {1, . . . , |E|} : M n ω {e1, ..., e b } t + n -1 .It follows from (41) that B < |E|, whence by maximality of B,M n ω {e1, ..., e B+1 } < t + n -1 .Therefore, if we write e B+1 = {x, y} with x ∈ ∂Λ(n) and y / ∈ ∂Λ(n), and if we consider the cluster which is disconnected from the boundary when closing this edge e B+1 , namely :C y = C Λ(n) y, ω {e1, ..., e B+1 } ,we have M n ω {e1, ..., e B+1 } = M n ω {e1, ..., e B } -|C y | , so that m = (t + n -1) -M n ω {e1, ..., e B+1 } satisfies 1 m |C y |. Hence, lemma 1 provides us with H 1 ⊂ E n , of size |H 1 | |C y | Writing H = H(ω) = {e 1 , . . . , e B } ∪ H 1 , we then have |H| (2d + 1)n d-1 and M n ω H = M n ω {e1, ..., e B+1 } + m = t + n -1 .

	(d-1)/d	n d-1 , such that
		C Λ(n) y, ω {e1, ..., e B+1 }∪H1	= m .
	Using lemma 6.3 in [CP00], we can deduce that

  . . , ε M ∈ E n and i.i.d. random variables X t,e t n d -1, e∈En again with Bernoulli law of parameter exp(-1/n a ). The second coupling of configurations is then defined by∀ t 0 ∈ 0, . . . , n d ω (t 0 ) : e ∈ E n -→ min 0 t<t0 X t,e if e ∈ {ε 1 , . . . , ε M } , ω(t 0 ) otherwise,with again intermediate configurations defined for all t ∈ 0, . . . , n d -1 and s 0 ∈ {0, . . . , r} by ω (t, s 0 ) : e s ∈ E n -→

	ω (t + 1)(e s )	if s s 0 ,
	ω (t)(e s )	otherwise.
	Hence, the two decreasing couplings have the same law, with ω(t)

d = ω (t)

  {ε 1 , . . . , ε M } = H 1 ∪ H 2 , ∀s > s 0 e s ∈ E B b n ω 0 ⇒ X t0, es = 1 , ∀e ∈ H 1 \H 2 ∀t ∈ {0, . . . , t 0 } X t, e = 1 ,which is independent of C t0, s0, ω0 because C t0, s0, ω0 depends only on the variables X t, es with (t, s) (t 0 , s 0 ). Therefore, we can write

			
			    	,
	∀e ∈ H 2	X 0, e = 0	    

* DMA, École normale supérieure, CNRS, PSL University, 75005, Paris. † Université Paris-Saclay, CNRS, Laboratoire de mathématiques d'Orsay, 91405, Orsay, France. ‡ Aix Marseille Univ, CNRS, Centrale Marseille, I2M, Marseille, France

Acknowledgments: We wish to thank two anonymous referees, one who suggested an improvement of our previous version of lemma 13, and another one for numerous remarks which helped us to improve the presentation of our paper.

In what follows, we will reason conditionally on the event {V = v 0 }. Until now, everything took place on the torus, which is translation-invariant. If v 0 / ∈ ∂Λ(n), we can apply a translation on the torus so as to have v 0 ∈ ∂Λ(n), modifying at the same time the orders considered on E T n , on Λ(n) and on the subsets of Λ(n). Therefore, without loss of generality, we can assume that v 0 ∈ ∂Λ(n). Hence, if V = v 0 , then we have C T max ω(T, S) ⊂ M n ω(T, S) , whence

following (26).

Construction of the second instant T : We now consider (T , S ) = min (t, s) (T, S) : ∃e ∈ E n M n ω(t, s) e t + 2 .

The fact that T n d -2 and M n ω(n d -2, r) n d ensures that (T , S ) is well-defined and that T n d -2. Let us show, by distinguishing several cases, that

• If (T , S ) = (T, S), then the claim follows from (27).

• If (T , S ) > (T, S) and S = 0, then the minimality of (T , S ) implies that

• Else if (T , S ) > (T, S) and S = 0, then by minimality of (T , S ), we know that for all e ∈ E n ,

M n ω(T , S -1) e > T + 2 , which entails in particular that M n ω(T , S ) > T + 2, because the configuration ω(T , S ) is obtained from the configuration ω(T , S -1) by closing at most one edge. We conclude that (28) holds in all cases.

Construction of the happy event: We now wish to define a set of edges H that we want to be closed between the configuration ω(T , S ) and the configuration ω(T + 2, 0) in order to have

We are only interested in situations where V = v 0 , thus we set arbitrarily H = ∅ if the event {V = v 0 } does not occur. We now assume that V = v 0 . By definition of (T , S ), there exists an edge e ∈ E n such that

We choose this edge e minimal (for the order e 1 , . . . , e r we have considered on E T n ) among the edges satisfying (29), which ensures that e only depends on T , ω(T , S ) and V . We then construct the set H by distinguishing two cases depending on whether the inequality (29) is strict or not.

• In case there is equality in (29), we take H = {e}.

• Assume that (29) is a strict inequality. It follows from (28) that M n ω(T , S ) e < M n ω(T , S ) , which means that closing the edge e changes the number of vertices connected to the boundary of the box. Consequently, one end of the edge e, say v, must be disconnected from ∂Λ(n) when closing e in the configuration ω(T , S ). Write (C v , E v ) for the graph of the open cluster of v in the configuration ω(T , S ) e . We have, using (28),

Combining this with the (strict) inequality (29) yields which, if it occurs and if V = v 0 , implies (34). Also, our expression (8) becomes

Conditional probability of the happy event: As in the proof of lemma 8, we consider (t 0 , t 0 , s 0 ) and a configuration ω 0 : E T n → {0, 1} such that P C t0, t 0 , s 0 , ω0 > 0 where C t0, t 0 , s 0 , ω0 = (T, T , S ) = (t 0 , t 0 , s 0

By definition of T and T , we have

The event C t0, t 0 , s 0 , ω0 depends only on the variables X t, es with (t, s) (t 0 , s 0 ) and, conditionally on this event, the event E only depends on the variables X t, es for (t 0 , s 0 ) < (t, s) < (t 0 + 2, 0). What's more, the set H only depends on T , ω(T , S ) and V , which allows us to write H = H (T , ω(T , S ), V ). Therefore, we have

Using the upper bound (36) on |M n (ω 0 )| and the upper bound (33) on |H| leads to

Taking the conditional expectation with respect to (T , V ) and using the fact that T T , we obtain

Upper bound on T : It follows from lemma 9 that

Therefore, we have

and thus, if we take τ + n defined as in (18) then, for n large enough,

We then have, using the fact that P

Given that An a = o(n d ), this implies that, for n large enough,

Therefore, we have the following implication :

The problem now is that the boxes on the boundaries might be truncated. However, the inside boxes are full, that is to say

Yet the number of boxes on the boundaries is

so that we have, for n large enough,

Using the independence of the sizes of the largest cluster inside disjoint boxes and the fact that the number of choices of at least |Λ(M n )| /4 boxes is at most 2 |Λ(Mn)| , we get

, which implies that

The result then follows from lemma 7, noting that n b θ(p)N d n /8.

Two lower bounds on the partition function

It remains to prove a lower bound on the normalization constant Z n . Adapting the technique of lemmas 8 and 11, we can easily obtain a bound with an exponent b. This is done in lemma 14, and the proof is much simpler than in the previous sections because, instead of performing a surgery step, we only "freeze" the edges of B b n during a certain number of steps. But this bound may not be sufficient to outweigh the bound in the supercritical phase, since it may be the case that b > d -b/d. To solve this problem, we show an other lower bound in lemma 15 with a different exponent, using a more geometrical technique. which completes the proof of this lemma.

The second geometrical lemma will tell us how many edges we need to close to adjust the size of B b n :

Lemma 17. Let b ∈ (0, d). There exists

then there exists H ⊂ E n such that

Sketch of the proof: If we have at our disposal a big enough cluster, then we may reach B b n = s by only closing edges of this cluster. In this case, using the geometrical results of section 3, the idea is to cut C max (ω) into one large piece of size m n b and remaining pieces all of size < n b (see the left part of figure 11). Adjusting the cutting so that m = |C max (ω)| -B b n (ω) + s yields the desired result. However, for this technique to work, we need m to be greater than n b , so that the large piece of size m still belongs to B b n after the cutting. This is the case if En and s ∈ N such that (45) holds. We distinguish between two cases :

and such that closing the edges of H 1 divides C max (ω) into one connected component of size exactly m, and one or several other pieces, whose total size is

Using the butcher's lemma (lemma 2), this remaining part can be cut into pieces smaller than 3n b . Using again the butcher's lemma on the connected subpieces which contain strictly more than (3/2)n b vertices (there are at most 3 such subpieces), we can cut them into pieces smaller than (3/2)n b . Repeating the operation on the pieces containing strictly more than (3/4)n b vertices (there are at most 7 such subpieces), we can cut them into pieces smaller than (3/4)n b .

Thus, using at most 11 times the butcher's lemma, we obtain

and such that in the configuration ω H1∪H2 , the vertices of C max (ω) are separated into one cluster of size exactly m and the remaining clusters which are all smaller than (3/4)n b < n b (see the left part of figure 11). Therefore, writing H = H 1 ∪ H 2 and using that

• Second case: Now assume that 2n b

|C max (ω)| < 7n b . The first step in this case is to find instead of constructing such a double coupling, we could also have used the standard estimate of, for example, lemma 6.3 in [START_REF] Cerf | On the Wulff crystal in the Ising model[END_REF], about the price to open or close specific edges.

Reconstitution of a big enough cluster: As in the proof of lemma 14, we consider the instant (T, S) defined by (42), which is such that

With τ + n defined as in (18) and τ - n given by τ - n = n a -ln

we have

thanks to lemmas 12 and 13. This allows us, in the sequel, to reason conditionally on the event that τ - n T τ + n . Thus, we have

for n large enough, given that a > b. This allows us to apply lemma 16, which provides us with H 1 = H 1 ω(T, S) ⊂ E n such that the configuration ω(T, S) H1 , where the edges of H 1 are reopened, contains a cluster with at least 2n b vertices, and such that

where K is a positive constant. We can choose this H 1 minimal in the sense of inclusion, so that either

where K is a positive constant, since τ + n = O(n a ).

The happy event:

We now consider the event (where

If this event occurs and τ - n T τ + n , then we have

Proof of the convergence result

We now obtain the third case of theorem 1, proceeding as in section 4.4.

Proof of theorem 1, case (iii). Let ε > 0 and 0 < b < a < d. The upper bound

follows from the exponential decay in the subcritical regime (lemma 12) and the lower bound on Z n given by lemma 14, using that b < a. Similarly, lemma 13 together with the other lower bound on Z n given by lemma 15 implies

using that

To obtain a lower bound on µ n p n < p c -ε , we use the same technique as in section 4.4, choosing this time a parameter a ∈ (a, 2a -b). Using the notations of the proof of lemma 14 and t - n = n a -ln(p c -ε) , we write

As in section 4.4, using that ϕ n (t - n ) p c -ε and ϕ n n a < p c /2 for n large enough, and using the exponential estimate of lemma 12, we have, for n large enough,

Combining this with the lower bound (44) on the conditional probability P E T , we obtain

We now turn to the other lower bound. With t + n given by (3) we have

We now consider a configuration ω ∈ {0, 1} En such that B b n (ω) t + n -1, and we use again the surgery procedure detailed in section 6.3 to force

(and hence of volume < n b ), and closing all the edges on the boundaries of these boxes, we can find

Rather than closing all this set of edges H 1 , we choose a maximal subset

Plugging this into (56) and recalling that the probability on the right-hand side tends to 1 according to lemma 13, we obtain lim inf

The last case of theorem 1 then follows from (53), ( 54), ( 55) and (57).

A control on the convergence speed

We now make the previous arguments more precise, in order to obtain an estimate on the convergence speed of p n towards the critical point p c . We assume that there exist real numbers β > 0, γ > 0 such that lim sup We also choose β and γ such that

Therefore, we can find ε 0 > 0 such that, for 0 < ε < ε 0 ,

Subcritical phase

Lemma 18. We have the upper bound

Proof. Take A > 0 and 0 < ε < p c . Without loss of generality, we can assume that ε < ε 0 . We repeat the proof of lemma 12, but replacing p with p c -ε/n c . To control P p C Λ(n) (x i ) n i , the upper bound (9) is no longer sufficient, because we would need to specify the dependence in n of λ(p c -ε/n c ). Thus, we use another inequality provided by the same theorem 6.75 in [START_REF] Grimmett | of Grundlehren der Mathematischen Wissenschaften[END_REF], which states that

With our choice of γ , we have that

Hence, the condition n b χ (p c -ε/n c ) 2 is satisfied for n large enough. This allows us to apply (58) to get, with N n = 1 + An a-b as in the proof of lemma 12,

Therefore, we obtain

which proves the desired upper bound.

Supercritical phase

Lemma 19. We have the upper bound

This bound is rougher than the bound we proved in lemma 13, but it presents the advantage of using only θ(p) which we have assumed to scale as (p -p c ) β . The counterpart is that it only works with b < 1.

Proof. We let N = n b , and we divide the box Λ(n) into smaller boxes of side 3N , leaving apart the remainder, meaning that we write

where the boxes B i = Λ(3N ) + τ i are disjoint translates of Λ(3N ). The boxes B i being disjoint, the variables in the above sum are pairwise independent. Besides, the expectation of this sum is

Using that d -cβ > a, we deduce that, for n large enough, we have

Therefore, applying Hoeffding's inequality (see [START_REF] Hoeffding | Probability inequalities for sums of bounded random variables[END_REF]) yields that, for n large enough, In fact, using also the lower bound of lemma 15, a slightly larger admissible window for c can be obtained, namely but we have preferred to present the simpler condition in the statement of the theorem, since none of them is optimal anyway.

An alternative model with cluster diameters

The variant obtained by replacing B b n with the function B b n defined by (2) can be dealt with using the same techniques. The main difference is that, instead of using theorem 6.75 of [START_REF] Grimmett | of Grundlehren der Mathematischen Wissenschaften[END_REF], we use the theorem 5.4 therein, which states that ∀p < p c ∃ ψ(p) > 0 ∀n 1 P p 0 ω ←→ ∂Λ(n) e -nψ(p) .