Michel Pain

The near-critical Gibbs measure of the branching random walk

Keywords: MSC 2010 60J80, 60F05, 60F17 Branching random walk, additive martingale, trajectories, phase transition

 in the strong disorder regime, with convergence towards the normalized Brownian excursion. We prove here the convergence for trajectories of particles chosen according to the nearcritical Gibbs measure and display continuous families of processes from the meander to the excursion or to the Brownian motion.

Introduction and main results

Definitions and assumptions

The branching random walk on the real line is a natural extension of the Galton-Watson process, with addition of a position to each individual, and is defined as follows. Initially, there is a single particle at the origin, forming the 0 th generation. It gives birth to children, scattered in R according to some point process L and forming the 1 st generation. Then, each particle of the 1 st generation produces its own children disposed around its position according to the law of L independently of others: this set of children forms the 2 nd generation. The system goes on indefinitely, unless there is no particle at some generation.

The genealogical tree of the branching random walk, denoted by T, is a Galton-Watson tree (where an individual can have an infinity of children). For z ∈ T, we denote by |z| the generation of the particle z and by V (z) its position in R. We denote by Ψ the log-Laplace transform of L: we set, for each

β ∈ R + , Ψ(β) := log E |z|=1 e -βV (z) ∈ (-∞, ∞],
noting that L has the same law as (V (z), |z| = 1).

Throughout the paper, we assume the following integrability conditions on the reproduction law L. First of all, we need to assume that the Galton-Watson tree T is supercritical, that is

E |z|=1 1 > 1, (1.1)
so that the survival event S has positive probability and, thus, we can introduce the new probability P * := P(• | S). Moreover, we work in the boundary case (Biggins and Kyprianou [START_REF] Biggins | Fixed points of the smoothing transform: the boundary case[END_REF]) by assuming

E |z|=1 e -V (z) = 1 and E |z|=1 V (z)e -V (z) = 0, (1.2)
which means Ψ(1) = 0 and Ψ (1) = 0. See the arXiv version of Jaffuel [START_REF] Jaffuel | The critical barrier for the survival of branching random walk with absorption[END_REF] for discussion on the cases where the branching random walk can be reduced to assumption (1.2). We assume also that

σ 2 := E |z|=1 V (z) 2 e -V (z) ∈ (0, ∞) and E X(log + X) 2 + X log + X < ∞, (1.3)
where we set, for y ≥ 0, log + y := max(0, log y) and z) and X :=

X := |z|=1 e -V (
|z|=1 V (z) + e -V (z) (1.4)
with V (z) + := max(0, V (z)). The first part of (1.3) gives Ψ (1) = σ 2 . We will say that L is (h, a)-lattice if h > 0 is the largest real number such that the support of L is contained by a + hZ and, then, h is called the span of L. In this paper, we work in both lattice and nonlattice cases, but we will need sometime to distinguish these cases. Finally, we set two last assumptions that are not supposed to hold in the whole paper, but only in specific cases of the results. The following assumption,

∃ 0 < δ 0 < 1 : E |z|=1 e -(1-δ 0 )V (z) < ∞, (1.5)
means that Ψ is finite on [1 -δ 0 , 1] and, thus, analytic on (1 -δ 0 , 1) and, using (1.2) and (1.3), we have Ψ(β) = σ 2 2 (β -1) 2 + o((β -1) 2 ) as β ↑ 1 by a Taylor expansion. The second one,

∃ 0 < δ 1 < δ 2 < 1/4 : E |z|=1 e -(1-δ 1 )V (z) 1+2δ 2 < ∞, (1.6)
comes from Madaule [START_REF] Madaule | First order transition for the branching random walk at the critical parameter[END_REF] and is probably not optimal for the results where it is used. Under assumption (1.6), Ψ is finite on [1 -δ 1 , 1 + δ 2 /2] and, therefore, analytic on a neighbourhood of 1, so we can improve the Taylor expansion: Ψ(β) = σ 2 2 (β -1) 2 + O((β -1) 3 ) as β → 1. For n ∈ N and β ∈ R + , we set W n,β := |z|=n e -βV (z) and ν n,β := 1 W n,β |z|=n e -βV (z) δ z , as soon as W n,β < ∞, which holds a.s. for β ≥ 1 under assumption (1.2). Then, ν n,β is a random probability measure on {z ∈ T : |z| = n}, which is called the Gibbs measure of parameter β on the n th generation of the branching random walk. It is also the law of a directed polymer on the tree T in a random environment, introduced by Derrida and Spohn [START_REF] Derrida | Polymers on disordered trees, spin glasses, and traveling waves[END_REF] as a mean field limit for directed polymer on a lattice as dimension goes to infinity: with this terminology, V (z) is the energy of the path leading from the root to particle z, β is the inverse temperature and W n,β is the partition function. In the case β = ∞, we can define ν n,∞ as the uniform measure on the random set {|x| = n : V (x) = min |z|=n V (z)}.

According to Derrida and Spohn [START_REF] Derrida | Polymers on disordered trees, spin glasses, and traveling waves[END_REF], there is a critical parameter β c > 0 for the directed polymer on a disordered tree (with our setting β c = 1, see Subsection 1.2 for more details) and our aim in this paper is to study the near-critical case, where β depends on n and tends from above and below to β c = 1 as n → ∞. The near-critical case has been recently studied for the directed polymer on the lattice in dimension 1 + 1 and 1 + 2 by Alberts, Khanin and Quastel [START_REF] Alberts | The intermediate disorder regime for directed polymers in dimension 1 + 1[END_REF] and Caravenna, Sun and Zygouras [START_REF] Caravenna | Polynomial chaos and scaling limits of disordered systems[END_REF][START_REF] Caravenna | Universality in marginally relevant disordered systems[END_REF], with the emergence of the so-called intermediate disorder regime. For the polymer on a tree, some work near criticality has been done by Alberts and Ortgiese [START_REF] Alberts | The near-critical scaling window for directed polymers on disordered trees[END_REF] and Madaule [START_REF] Madaule | First order transition for the branching random walk at the critical parameter[END_REF], mostly on the partition function.

Before stating our results, we recall some well-known properties of the branching random walk, that hold under assumptions (1.1), (1.2) and (1.3). First, the sequence

D n := |z|=n V (z)e -V (z) , n ∈ N,
is a martingale, called the derivative martingale, and Biggins and Kyprianou [START_REF] Biggins | Measure change in multitype branching[END_REF] (under slightly stronger assumptions) and Aïdékon [START_REF] Aïdékon | Convergence in law of the minimum of a branching random walk[END_REF] showed that we have

D n ---→ n→∞ D ∞ > 0, P * -a.s.
(1.7)

Moreover, Chen [START_REF] Chen | A necessary and sufficient condition for the nontrivial limit of the derivative martingale in a branching random walk[END_REF] proved that these assumptions are optimal for the nontriviality of D ∞ . Furthermore, Aïdékon [START_REF] Aïdékon | Convergence in law of the minimum of a branching random walk[END_REF] also showed that, in the nonlattice case, min |z|=n V (z) -3 2 log n converges in law under P * and described the limit as a random shift (depending on D ∞ ) of a Gumbel distribution. In the lattice case, we do not have this convergence, but the tightness still holds (see Equation (4.20) of Chen [START_REF] Chen | Scaling limit of the path leading to the leftmost particle in a branching random walk[END_REF] or Mallein [START_REF] Mallein | Asymptotic of the maximal displacement in a branching random walk[END_REF]): for each ε > 0, it exists C > 0 such that

P min |z|=n V (z) - 3 2 log n ∈ [-C, C] ≥ 1 -ε, (1.8)
for n large enough.

The partition function

The process (W n,β ) n∈N for some fixed β ∈ R + has been intensively studied because, if Ψ(β) is finite, then the renormalized process ( W n,β ) n∈N := (e -nΨ(β) W n,β ) n∈N is a nonnegative martingale, called additive martingale, and, therefore, converges a.s. to some limit W ∞,β . Kahane and Peyrière [START_REF] Kahane | Sur certaines martingales de Benoit Mandelbrot[END_REF], Biggins [START_REF] Biggins | Martingale convergence in the branching random walk[END_REF] and Lyons [START_REF] Lyons | A simple path to Biggins' martingale convergence for branching random walk[END_REF] have determined when this limit is nontrivial: under the additional assumption that the expectation E[W 1,β log + W 1,β ] is finite, we have the following dichotomy

W ∞,β > 0 P * -a.s. if β ∈ [0, 1), W ∞,β = 0 P * -a.s. if β ≥ 1. (1.9)
With the terminology of polymers' literature (see [START_REF] Comets | Directed polymers in random environment are diffusive at weak disorder[END_REF]), the region β ∈ [0, 1) is thus called the weak disorder regime and the region β ≥ 1 the strong disorder regime.

In the strong disorder regime β ≥ 1, it is natural to seek a proper renormalization of W n,β , so that it converges towards a nontrivial limit. This question has already been answered when β does not depend on n. In the critical case β = 1, Aïdékon and Shi [START_REF] Aïdékon | The Seneta-Heyde scaling for the branching random walk[END_REF] proved that we have

√ nW n,1 ---→ n→∞ 1 σ 2 π D ∞ , in P * -probability, (1.10)
and that the particles that contribute mainly to W n,1 are those of order √ n. On the other hand, Theorem 2.3 of Madaule [START_REF] Madaule | Convergence in law for the branching random walk seen from its tip[END_REF] shows (in the nonlattice case) that, in the case β > 1, we have

n 3β/2 W n,β ---→ n→∞ Z β D ∞ , in law,
where Z β is a random variable independent of D ∞ , whose law have been described by Barral, Rhodes and Vargas [START_REF] Barral | Limiting laws of supercritical branching random walks[END_REF]. Moreover, the particles that contribute mainly to W n,β are in [ 3 2 log n -C, 3 2 log n + C] with C some large constant and, thus, are close to the lowest particle at time n (see (1.8)).

Since there is a discontinuity in the size of the partition function W n,β as β ↓ 1 between n -3β/2 and n -1/2 , and also as β ↑ 1 between exponential and polynomial size, we try to understand this transition by considering the near-critical case where β n → 1 as n → ∞. Some work has been done to this end by Alberts and Ortgiese [START_REF] Alberts | The near-critical scaling window for directed polymers on disordered trees[END_REF], who considered the case where β n = 1 ± n -δ for δ > 0 and proved, under stronger assumptions, that (1) if β n = 1 + n -δ with 0 < δ < 1/2, n -1/2+o (1) if

W n,βn =      n 2δ-3/2+o
β n = 1 ± n -δ with δ ≥ 1/2, exp σ 2 2 n 1-2δ (1 + o(1)) if β n = 1 -n -δ with 0 < δ < 1/2.
in P * -probability.

Moreover, the behavior of the limit W ∞,β of the additive martingale has been studied near criticality by Madaule [START_REF] Madaule | First order transition for the branching random walk at the critical parameter[END_REF], who showed under assumption (1.6) the following convergence

W ∞,β 1 -β --→ β↑1
2D ∞ , in P * -probability, (1.11) and, although this is not exactly our setting where n → ∞ and β → 1 simultaneously, this result and its proof are useful in this paper. The following theorem improves Alberts and Ortgiese's result, showing convergence to a nontrivial limit of W n,βn after rescaling, for every sequence (iii) If (1.5) holds,

β n := 1 ± 1/α n .
β n := 1 -1/α n and √ n/α n → γ ∈ [0, ∞) as n → ∞, then we have √ nW n,βn ---→ n→∞ 1 σ 2 π E e σγM(1) D ∞ , in P * -probability. (iv) If (1.6) holds, β n := 1 -1/α n and √ n/α n → ∞ as n → ∞, then we have α n e -nΨ(βn) W n,βn ---→ n→∞ 2D ∞ , in P * -probability.
Note that, in case (i), the size of W n,βn can be a o(n -3/2 ) as soon as α n log α n log n: this possibility does not appear in Alberts and Ortgiese's result.

Trajectory of particles under the Gibbs measure

The second main result of this paper concerns the trajectory of particles chosen according to the Gibbs measure. We first need to introduce some additional notation. For a particle z at the n th generation and 0 ≤ i ≤ n, we denote by z i its ancestor at the i th generation and we set

V(z) := V (z tn ) σ √ n , t ∈ [0, 1]
the rescaled trajectory of z's lineage. We work here in the set D([0, 1]) of the càdlàg functions from [0, 1] to R, with the Skorokhod distance (see Section A.2). For n ∈ N and β ≥ 1, we denote by µ n,β the image measure of the Gibbs measure ν n,β by V, that is the random measure on

D([0, 1]) such that, for each F ∈ C b (D([0, 1]
)), we have

µ n,β (F ) := 1 W n,β |z|=n e -βV (z) F (V(z)),
where C b (D([0, 1])) denotes the set of continuous bounded functions from D([0, 1]) to R.

Convergence of µ n,β has already been studied in the strong disorder regime when β does not depend on n, under assumptions (1.1), (1.2) and (1.3). In the critical case β = 1, Theorem 1.2 of Madaule [START_REF] Madaule | First order transition for the branching random walk at the critical parameter[END_REF] shows1 that, for all

F ∈ C b (D([0, 1])), µ n,1 (F ) ---→ n→∞ E[F (M)], in P * -probability.
(1.12)

On the other hand, in the case β > 1, Chen, Madaule and Mallein [START_REF] Chen | On the trajectory of an individual chosen according to supercritical gibbs measure in the branching random walk[END_REF] proved (in the nonlattice case) that, under P * , we have, for all uniformly continuous

F ∈ C b (D([0, 1])), µ n,β (F ) ---→ n→∞ k∈N p k F (e k ), in law, (1.13)
where (e k ) k∈N is a sequence of i.i.d. normalized Brownian excursions and (p k ) k∈N follows an independent Poisson-Dirichlet distribution with parameter (β -1 , 0). The convergence in (1.13) is believed to hold for all F ∈ C b (D([0, 1])). Moreover, Chen [START_REF] Chen | Scaling limit of the path leading to the leftmost particle in a branching random walk[END_REF] considered the case β = ∞ and showed that, for all

F ∈ C b (D([0, 1])), we have E * [µ n,∞ (F )] → E[F (e)] as n → ∞
, where e denotes the normalized Brownian excursion. Finally, in the weak disorder regime β < 1, if there is some p > 1 such that E[W p 1,β ] < ∞, we have the following convergence, with σ2

β := Ψ (β), for all F ∈ C b (D([0, 1])), 1 W n,β |z|=n e -βV (z) F V (z tn ) + tnΨ (β) σ β √ n , t ∈ [0, 1] ---→ n→∞ E[F (B)], (1.14)
in P * -probability 2 . It means that the trajectory is a straight line of slope -Ψ (β) > 0 at first order and around which Brownian fluctuations occur at second order. Our aim is to prove the convergence for trajectories of particles chosen according to the Gibbs measure in the near-critical case, in order to explain how happens the transition between the Brownian excursion, the Brownian meander and the straight line with Brownian fluctuations. 1) E e -σγM(1) F (M) , in P * -probability. 1) E e σγM (1)

(i) If β n := 1 + 1/α n and √ n/α n → ∞ as n → ∞, then we have, for all F ∈ C b (D([0, 1])), µ n,βn (F ) ---→ n→∞ E[F (e)], in P * -probability. (ii) If β n := 1 + 1/α n and √ n/α n → γ ∈ [0, ∞) as n → ∞, then we have, for all F ∈ C b (D([0, 1])), µ n,βn (F ) ---→ n→∞ 1 E e -σγM(
(iii) If (1.5) holds, β n := 1 -1/α n and √ n/α n → γ ∈ [0, ∞) as n → ∞, then we have, for all F ∈ C b (D([0, 1])), µ n,βn (F ) ---→ n→∞ 1 E e σγM(
F (M) , in P * -probability. (iv) If (1.6) holds, β n := 1 -1/α n and √ n/α n → ∞ as n → ∞, then we have, for all F ∈ C b (D([0, 1])), in P * -probability, 1 W n,βn |z|=n e -βnV (z) F V (z tn ) + tnΨ (β n ) σ √ n , t ∈ [0, 1] ---→ n→∞ E[F (B)].
We now state a corollary of this theorem, concerning the location of the mass of the Gibbs measure. Note that, for the terminology of the polymers' literature, this position is the typical energy of the polymer in the near-critical case. (i) If β n := 1 + 1/α n and √ n/α n → ∞ as n → ∞, then, for all ε > 0, it exists C > 0 such that for n large enough

P * ν n,βn 3 2 log n + C -1 α n , 3 2 log n + Cα n ≥ 1 -ε ≥ 1 -ε. (ii) If β n := 1 + 1/α n and √ n/α n → γ ∈ [0, ∞)
as n → ∞, then, for all ε > 0, it exists C > 0 such that for n large enough

P * ν n,βn C -1 √ n, C √ n ≥ 1 -ε ≥ 1 -ε. (iii) If (1.5) holds, β n := 1 -1/α n and √ n/α n → γ ∈ [0, ∞) as n → ∞, then the same property as in case (ii) holds. (iv) If (1.6) holds, β n := 1 -1/α n and √ n/α n → ∞ as n → ∞, then, for all ε > 0, it exists C > 0 such that for n large enough P * ν n,βn -Ψ (β n )n -C √ n, -Ψ (β n )n + C √ n ≥ 1 -ε ≥ 1 -ε.
Proof. In cases (ii), (iii) and (iv), it is a direct consequence of Theorem 1.2. For case (i), the assertion will be proved at the end of Subsection 5.2.

Genealogy under the Gibbs measure

We state here a direct consequence of Theorem 1.2 concerning the overlap in the branching random walk, introduced by Derrida and Spohn [START_REF] Derrida | Polymers on disordered trees, spin glasses, and traveling waves[END_REF] in the context of polymers on trees. We set, for x, y ∈ T,

|x ∧ y| := max{k ≤ min(|x|, |y|) : x k = y k },
that is the generation of the most recent common ancestor of x and y. For some couple of particles (x, y) chosen according to ν ⊗2 n,β , we are interested in the overlap between x and y defined by |x ∧ y|/n. Thus, we set, for a Borel set A ⊂ [0, 1],

ω n,β (A) := ν ⊗2 n,β |x ∧ y| n ∈ A so that ω n,β is a random probability measure on [0, 1].
Madaule [START_REF] Madaule | First order transition for the branching random walk at the critical parameter[END_REF] gives the following consequence of (1.12) in the case β = 1:

ω n,1 ---→ n→∞ δ 0 , in P * -probability.
For the other extremal case β = ∞, one can prove in the nonlattice case only 3 that, .15) The transition between this two cases appears with case β ∈ (1, ∞), with which Chen, Madaule and Mallein [START_REF] Chen | On the trajectory of an individual chosen according to supercritical gibbs measure in the branching random walk[END_REF] deal, but their result (1.13) is only proved for F ∈ C u b (D([0, 1])) and, thus, the convergence in law of ω n,β cannot be obtained as a corollary. However, Mallein [START_REF] Mallein | Genealogy of the extremal process of the branching random walk[END_REF] shows that, under P * , we have

ω n,∞ ---→ n→∞ δ 1 , in P * -probability. ( 1
ω n,β ---→ n→∞ (1 -π β )δ 0 + π β δ 1 , in law,
where π β := k∈N p 2 k and (p k ) k∈N follows a Poisson-Dirichlet distribution with parameter (β -1 , 0). It confirms a conjecture of Derrida and Spohn [START_REF] Derrida | Polymers on disordered trees, spin glasses, and traveling waves[END_REF].

In the near critical case, we can state the following consequence of Theorem 1.2.

Corollary 1.4. In each case of Theorem 1.1, under the same assumptions, we have

ω n,βn ---→ n→∞ δ 0 , in P * -probability.
Proof. We give the proof for the case (i), but it is exactly the same for other cases (only the limiting trajectory changes). First note that, for all F ∈ C b (D([0, 1]) 2 ), we have, with e and e independent normalized Brownian excursions,

µ ⊗2 n,βn (F ) = 1 W 2 n,βn |x|=|y|=n e -βnV (x)-βnV (y) F (V(x), V(y)) ---→ n→∞ E F (e, e ) , ( 1.16)
in P * -probability and therefore in L 1 , because µ ⊗2 n,βn (F ) is bounded. Indeed, by Theorem 1.2, (1.16) holds when F is of the form

F (x, y) = G 1 (x)G 2 (y) for some G 1 , G 2 ∈ C b (D([0, 1]
)) and, the general case follows. Then, we consider some ε > 0 and the closed set

A := {(x, y) ∈ D([0, 1]) 2 : ∀ t ≤ ε, x t = y t } of D([0, 1]) 2 . We have E[ω n,βn ([ε, 1])] ≤ E µ ⊗2 n,βn (A) ---→ n→∞ P ∀ t ≤ ε, e t = e t = 0,
using (1.16) and the Portmanteau theorem.

3 For example, using the stopping line Z[A] := {z ∈ T : V (z) ≥ A > max k<|z| V (z k )} for large A, it is possible to show that, with high probability for n large, all particles under νn,∞ have the same ancestor in Z[A]. Then (1.15) follows from Lemma 3.3 of Chen [START_REF] Chen | Scaling limit of the path leading to the leftmost particle in a branching random walk[END_REF]. In the lattice case, it is clear that (1.15) cannot hold.

Comments on the results

Theorem 1.1 fully describes the transition from the size n -3β/2 when β > 1 to the size e nΨ(β) when β < 1 and Corollary 1.3 shows the transition in the location of the mass of the Gibbs measure from [ (1.14)). Secondly, Theorem 1.2 describes the transition between the Brownian excursion in case (i) and the straight line with Brownian fluctuations in case (iv). Note that, since min z∈T V (z) > -∞ a.s., it is natural that the limiting trajectory stays nonnegative on [0, 1] in cases (i) to (iii). In case (iv), this constraint disappears in the limit due to the drift. Indeed, staying above a constant for a random walk with drift approximately σ √ n/α n needs effort until times of order α 2 n , so it disappears in the trajectory after scaling by n. However, this effort has a cost of order 1/α n , which explains the presence of this factor in the size e nΨ(βn) /α n of the partition function.

3 2 log n -C, 3 2 log n + C] when β > 1, to [-Ψ (β)n -C √ n, -Ψ (β)n + C √ n] when β < 1 (this follows from
In cases (ii) and (iii), when α n is of order √ n or larger, the perturbation is too small to change radically the behavior of the Gibbs measure in regards to the critical case β = 1: the size of the partition function is still n -1/2 , the typical energy is of order √ n and the limiting trajectory has a density w.r.t. the Brownian meander. Therefore, cases (ii) and (iii) are called the critical window by Alberts and Ortgiese [START_REF] Alberts | The near-critical scaling window for directed polymers on disordered trees[END_REF]. It brings to light a family of laws (P γ ) γ∈R on D([0, 1]) defined by P γ (F ) := E[e -σγM(1) F (M)]/E[e -σγM (1) ] for all F ∈ C b (D([0, 1])), including the law of the Brownian meander of length 1 for γ = 0 and such that P γ ⇒ L(e) as γ → ∞, where L(e) denotes the law of e. But there is no convergence as γ → -∞, because the trajectory is sent to infinity. However, we can consider another family (Q γ ) γ∈R defined by Q γ (F ) := E[e σγM (1) (1) ], so that, in case (iii), we have

F (M t -σγt, t ∈ [0, 1])]/E[e σγM
1 W n,βn |z|=n e -βnV (z) F V (z tn ) + tnΨ (β n ) σ √ n , t ∈ [0, 1] ---→ n→∞ Q γ (F ) in P * -probability,
This family includes also the Brownian meander's law and we have Q γ ⇒ L(B) as γ → ∞.

As opposed to this, cases (i) and (iv) are called the near-critical window and highlight some new behaviors. In case (i), the transition between the critical size n -1/2 and the strong disorder size n -3β/2 appears. The factor n 3βn/2 starts to behave differently than n 3/2 when α n = O(log n) and this is also the region where the particles mainly contributing to W n,βn are not simply those in [C -1 α n , Cα n ]: the fact that the lowest particle at time n is around 3 2 log n plays a role only in this region (see Lemma 5.2). Nevertheless, in the near-critical regime, the lowest particle at time n never has a positive weight in ν n,βn in the limit, unlike in the case β > 1. Since the particles mainly contributing to W n,βn are far below √ n, the endpoint of the limiting trajectory has to be 0 and, therefore, the excursion appears.

In case (iv), since β n tends to 1 slowly enough, we find the same asymptotic behavior for the partition function as in (1.11) when we first take n → ∞ and then β ↑ 1. For the limiting trajectory, the result is also similar to (1.14) in the case β > 1. Moreover, when α n is not too small, the different results in case (iv) can be rewritten only in terms of σ 2 : if α n n 1/3 , the size of the partition function is e σ 2 n/2α 2 n /α n and, if

n 1/4 = O(α n ), the location of the mass is in [σ 2 n/α n -C √ n, σ 2 n/α n + C √ n].
But, on the contrary, if α n is too small, there is a break of universality.

Finally, we stress that there is no discontinuity between the different cases of the results. Indeed, using that E[e -σγM (1) ] ∼ 1/(σγ) 2 and E[e σγM (1) ] ∼ √ 2πσγe (σγ) 2 /2 as γ → ∞, all cases of Theorem 1.1 (requiring α n n 1/3 in case (iv)) can be written

n 3(1-βn)/2
√ n E e (1-βn)σM(1)

√ n W n,βn ---→ n→∞ 1 σ 2 π D ∞ , in P * -probability, (1.17)
noting that n 3(1-βn)/2 → 1 as soon as α n log n. For Theorem 1.2, the continuity between the different cases follows from the convergences P γ ⇒ L(e) and Q γ ⇒ L(B) as γ → ∞.

Organization of the paper

Sections of this paper correspond to the different cases in the results: case (i) is treated in Section 5, cases (ii) and (iii) in Section 3 and case (iv) in Section 4. The behavior in the critical window (cases (ii) and (iii)) is a direct consequence of the analogue results (1.10) and (1.12) in the critical case (apart from some technical details). The near-critical window in the weak disorder regime (case (iv)) needs slightly more work, but relies mainly on Madaule's [START_REF] Madaule | First order transition for the branching random walk at the critical parameter[END_REF] results and on L p inequalities techniques (see [START_REF] Biggins | Uniform convergence of martingales in the branching random walk[END_REF]). Finally, the main part of this paper is dedicated to the proof of case (ii), which follows some ideas of Aïdékon and Shi [START_REF] Aïdékon | The Seneta-Heyde scaling for the branching random walk[END_REF], with change of measure and spine decomposition techniques. One main difference with the previous literature on the branching random walk is that we need here to consider particles that are far below √ n but also far above the lowest particle. Note that we prove in Subsection 5.6 a new version of the so-called peeling lemma (see Shi [START_REF] Shi | Branching random walks[END_REF]) with a more general setting than what is needed for the aim of this paper and that could be of independent interest.

On the other hand, Section 2 regroups some well-known results on the branching random walk and on classical random walk. Some new results are stated in this section and proved in the appendix. Note that none of the results of Section 2 are needed for the proof of cases (ii) and (iii) and only a few of them for case (iv). The appendix contains some other technical results.

Throughout the paper, the c i 's denote positive constants, we set N := {0, 1, 2, . . .} and, for a, b ∈ N, a, b := [a, b] ∩ N. For two sequences (u n ) n∈N and (v n ) n∈N of positive real numbers, we say that

u n ∼ v n as n → ∞ if lim n→∞ u n /v n = 1, that u n = O(v n ) as n → ∞ if lim sup n→∞ u n /v n < ∞, and that u n = o(v n ) or u n v n if lim n→∞ u n /v n = 0.
For (S, d) a metric space, let C b (S) be the set of bounded continuous functions from S → R and C u b (S) be its subset of uniformly continuous functions. For F ∈ C b (S), we set F := sup x∈S |f (x)|. For F ∈ C u b (S), we will denote by ω F a modulus of continuity for function F : ω F is a continuous bounded nondecreasing function from R + → R + such that ω F (0) = 0 and ∀ x, y ∈ S, |F (x) -F (y)| ≤ ω F (d(x, y)).

Preliminary results

In this section, we state some preliminary results that are mostly needed in Section 5. In Subsection 2.1, we present some well-known tools to study the branching random walk and the next subsections contain results concerning one-dimensional random walk.

Many-to-one lemma and changes of probabilities

For a ∈ R, let P a denote a probability measure under which (V (z), z ∈ T) is the branching random walk starting from a, and E a the associated expectation (for brevity we will write P and E instead of P 0 and E 0 ). We define a random walk (S n ) n≥0 associated to the branching random walk: under P a , S 0 = a a.s. and the law of the increments is given by

E a [h(S 1 -S 0 )] = E |z|=1 h(V (z))e -V (z) ,
for all measurable h : R → R + . This random walk is well-defined and centred thanks to assumption (1.2). Moreover, by assumption (1.3), we have E[S 2 1 ] = σ 2 ∈ (0, ∞). Then, by induction, one gets the following result (see Biggins and Kyprianou [START_REF] Biggins | Seneta-Heyde norming in the branching random walk[END_REF]). It is also a corollary of the forthcoming Proposition 2.2. Lemma 2.1 (Many-to-one lemma). For all n ≥ 1, a ∈ R and all measurable function g : R n+1 → R + , we have

E a |z|=n g(V (z 0 ), . . . , V (z n )) = E a e Sn-a g(S 0 , . . . , S n ) .
We now state some well-known change of probabilities and spinal decomposition results. This method dates back at least to Kahane and Peyrière [START_REF] Kahane | Sur certaines martingales de Benoit Mandelbrot[END_REF], Rouault [START_REF] Rouault | Lois empiriques dans les processus de branchement spatiaux homogènes supercritiques[END_REF] and Chauvin and Rouault [START_REF] Chauvin | KPP equation and supercritical branching Brownian motion in the subcritical speed area. Application to spatial trees[END_REF]. See also Biggins and Kyprianou [START_REF] Biggins | Measure change in multitype branching[END_REF] for spinal decomposition in more general type of branching structures and Shi [START_REF] Shi | Branching random walks[END_REF] for a survey on this topic. Let F n denote the σ-algebra generated by (V (z), |z| ≤ n) and

F ∞ := σ( n∈N F n ).
We first introduce Lyons' change of measure [START_REF] Lyons | A simple path to Biggins' martingale convergence for branching random walk[END_REF]: since (W n,1 ) n∈N is a nonnegative martingale of mean e -a under P a , we can define a new probability measure Q a on F ∞ , by letting for all n ∈ N,

Q a | Fn := e a W n,1 • P a | Fn .
We will denote by E Qa the associated expectation and we will write Q and E Q instead of Q 0 and E Q 0 . Let L be a point process on R which has the law of (V (z), |z| = 1) under Q.

Lyons [START_REF] Lyons | A simple path to Biggins' martingale convergence for branching random walk[END_REF] proved the following description for the branching random walk under Q a , with a decomposition along a spine (w n ) n∈N which is a marked ray in the the genealogical tree T (in order to be mathematically rigorous, one should enlarge the probability space and work on a product space, see Lyons [START_REF] Lyons | A simple path to Biggins' martingale convergence for branching random walk[END_REF]). The system starts with one particle w 0 at position a, forming the 0 th generation. For each n ∈ N, individuals of the n th generation give birth independently of each other and of the foregoing. Individuals other than w n generate offspring around their position according to the law of L and w n breeds according to the law of L. Then, w n+1 is chosen independently among w n 's children, with probability proportional to e -V (z) for each child z. Moreover, Lyons showed the following result concerning the spine (w n ) n∈N under Q a . Proposition 2.2. Let a ∈ R.

(i) For each n ∈ N and |z| = n, we have

Q a (w n = z | F n ) = e -V (z) W n,1
.

(ii) The process (V (w n )) n∈N under Q a has the same law as (S n ) n∈N under P a . Now, we present another change of measure, that was first introduced by Biggins and Kyprianou [START_REF] Biggins | Measure change in multitype branching[END_REF]. For this, we need to define R the renewal function in the first strict descending ladder height process of the random walk (S n ) n∈N . For u ≥ 0,

R(u) := ∞ k=0 P(H k ≤ u),
where (H k ) k∈N is the first strict descending ladder height process: we set τ 0 := 0, H 0 := 0 and, for k ≥ 1, τ k := inf n > τ k-1 : S n < S τ k-1 and H k := -S τ k . Then, we introduce the truncated derivative martingale: for L ≥ 0 and n ∈ N,

D (L) n := |z|=n R L (V (z))e -V (z) 1 V (z)≥-L ,
where, for u ≥ -L, R L (u) := R(L + u) and, for |z| = n, V (z) := min 0≤i≤n V (z i ). Fix now some L ≥ 0. For a ≥ -L, under P a , (D

(L)
n ) n∈N is a nonnegative martingale of mean R L (a) and therefore we can define another probability measure

Q (L) a by Q (L) a | Fn := D (L) n R L (a) • P a | Fn .
We will denote by

E Q (L) a
the associated expectation and write

Q (L) and E (L) Q instead of Q (L) 0 and E Q (L) 0
. For a ≥ -L, let L(L) a be a point process on R with the law of (V (z), |z| = 1) under

Q (L)
a . Biggins and Kyprianou [START_REF] Biggins | Measure change in multitype branching[END_REF] proved the following spinal decomposition description for the branching random walk under Q (L) a , where the spine is denoted by (w

(L) n ) n∈N .
The description is similar to the previous one, but here w (L) n have offspring according to L(L)

V (w (L) n )
and w

(L)
n+1 is chosen among these children, with probability proportional to R L (V (z))e -V (z) 1 V (z)≥-L for each child z. Moreover, we get the following analogue of Proposition 2.2.

Proposition 2.3. Let L ≥ 0 and a ≥ -L.

(i) For each n ∈ N and |z| = n, we have

Q (L) w (L) n = z F n = R L (V (z))e -V (z) 1 V (z)≥-L D (L) n .
(ii) The process (V (w

(L) n )) n∈N under Q (L) a
has the same law as (S n ) n∈N under P a conditioned to stay in [-L, ∞): for all n ∈ N and all measurable function g :

R n+1 → R + , E Q (L) a g V (w (L) 0 ), . . . , V (w (L) n ) = 1 R L (a) E a g(S 0 , . . . , S n )R L (S n )1 S n ≥-L .

One-dimensional random walks

Up to the end of this section, we consider a centred random walk (S n ) n∈N with finite variance

E[S 2 1 ] = σ 2 ∈ (0, ∞).
In this subsection, we state various results concerning this one-dimensional random walk and the associated renewal function R. For n ∈ N, we set S n := min 0≤i≤n S i .

Recall that R is the renewal function associated to the first strict descending ladder height process (H k ) k∈N . Since E[S 1 ] = 0 and E[S 2 1 ] < ∞, by Feller [27, Theorem XVIII.5.1 (5.

2)], we have E[H 1 ] < ∞. Thus, it follows from Feller's [27, p. 360] renewal theorem that it exists a constant c 0 > 0 such that

R(u) u ---→ u→∞ c 0 (2.1)
and so there exist also c 1 , c 2 > 0 such that, for all u ≥ 0,

c 1 (1 + u) ≤ R(u) ≤ c 2 (1 + u). (2.2)
We are interested in the behavior of random walks staying above a barrier. First, we recall the following estimate for the probability of a random walk to stay above -a: by Kozlov [32, Theorem A], it exists a constant θ > 0 such that for all u ≥ 0,

P(S n ≥ -u) ∼ n→∞ θR(u) √ n , ( 2.3)
and it exists c 3 > 0 such that, for all n ∈ N and u ≥ 0, we have the uniform bound

P(S n ≥ -u) ≤ c 3 (1 + u) √ n . (2.4)
Constants c 0 and θ will appear all along the paper and they are related by the following equation, from Aïdékon and Shi [3, Lemma 2.1],

θc 0 = 2 πσ 2 1/2 . (2.5)
The following result states the convergence in law of S n /σ √ n conditioned to stay above -u towards a Rayleigh distribution, with uniformity in the position of the barrier -u: by Aïdékon and Jaffuel [2, Lemma 2.2], if (γ n ) n∈N is a sequence of positive numbers such that γ n √ n as n → ∞, then we have, for all continuous bounded function g : R + → R,

E g S n + u σ √ n 1 S n ≥-u = θR(u) √ n ∞ 0 g(t)te -t 2 /2 dt + o(1) , (2.6) uniformly in u ∈ [0, γ n ].
From Lemmas 2.2 and 2.4 of Aïdékon and Shi [START_REF] Aïdékon | The Seneta-Heyde scaling for the branching random walk[END_REF], we have the following inequalities, sometimes called ballot theorems: it exists c 4 > 0 such that, for all b > a ≥ 0, u ≥ 0 and n ≥ 1,

P(S n ∈ [a -u, b -u], S n ≥ -u) ≤ c 4 (u + 1)(b + 1)(b -a + 1) n 3/2 , ( 2.7)
and, for λ ∈ (0, 1), it exists c 5 = c 5 (λ) > 0 such that for all b > a ≥ 0, u ≥ 0, v ∈ R and n ≥ 1, we have

P S n ∈ [a + v, b + v], S λn ≥ -u, min λn ≤j≤n S j ≥ v ≤ c 5 (u + 1)(b + 1)(b -a + 1) n 3/2 , ( 2.8)
where we added a second barrier between times λn and n. From the previous results, it follows (see Aïdékon [START_REF] Aïdékon | Convergence in law of the minimum of a branching random walk[END_REF]Lemma B.2]) that it exists c 6 > 0 such that, for all a, u ≥ 0,

i≥0 P(S i ≥ -u, S i ≤ a -u) ≤ c 6 (1 + a)(1 + min(a, u)). ( 2.9)
Finally, we state a last result that is used for the proof of the peeling lemma in Subsection 5.6 and is proved in Subsection B.2. Lemma 2.4. Let (r n ) n∈N be an increasing sequence of positive numbers such that we have

n∈N r n n -3/2 < ∞. We set, for , i ∈ 0, n , u, µ ≥ 0 and v ∈ R, m (n, ) i := -u + r i -µ if 0 ≤ i < , v + r n-i -µ if ≤ i ≤ n.
For any ε > 0 and λ

∈ (0, 1/2), it exists µ = µ(ε, λ) > 0 such that for all b, u ≥ 0, v ∈ R, n ∈ N and ∈ [λn, (1 -λ)n], P S ≥ -u, min ≤j≤n S j ≥ v, S n ∈ [b + v, b + v + 1], ∃ i ∈ 0, n : S i ≤ m (n, ) i ≤ ε(1 + u)(1 + b) n 3/2 .
Remark 2.5. This kind of lemma is useful in the proof of peeling lemmas: see Lemma B.3 of [1], Lemma 6.1 of [START_REF] Madaule | Convergence in law for the branching random walk seen from its tip[END_REF] and Lemma A.6 of [START_REF] Shi | Branching random walks[END_REF]. A slight difference here is that the terminal interval is

[b + v, b + v + 1] instead of [b, b + v]
in the previous results. Furthermore, in these papers, they take r i = i α with α ∈ (0, 1/6). This would have been sufficient for the proof of our peeling lemma, but an anonymous referee asked us whether the result holds for any α ∈ (0, 1/2) and this lemma answers in the affirmative.

Convergence towards the Brownian meander

We define the rescaled trajectory of the random walk until time n: for each n ∈ N * ,

S (n) := S nt σ √ n , t ∈ [0, 1] .
We state the following convergence result for the trajectory S (n) conditioned to stay nonnegative, with uniformity in the starting point of the random walk.

Proposition 2.6. Let (γ n ) n∈N be a sequence of positive numbers such that γ n

√ n as n → ∞. Then, for all F ∈ C b (D([0, 1])), we have E u F (S (n) )1 S n ≥0 = θR(u) √ n (E[F (M)] + o(1)), as n → ∞, uniformly in u ∈ [0, γ n ],
where M denotes the Brownian meander of length 1.

This invariance principle has been proved in the case u = 0 by Iglehart [START_REF] Iglehart | Functional central limit theorems for random walks conditioned to stay positive[END_REF], Bolthausen [START_REF] Bolthausen | On a functional central limit theorem for random walks conditioned to stay positive[END_REF] and Doney [START_REF] Doney | Conditional limit theorems for asymptotically stable random walks[END_REF]. The case where F is a function of the terminal value of the trajectory is already showed in (2.6). We give a short proof of this generalization in Subsection B.1, that relies on the invariance principle by Caravenna and Chaumont [START_REF] Caravenna | Invariance principles for random walks conditioned to stay positive[END_REF], for random walk conditioned to stay nonnegative for all time.

We present also a corollary of Proposition 2.6, which holds under an additional assumption on the random walk S that is equivalent to assumption (1.5) by the many-to-one lemma. It will be used in Section 4 and its proof is postponed to Subsection B.1.

Corollary 2.7. Assume that it exists

δ 0 > 0 such that E[e δ 0 S 1 ] < ∞. Then, for all C, L > 0 and F ∈ C b (D([0, 1])), we have E e CSn/ √ n F (S (n) )1 S n ≥-L = θR(L) √ n E e CσM(1) F (M) + o(1) ,
as n → ∞, where M denotes the Brownian meander of length 1.

Convergence towards the Brownian excursion

In this subsection, our interest is the convergence of S (n) , conditioned to stay above two successive barriers and to end up in a small interval, towards the normalized Brownian excursion, with uniformity with respect to the barriers' positions and to the endpoint as long as they are much smaller than √ n. This result will be used repetitively in Section 5 for the proof of part (ii) of Theorems 1.1 and 1.2.

At this scale for the endpoint S n , the random walk behaves differently in the lattice and nonlattice cases, so they have to be distinguished. Moreover, we need some new notation: let (S - n ) n∈N be a random walk such that, under P a , S - 0 = a a.s. and S - 1 -S - 0 has the same law as S 0 -S 1 . All objects referring to S (R, c 0 , θ, . . . ) have their analogue for S -denoted with asuperscript (R -, c - 0 , θ -, . . . ). We can now state our result, which generalizes Lemma 2.6 of Chen, Madaule and Mallein [START_REF] Chen | On the trajectory of an individual chosen according to supercritical gibbs measure in the branching random walk[END_REF] and is proved in Subsection B.5. Proposition 2.8. Let (γ n ) n∈N be a sequence of positive numbers such that γ n √ n as n → ∞ and e be the normalized Brownian excursion.

(i) If the law of S 1 is nonlattice, then for all h > 0, λ ∈ (0, 1) and F ∈ C b (D([0, 1])), E F (S (n) )1 S λn ≥-u,min λn ≤i≤n S i ≥v,Sn∈[v+b,v+b+h) = π 2 θθ - σ R(u) n 3/2 (E[F (e)] + o(1)) b+h b R -(t) dt, as n → ∞, uniformly in b, u ∈ [0, γ n ] and v ∈ [-γ n , γ n ].
(ii) If the law of S 1 is (h, a)-lattice, then for all λ ∈ (0, 1) and

F ∈ C u b (D([0, 1])), E F (S (n) )1 S λn ≥-u,min λn ≤i≤n S i ≥v,Sn=v+b = π 2 θθ - σ R(u) n 3/2 (E[F (e)] + o(1))hR -(b), as n → ∞, uniformly in u ∈ [0, γ n ], v ∈ [-γ n , γ n ] and b ∈ [0, γ n ] ∩ (-v + an + hZ).

The critical window

We prove here cases (ii) and (iii) of Theorems 1.1 and 1.2, where

β n = 1 ± 1/α n with √ n/α n → γ ∈ [0, ∞).
This proof is based on Theorem 1.2 of Madaule [START_REF] Madaule | First order transition for the branching random walk at the critical parameter[END_REF], recalled in (1.12). 1) F (M) , in P * -probability.

Proof of part (ii) of Theorems 1.1 and 1.2. Our aim is to show that, for all

F ∈ C b (D([0, 1])), we have 1 W n,1 |z|=n e -βnV (z) F (V(z)) ---→ n→∞ E e -σγM(
(3.1)

Then, using (1.10) and the case F ≡ 1, part (ii) of Theorems 1.1 and 1.2 follows. Note also that it is sufficient to show (3.1) for F nonnegative. We first deal with the case γ ∈ (0, ∞). We consider some nonnegative function

F ∈ C b (D([0, 1])) and ε > 0. By dominated convergence, we have that E[e -σγ M(1) F (M)] tends towards E[e -σγM(1) F (M)] as γ → γ, so we can choose 0 < γ -< γ < γ + such that E e -σγM(1) F (M) - ε 2 ≤ E e -σγ ± M(1) F (M) ≤ E e -σγM(1) F (M) + ε 2 . (3.2) Since √ n/α n → γ and, under P * , min |z|=n V (z) → ∞ a.s., it exists n 0 ∈ N such that, for all n ≥ n 0 , we have √ n/α n ∈ [γ -, γ + ] and also P * (min |z|=n V (z) < 0) ≤ ε. Moreover, we set, for x ∈ D([0, 1]), G + (x) := F (x)(e -σγ + x 1 ∧ 1) and G -(x) := F (x)(e -σγ -x 1 ∧ 1): if x 1 ≥ 0, then we have G + (x) ≤ e (1-βn)x 1 F (x) ≤ G -(x)
. Thus, we get, for all n ≥ n 0 ,

P * 1 W n,1 |z|=n e -βnV (z) F (V(z)) / ∈ µ n,1 (G + ), µ n,1 (G -) ≤ P * min |z|=n V (z) < 0 ≤ ε.
Therefore, we have

P * 1 W n,1 |z|=n e -βnV (z) F (V(z)) -E e -σγM(1) F (M) ≤ ε ≤ ε + P * µ n,1 (G -) ≥ E e -σγM(1) F (M) + ε + P * µ n,1 (G + ) ≤ E e -σγM(1) F (M) -ε ≤ ε + P * µ n,1 (G -) ≥ E G + (M) + ε 2 + P * µ n,1 (G + ) ≤ E G -(M) - ε 2 , ( 3.3)
using (3.2). Then, applying (1.12), we get that both probabilities in (3.3) tends to 0, because

G -, G + ∈ C b (D([0, 1]
)) and it concludes the proof of (3.1) in the case γ ∈ (0, ∞).

Finally, for the case γ = 0, we proceed in the same way as for γ ∈ (0, ∞), taking here

γ + > 0 such that E[F (M)] -ε 2 ≤ E[e -σγ + M(1) F (M)],
G + defined as before and G -:= F . Then, the same inequalities hold. Remark 3.1. In case (iii), we work under assumption (1.5) and we will use Proposition 3.8 of Madaule [START_REF] Madaule | First order transition for the branching random walk at the critical parameter[END_REF], whereas Madaule works in [START_REF] Madaule | First order transition for the branching random walk at the critical parameter[END_REF] under the stronger assumption (1.6). But, for the proof of his Proposition 3.8, he only uses Assumption (1.6) in the proof of his Lemma A.2, in order to have that Ψ is finite on a left-neighbourhood of 1 and Ψ

(β) = σ 2 2 (β -1) 2 + o((β -1)
2 ) as β ↑ 1, and this holds also under our assumption (1.5).

Proof of part (iii) of Theorems 1.1 and 1.2. Applying Proposition 3.8 of Madaule [START_REF] Madaule | First order transition for the branching random walk at the critical parameter[END_REF] (see Remark 3.1 above), we get that

µ n,1 (G) → E[G(M)] in P * -probability with G : x ∈ D([0, 1]) → e Cx 1
for any C > 0, although G is not bounded. Combining this with (1.12), it is straightforward to extend this convergence to functions of the type G :

x ∈ D([0, 1]) → e Cx 1 F (x) with C > 0 and F ∈ C b (D([0, 1])).
Then, we prove that, for all 1) F (M) , in P * -probability, (3.4) using the same method as for the proof of case (ii): we approach function x → e σx 1 √ n/αn F (x) from above and from below, by considering here G + (x) := e σγ + x 1 F (x) and G -(x) := e σγ -x 1 F (x) when γ ∈ (0, ∞) and the same function G + but with G -:= F when γ = 0. Finally, part (iii) of Theorems 1.1 and 1.2 follows from (3.4).

F ∈ C b (D([0, 1])) nonnegative, 1 W n,1 |z|=n e -βnV (z) F (V(z)) ---→ n→∞ E e -σγM(

The near-critical window in the weak disorder regime

In this section, we deal with the case where β n = 1 -1/α n and α n √ n and prove successively convergence of the rescaled partition function and then convergence of the trajectories. We work here under assumption (1.6) so Ψ is analytic on an open interval I containing 1. Moreover, by Lemma 4.3 of Madaule [START_REF] Madaule | First order transition for the branching random walk at the critical parameter[END_REF], assumption (1.6) implies that there exist c 7 > 0 and

η 0 < 1 such that E[( W ∞,β ) 1+ η
2 ] ≤ c 7 for any 0 < η < η 0 and β = 1 -η. Thus, for any n ∈ N and 0 < η < η 0 , we have, with β = 1 -η,

E ( W n,β ) 1+ η 2 = E E W ∞,β F n 1+ η 2 ≤ E ( W ∞,β ) 1+ η 2 ≤ c 7 , ( 4.1)
by using Jensen's inequality. We need to introduce a more general statement of the many-to-one lemma. For β ∈ I, we define another random walk (S n,β ) n∈N starting at 0 under P and such that

E[h(S 1,β )] = e -Ψ(β) E |z|=1 h(V (z) + Ψ (β))e -βV (z) .
Then, S 1,β is centred and has variance σ 2 β := E[S 2 1,β ] = Ψ (β) ∈ (0, ∞). Moreover, we have the following analogue of the many-to-one lemma (see Shi [START_REF] Shi | Branching random walks[END_REF]): for all n ≥ 1 and all measurable function g : R n+1 → R + , we have

E |z|=n g(V (z i ), i ∈ 0, n ) = e nΨ(β) E e β(S n,β -nΨ (β)) g(S i,β -iΨ (β), i ∈ 0, n ) . (4.2)
One can see (S n,β ) n∈N as a discrete Girsanov transform of (S n ) n∈N . We first establish a preliminary lemma. (1) .

Lemma 4.1. Let (β n ) n∈N ∈ (0, 1) N and (k n ) n∈N ∈ N N be sequences such that β n → 1 and k n ≥ 1/(1 -β n ) 2 for any n ∈ N. Then, for any L > 0, lim sup n→∞ 1 1 -β n E W kn,β n 1 inf x∈T V (x)≥-L ≤ θR(L)E e σM
Proof. We first apply the many-to-one lemma to get that 1

1 -β n E W kn,β n 1 inf x∈T V (x)≥-L ≤ 1 1 -β n e -knΨ(β n ) E e (1-β n )S kn 1 S kn ≥-L .
Then, we set m n := (1 -β n ) -2 ≤ k n and, applying the Markov property at time m n , we have (1) + o( 1) e (kn-mn)Ψ(β n ) , (4.3) using Corollary 2.7 to bound the first expectation in the middle part of (4.3) and the many-to-one lemma for the second expectation. Using that e -mnΨ(β n ) ≤ 1, it proves the lemma.

E e (1-β n )S kn 1 S kn ≥-L ≤ E e (1-β n )Sm n 1 S mn ≥-L E e (1-β n )S kn-mn = θR(L) √ m n E e σM
Proof of part (iv) of Theorem 1.1. We set ξ n := α n | W ∞,βn -W n,βn | and want to show that ξ n → 0 in P * -probability. It will prove part (iv) of Theorem 1.1, since α n W ∞,βn → 2D ∞ in P * -probability by (1.11). We first follow the proof of Lemma 4.2 of Madaule [START_REF] Madaule | First order transition for the branching random walk at the critical parameter[END_REF]. We set p n := 1 + 1/2α n and

ξ n := E[ξ pn n |F n ].
For ε > 0, we have

P * (ξ n ≥ ε) ≤ 1 P(S) E 1 ξ pn n ≥ε pn 1 ξ n <ε 1+pn + P * ξ n ≥ ε 1+pn ≤ ε P(S) + P * ξ n ≥ ε 1+pn , (4.4) using that P(ξ pn n ≥ ε pn | F n ) ≤ ε -pn E[ξ pn n |F n ] = ε -pn ξ n .
By the branching property at time n, we have

ξ n = α n |x|=n e -βnV (x)-nΨ(βn) W (x) ∞,βn -1 ,
where, conditionally on F n , the W (x) ∞,βn for |x| = n are independent variables with the same law as W ∞,βn . Then, using that for any sequence (X i ) i∈N of independent centred variables and any [START_REF] Bahr | Inequalities for the rth absolute moment of a sum of random variables, 1 ≤ r ≤ 2[END_REF]), we get

γ ∈ [1, 2] we have E[| X i | γ ] ≤ 2 E[|X i | γ ] (see
ξ n = E[ξ pn n | F n ] ≤ 2α pn n |x|=n e -pnβnV (x)-npnΨ(βn) E W ∞,βn -1 pn ≤ 2α pn n W n,pnβn e n[Ψ(pnβn)-pnΨ(βn)] 2 pn (c 7 + 1), (4.5)
by using (4.1) for n large enough such that 1 -η 0 < 1 -β n < 1. Now, we choose L > 0 such that P * (inf x∈T V (x) < -L) ≤ ε and, by (4.5) and Markov's inequality, we get

P * ξ n ≥ ε 1+pn ≤ ε + ε -(1+pn) c 8 P(S) (2α n ) pn e n[Ψ(pnβn)-pnΨ(βn)] E W n,pnβn 1 inf x∈T V (x)≥-L .
As n → ∞, we have (2α n ) pn ∼ 2α n , 1 -p n β n ∼ 1/2α n and, by a Taylor expansion,

n[Ψ(p n β n ) - p n Ψ(β n )] ∼ -3σ 2 n/8α 2 n → -∞.
Thus, applying Lemma 4.1 with k n = n and β n = p n β n , we showed that lim sup n→∞ P * (ξ n ≥ ε 1+pn ) ≤ ε. Coming back to (4.4), it concludes the proof.

Proof of part (iv) of Theorem 1.2. By Lemma A.1, we can reduce the proof to the case F ∈ C u b (D([0, 1])). Moreover, by considering F -E[F (B)] instead of F , we can assume that E[F (B)] = 0. By Theorem 1.1, we have α n e -nΨ(βn) W n,βn → 2D ∞ in P * -probability with D ∞ > 0 P * -a.s., so it is sufficient to prove that

U n (F ) := α n e -nΨ(βn) |z|=n e -βnV (z) F V (n) (z) ---→ n→∞ 2D ∞ E[F (B)] = 0, (4.6)
in P * -probability, where

V (n) t (z) := [V (z tn ) + tnΨ (β n )]/σ √ n for t ∈ [0, 1]. For some C > 0, we set k n := Cα n 2 and, for each x ∈ D([0, 1]), F n (x) := F (x ((n-kn)t+kn)/n -x kn/n , t ∈ [0, 1]
). Let ε > 0. In order to prove (4.6), it is sufficient to prove that we can choose C such that lim sup

n→∞ P * (|U n (F ) -U n (F n )| ≥ ε) ≤ 2ε, (4.7) lim sup n→∞ P * (|U n (F n ) -E[U n (F n ) | F kn ]| ≥ ε) ≤ (2 + P(S) -1 )ε, (4.8) lim sup n→∞ P * (|E[U n (F n ) | F kn ]| ≥ ε) ≤ ε. (4.9)
The assumption that E[F (B)] = 0 is only needed for (4.9). For the sequel, we fix some L > 0 such that

P * (inf x∈T V (x) < -L) ≤ ε.
We first prove (4.8). We set

ζ n := U n (F n ) -E[U n (F n ) | F kn ]
and proceed in a similar way as for the proof of part (iv) of Theorem 1.1, by setting p n := 1 + 1/2α n and

ζ n := E[|ζ n | pn |F n ]. By (4.4), we have P * (|ζ n | ≥ ε) ≤ εP(S) -1 + P * (ζ n ≥ ε 1+pn
). By the branching property at time k n , we have

ζ n = α n |x|=kn e -βnV (x)-knΨ(βn) Υ (x) n -E[Υ n ] , (4.10)
where, conditionally on F kn , the Υ

(x)
n for |x| = k n are independent variables with the same law as Υ n defined by

Υ n := |z|=n-kn e -βnV (z)-(n-kn)Ψ(βn) F V (z t(n-kn) ) + t(n -k n )Ψ (β n ) σ √ n , t ∈ [0, 1] . (4.11) Since the Υ (x)
n -E[Υ n ] are also centred, we get, in the same way as for (4.5),

ζ n = E[|ζ n | pn | F kn ] ≤ 2α pn n |x|=kn e -pnβnV (x)-knpnΨ(βn) E[|Υ n -E[Υ n ]| pn ] ≤ 4c 7 (2 F α n ) pn W kn
,pnβn e kn(Ψ(pnβn)-pnΨ(βn)) , (4.12)

by using the following bound

E[|Υ n -E[Υ n ]| pn ] ≤ 2 pn+1 E[|Υ n | pn ] ≤ 2 pn+1 F pn E W n-kn,βn pn ≤ 2 pn+1 F pn c 7 ,
where we used (4.1) for n large enough such that 1 -η 0 < 1 -β n < 1. Now, using that P * (inf x∈T V (x) < -L) ≤ ε and Markov's inequality, we get

P * ζ n ≥ ε 1+pn ≤ ε + ε -(1+pn) c 9 P(S)
(2 F α n ) pn e kn[Ψ(pnβn)-pnΨ(βn)] E W kn,pnβn 1 inf x∈T V (x)≥-L so, using that, as n → ∞, we have (2

F α n /ε) pn ∼ 2 F α n /ε, 1 -p n β n ∼ 1/2α n and k n [Ψ(p n β n ) -p n Ψ(β n )] ∼ -3σ 2 k n /8α
P * ζ n ≥ ε 1+pn ≤ ε + c 10 R(L) ε 2 e -3σ 2 C 2 /8 ≤ 2ε,
by choosing C large enough. This proves (4.8). The constant C is now fixed. We now prove (4.7). Using that P * (inf x∈T V (x) < -L) ≤ ε and the Markov inequality, we get, with c 11 := 1/P(S),

P * (|U n (F ) -U n (F n )| ≥ ε) ≤ ε + c 11 α n εe nΨ(βn) E |z|=n e -βnV (z) 1 V (z)≥-L |F -F n | V (n) (z) . (4.13) For t ∈ [0, 1], we define S (n) t := [S tn + tnΨ (β n )]/σ √ n.
Then, using the many-to-one lemma and the triangle inequality, we get that, for any M > 0, the expectation in (4.13) is smaller than

E e Sn/αn 1 S n ≥-L,max 0≤k≤kn S k ≤M αn |F -F n |( S (n) ) + 2 F E e Sn/αn 1 S n ≥-L,max 0≤k≤kn S k >M αn . (4.14)
Note that, using (A.7) with here κ n = k n /n, we have, for any x ∈ D([0, 1]),

|F -F n |(x) ≤ ω F k n n ∨ 3
P * (|U n (F ) -U n (F n )| ≥ ε) ≤ ε + c 12 R(L) εC E e CσM(1) 1 max M>M/(Cσ) ≤ 2ε, (4.15)
by choosing M large enough (L and C being fixed). It proves (4.7). Finally, we prove (4.9). Using the branching property in the same way as for (4.10), we have

E[U n (F n ) | F kn ] = α n W kn,βn E[Υ n ],
where Υ n is defined in (4.11). By (4.2), we get

E[Υ n ] = E F S t(n-kn) ,βn + (t(n -k n ) -t(n -k n ) )Ψ (β n ) σ √ n , t ∈ [0, 1] = E F u n S (n-kn,βn) + v n ,
where we introduced S (n,β) := (S tn ,β /σ β √ n) t∈[0,1] and where (u n ) n∈N ∈ (R * + ) N and (v n ) n∈N ∈ D([0, 1]) N satisfy u n → 1 and v n ∞ → 0, using that k n n, Ψ (β n ) → 0 and σ βn → σ. Now, note that S (n-kn,βn) → B in law. This is not a direct consequence of Donsker's theorem because here the law of the random walk changes for each n. However, we can apply a strong invariance principle like Equation [START_REF] Biggins | Measure change in multitype branching[END_REF] of Sakhanenko [START_REF] Sakhanenko | Estimates in the invariance principle in terms of truncated power moments[END_REF]: it exists c 13 > 0 such that, for any n ≥ 1 and β ∈ I, it exists a Brownian motion B (n,β) such that

P sup t∈[0,1] S (n,β) t -B (n,β) t ≥ c 13 n 2/5 σ β √ n ≤ nE |S 1,β | 3 (n 2/5 ) 3 . Since σ βn → σ and E[|S 1,βn | 3 ] → E[|S 1 | 3 ] < ∞ as n → ∞, this proves that S (n-kn,βn) → B in law. Then, applying Lemma A.3, we get that E[Υ n ] → E[F (B)] = 0 as n → ∞.
On the other hand, using that P * (inf x∈T V (x) < -L) ≤ ε and the Markov inequality, we get

P * (|E[U n (F n ) | F kn ]| ≥ ε) ≤ ε + 1 ε α n E W kn,βn 1 inf x∈T V (x)≥-L |E[Υ n ]|,
and it proves (4.9) by using Lemma 4.1 and that E[Υ n ] → 0 as n → ∞.

The near-critical window in the strong disorder regime

In this section, we prove case (i) of Theorems 1.1 and 1.2 and of Corollary 1.3, where β n = 1 + 1/α n with α n √ n. This case constitutes the main part of this paper.

Change of probabilities

We introduce a first barrier by setting, for L > 0 and

F ∈ C b (D([0, 1])), W n,βn (F ) := |z|=n e -βnV (z) F (V(z)), W (L) n,βn (F ) := |z|=n e -βnV (z) F (V(z))1 V (z)≥-L ,
so that for L large this two variables are equal with high probability. Moreover, recall that, for u ≥ 0, R L (u) = R(L + u) and

D (L) n = |z|=n e -V (z) R L (V (z))1 V (z)≥-L
and that, using the martingale (D

(L)
n ) n∈N , we defined the modified probability measure Q (L) . We will work under this measure to study the asymptotic behaviour of W n,βn (F ). )), we have the convergence

n 3βn/2 α 2 n W (L) n,βn (F ) D (L) n ---→ n→∞ θ σ 2 E[F (e)], in Q (L) -probability.
This proposition will be proved in the following subsections, using a second moment argument similar to the one used by Aïdekon and Shi [START_REF] Aïdékon | The Seneta-Heyde scaling for the branching random walk[END_REF].

Proof of part (i) of Theorems 1.1 and 1.2. We consider

F ∈ C u b (D([0, 1]
)) and we are going to show here that )). The general case follows by Lemma A.1. Thus, it is now sufficient to prove (5.1) and we can for this purpose assume that F is nonnegative. Let ε, η > 0. We first fix L > 0 such that P * (inf x∈T V (x) < -L) ≤ η. Combining that min |x|=n V (x) → ∞ P * -a.s. as n → ∞ and that R L (u) ∼ c 0 u as u → ∞, we get that, on the

n 3βn/2 α 2 n W n,βn (F ) ---→ n→∞ c 0 θ σ 2 E[F (
event {inf x∈T V (x) ≥ -L}, lim n→∞ D (L) n = c 0 D ∞ > 0 P * -a.s. Thus, considering the event Ω 0 := S ∩ inf x∈T V (x) ≥ -L ∩ {∀n ≥ n 0 , 0 < c 0 (1 -ε)D ∞ ≤ D (L) n ≤ c 0 (1 + ε)D ∞ },
we can fix n 0 ∈ N such that P * (Ω 0 ) ≥ 1 -2η. We now introduce the event

E n := n 3βn/2 α 2 n W n,βn (F ) / ∈ c 0 θ σ 2 D ∞ (E[F (e)] -ε)(1 -ε), c 0 θ σ 2 D ∞ (E[F (e)] + ε)(1 + ε) .
Then, using that on Ω 0 we have

W (L) n,βn (F ) = W n,βn (F ), we get, for n ≥ n 0 , E D (L) n 1 En∩Ω 0 = Q (L) (E n ∩ Ω 0 ) ≤ Q (L)   n 3βn/2 α 2 n W (L) n,βn (F ) D (L) n - θ σ 2 E[F (e)] > θε σ 2   ---→ n→∞ 0, applying Proposition 5.1. Since D (L) n 1 En∩Ω 0 ≥ 0, it follows that D (L) n 1 En∩Ω 0 → 0 in P- probability. Using again that, on Ω 0 , lim n→∞ D (L) n = c 0 D ∞ > 0 P * -a.
s., we get that P * (E n ∩ Ω 0 ) → 0. Recalling that P * (Ω 0 ) ≥ 1 -2η, we showed that lim sup n→∞ P * (E n ) ≤ 2η and, therefore, it proves (5.1).

Proof of Proposition 5.1 and of part (i) of Corollary 1.3

The aim of this section is to break down the proof of Proposition 5.1 into the proof of several lemmas. Our goal is to use a second moment argument, but the first moment of W (L) n,βn (F )/D (L) n under Q (L) does not have the right order and the second moment is not even necessarily finite. Thus, we first need to come down to another random variable Y n (F ) that is close to W (L) n,βn (F ) with high probability and that has first and second moments of the right order, by eliminating some rare particles with a too strong weight in the expectations. The lemmas stated in this subsection will also allow us to prove part (i) of Corollary 1.3.

Until the end of the paper, L is a fixed positive constant. We first add a second barrier between times n/2 and n at position (3/2) log n -K, by setting, for K > 0,

W (L,K) n,βn (F ) := |z|=n e -βnV (z) F (V(z))1 V (z)≥-L,min n/2 ≤j≤n V (z j )≥ 3 2 log n-K .
This first lemma shows that W

(L)
n,βn (F ) and W

(L,K)
n,βn (F ) are close with high probability and will be proved in Subsection 5.4. Note that this step is superfluous when log n α n √ n : since the particles contributing to W n,βn are of order α n , it is approximately the same difficulty for them to stay above the two barriers than only above the first (see Proposition 2.8) and, thus,

W (L)
n,βn (F ) and W (L,K) n,βn (F ) have asymptotically the same first moment in that case. Lemma 5.2. For all L > 0 and ε, η > 0, it exists K > 0 such that, for each

F ∈ C b (D([0, 1])), lim sup n→∞ Q (L)   n 3βn/2 α 2 n W (L) n,βn (F ) -W (L,K) n,βn (F ) D (L) n > ε F   ≤ η.
We now consider a fixed K > 0. We will see in Lemma 5.4 that W (L,K) n,βn (F ) has the right first moment, but we still need to remove other particles for the second moment. Let (α + n ) n∈N and (α - n ) n∈N be sequences of positive real numbers and (k n ) n∈N be a sequence of integers such that 1 α -

n α n α + n √ n, (log n) 6 k n √ n,
when n → ∞. We add some controls on the trajectory of the particle's lineage, by considering

Y n (F ) := |z|=n e -βnV (z) F (V(z))1 z∈An ,
where we set A n := {|z| = n : ∀ j ∈ 0, n , V (z j ) ∈ I n,j } (see Figure 1), with

I n,j :=          [-L, ∞) if 0 ≤ j < n/2 and j = k n , [k 1/3 n , k n ] if j = k n , [(3/2) log n, ∞) if n/2 ≤ j ≤ n, [(3/2) log n + α - n , (3/2) log n + α + n ] if j = n.
Note that the second barrier is here simply at (3/2) log n: indeed, Lemma 5.4 shows that it does not change the first moment (and we could even have taken (3/2) log n + C with any C > 0).

But, in order to compute the first moment of Y n (F ), we will first need to take the conditional expectation given F kn and, thus, we want to show that F (V(z)) does not asymptotically depend on what happen before time k n . For this, we consider for each n ∈ N * a slight modification F n of function F such that More formally, we set, for each

F n (V(z)) := F V (z (n-kn)t +kn ) -V (z kn ) σ √ n -k n , t ∈ [0, 1] . V 0 k n n/2 n 3 2 log n -L k 1/3 n k n 3 2 log n + α + n 3 2 log n + α - n
x ∈ D([0, 1]), F n (x) := F ([ √ n/ √ n -k n ]•[x ((n-kn)t+kn)/n -x kn/n ], t ∈ [0, 1]
). The following lemma, proved in Subsection 5.5, shows that we can replace F with F n . It does not play a crucial role, but makes the calculations easier for the next lemmas. )), we have

E Q (L)   W (L,K) n,βn (|F -F n |) D (L) n   = o α 2 n n 3βn/2 ,
as n → ∞.

Noting that, for F nonnegative, Y n (F ) ≤ W

lim sup n→∞ n 3βn/2 α 2 n E Q (L)   W (L,K) n,βn (F ) D (L) n   ≤ θ σ 2 E[F (e)] ≤ lim inf n→∞ n 3βn/2 α 2 n E Q (L) Y n (F n ) D (L) n .
However, this is still not sufficient for controlling the second moment of Y n (F ) and we need to introduce a new random variable Y n (F ). We consider the sequences

a (n) j := -L if 0 ≤ j < n/2 , 3 2 log n if n/2 ≤ j ≤ n,

and

(n)

j := j 1/7 if 0 ≤ j < n/2 , (n -j) 1/7 if n/2 ≤ j ≤ n.
Then, for some fixed sequence (ρ n ) n∈N that tends to infinity such that ρ n α 2 n , we define the following set, that will allow us to control the offspring of the spine in the second moment calculation (see Aïdékon [START_REF] Aïdékon | Convergence in law of the minimum of a branching random walk[END_REF] for the first use of this method),

B n := |z| = n : ∀ j ∈ 0, n -1 , y∈Ω(z j+1 ) e -[V (y)-a (n) j ] ≤ ρ n e -(n) j ,
where Ω(x) is the set of brothers of x, and we set

Y n (F ) := |z|=n e -βnV (z) F n (V(z))1 z∈An∩Bn .
The following lemma shows that this new random variable Y n (F ) is close to Y n (F ). Its proof relies on the peeling lemma stated in Subsection 5.6 in a more general feature.

Lemma 5.5. For all L, K > 0 and F ∈ C b (D([0, 1])), we have

E Q (L) |Y n (F n ) -Y n (F n )| D (L) n = o α 2 n n 3βn/2 ,
as n → ∞.

By considering Y n (F n ), we can now control the second moment properly, as stated in this last lemma, proved in Subsection 5.7. This second moment is exactly the square of the first moment in Lemma 5.4. Lemma 5.6. For all L, K > 0 and F ∈ C b (D([0, 1])) nonnegative, we have

lim sup n→∞ n 3βn/2 α 2 n 2 E Q (L)   Y n (F n ) D (L) n 2   ≤ θ σ 2 E[F (e)] 2 .
Using these lemmas, we can now prove Proposition 5.1.

Proof of Proposition 5.1. We consider here the case F ≥ 0 and the general case follows by taking the positive and negative parts of F . We first fix K > 0. Using Lemmas 5.4 et 5.5, we have

lim inf n→∞ n 3βn/2 α 2 n E Q (L) Y n (F n ) D (L) n ≥ θ σ 2 E[F (e)], (5.2)
and so, by applying Bienaymé-Chebyshev inequality and Lemma 5.6, we get that

n 3βn/2 α 2 n Y n (F n ) D (L) n ---→ n→∞ θ σ 2 E[F (e)]
, in Q (L) -probability.

(5.3) Combining (5.3) with Lemmas 5.4, 5.3 and 5.5, we deduce that for all K > 0,

n 3βn/2 α 2 n W (L,K) n,βn (F ) D (L) n ---→ n→∞ θ σ 2 E[F (e)], in Q (L) -probability. ( 5.4)
Now, for ε, η > 0, by Lemma 5.2, we can choose K > 0 such that lim sup n→∞

Q (L)   n 3βn/2 α 2 n W (L) n,βn (F ) -W (L,K) n,βn (F ) D (L) n > ε 2   ≤ η (5.5)
and, thus, we get, using the triangle inequality, (5.4) and (5.5),

lim sup n→∞ Q (L)   n 3βn/2 α 2 n W (L) n,βn (F ) D (L) n - θ σ 2 E[F (e)] > ε   ≤ η.
This concludes the proof of Proposition 5.1.

We conclude this subsection by proving part (i) of Corollary 1.3, which is a consequence of the fact that considering only particles in A n does not change the first moment asymptotic. Indeed, if we assume that part (i) of Corollary 1.3 is false, then it exists ε > 0 such that for any k ≥ 1, it exists n k > n k-1 (with n 0 := 0) such that

Proof of part (i) of

P * ν n k ,βn k (3/2) log n k + k -1 α n k , (3/2) log n k + kα n k ≤ 1 -ε ≥ 1 -ε. (5.7)
Setting e n := inf{k ∈ N : n k ≥ n}, we have e n → ∞ and e n k = k. Thus, with α + n := e n α n and α - n := e -1 n α n , (5.7) implies the negation of (5.6). Therefore, we now want to prove (5.6). The left-hand side of (5.6) is larger than Y n (1)/W n,βn , therefore, it is sufficient to show that Y n (1)/W n,βn → 1 in P * -probability. Combining (5.3) with Lemma 5.5, we first have

n 3βn/2 α 2 n Y n (1) D (L) n ---→ n→∞ θ σ 2 , in Q (L) -probability, (5.8)
and, in the same way as in the proof of part (i) of Theorems 1.1 and 1.2, where we showed (5.1) from Proposition 5.1, it follows from (5.8) that

n 3βn/2 α 2 n Y n (1) D n ---→ n→∞ c 0 θ σ 2
, in P * -probability.

(5.9)

Using (5.9) and (5.1) with F ≡ 1, we get that Y n (1)/W n,βn → 1 in P * -probability and so (5.6) is proved.

First moments of W (L,K)

n,βn (F ) and Y n (F n )

We start with the proof of Lemma 5.4 in this subsection, because this first moment calculation will be a kind of routine at which we will refer for the proof of other lemmas. In this calculations, sums of general term (i + C)e -i/γn appear regularly, with C ∈ R a constant and γ n → ∞ as n → ∞.

Q (L) [W (L,K) n,βn (F )/D (L) n ]. Note that E Q (L) [W (L,K) n,βn (F )/D (L) n ] = E[W (L,K)
n,βn (F )]/R(L). Then, using the many-to-one lemma, we get (F )]. This time, we cut the segment [0, α + n + K] in pieces of length h > 0, where h is any real number if S 1 is nonlattice and is the span of the lattice if S 1 is lattice, and thus we get

E W (L,K) n,βn (F ) = E e -Sn/αn F (S (n) )1 S n ≥-L,min n/2 ≤j≤n V (z j )≥ 3 2 log n-K . We cut
+ n + K, ∞) in pieces of length 1, E e -Sn/αn F (S (n) )1 S n ≥-L,min n/2 ≤j≤n S j ≥ 3 2 log n-K,Sn> 3 2 log n+α + n ≤ i≥ α + n +K E e -( 3 2 log n-K+i)/αn F 1 S n ≥-L,min n/2 ≤j≤n S j ≥ 3 2 log n-K,Sn-( 3 2 log n-K)∈[i,i+1) ≤ F e K/
E e -Sn/αn F (S (n) )1 S n ≥-L,min n/2 ≤j≤n S j ≥ 3 2 log n-K,Sn≤ 3 2 log n+α + n ≤ (α + n +K)/h -1 i=0 e (K-ih)/αn n 3/2αn E F (S (n) )1 S n ≥-L,min n/2 ≤j≤n S j ≥ 3 2 log n-K,Sn-( 3 2 log n-K)∈[ih,(i+1)h) ≤ (α + n +K)/h -1 i=0 e -ih/αn n 3/2αn π 2 θθ - σ R(L) n 3/2 (E[F (e)] + o(1))hR -((i + 1)h), (5.12)
by using Proposition 2.8 in both lattice and nonlattice cases, with uniformity in i because h( (α

+ n + K)/h -1)
√ n. Then, by applying (2.1) to R -, for ε > 0, it exists M > 0 such that, for all u ≥ 0, R -(u) ≤ c - 0 (1 + ε)(M + u) and thus we get

(α + n +K)/h -1 i=0 R -((i + 1)h)e -ih/αn ≤ ∞ i=0 c - 0 (1 + ε) i + 1 + M h he -ih/αn = (1 + o(1))c - 0 (1 + ε)h α n h 2 , ( 5.13)
by applying (5.10). Coming back to (5.12), we get lim sup

n→∞ n 3βn/2 α 2 n E e -Sn/αn F (S (n) )1 S n ≥-L,min n/2 ≤j≤n S j ≥ 3 2 log n-K,Sn≤ 3 2 log n+α + n ≤ π 2 θθ - σ R(L)E[F (e)]h c - 0 (1 + ε) h ---→ ε→0 θ σ 2 R(L)E[F (e)], (5.14)
by applying (2.5) to constants c - 0 and θ -. Combining (5.11) and (5.14), we conclude that

E Q (L)   W (L,K) n,βn D (L) n   = 1 R(L) E W (L,K) n,βn ≤ θ σ 2 α 2 n n 3βn/2 (E[F (e)] + o(1)),
and it shows the first part of Lemma 5.4. We now want to prove the lower bound for

E Q (L) [Y n (F n )/D (L) n ] = E[Y n (F n )]/R(L).
We use the branching property at time k n to get

E[Y n (F n )] = E |x|=kn 1 V (x)≥-L,V (x)∈[k 1/3 n ,kn] ψ(V (x)) , (5.15)
where we set, for all b ∈ [k

1/3 n , k n ], ψ(b) := E b |z|=n-kn e -βnV (z) F V(z) - b σ √ n -k n 1 ∀ 0≤i≤n-kn,V (z i )∈I n,i+kn .
We note m := n -k n and fix λ ∈ (0, 1/2), then we have λm ≤ n/2 -k n for n large enough because k n n. Using the many-to-one lemma, we get, for b ∈ [k

1/3 n , k n ], ψ(b) ≥ e -b E b e -Sm/αn F S (m) - b σ √ m 1 Sm-3 2 log n∈[α - n ,α + n ],S m ≥-L,min λm ≤j≤m S j ≥ 3 2 log n = e -b E e -(Sm+b)/αn F (S (m) )1 Sm-( 3
Then, we cut the segment [α - n , α + n ] in pieces of length h > 0, where h is any real number if S 1 is nonlattice and is the span of the lattice if S 1 is lattice, and we get that ψ(b) is larger than

e -b α + n /h -1 i= α - n /h e -(i+1)h/αn n 3/2αn E F (S (m) )1 Sm-( 3 2 log n-b)∈[ih,(i+1)h),S m ≥-(L+b),min λm ≤j≤m S j ≥ 3 2 log n-b ≥ e -b n 3/2αn α + n /h -1 i= α - n /h e -(i+1)h/αn π 2 θθ - σ R(L + b) m 3/2 (E[F (e)] + o(1))hR -(ih),
where the o( 1)

is uniform in b ∈ [k 1/3 n , k n ] and i ∈ α - n /h , α + n /h -1 , by using Proposition 2.8, because we have L + k n √ n and h( α + n /h -1)
√ n. Thus, we get, using that

R(L + b) ∼ c 0 (L + b) and R -(ih) ∼ c - 0 ih uniformly in b and i, ψ(b) ≥ e -b n 3/2αn π 2 θθ - σ c 0 (L + b) n 3/2 h(E[F (e)] + o(1)) α + n /h -1 i= α - n /h e -(i+1)h/αn c - 0 ih = (L + b)e -b 2 π 1 σ 3 α 2 n n 3βn/2 (E[F (e)] + o(1)), uniformly in b ∈ [k 1/3 n , k n ],
where we used (2.5) twice and also (5.10). Coming back to (5.15), we get that

E[Y n (F n )] is larger than 2 π 1 σ 3 α 2 n n 3βn/2 (E[F (e)] + o(1))E |x|=kn 1 V (x)≥-L,V (x)∈[k 1/3 n ,kn] (L + V (x))e -V (x) .
(5. [START_REF] Caravenna | An invariance principle for random walk bridges conditioned to stay positive[END_REF] Using the many-to-one lemma, the expectation in (5.16) is equal to

E (S kn + L)1 S kn ≥-L,S kn ∈[k 1/3 n ,kn] ≥ E (S kn + L)1 S kn ≥-L,(S kn +L)/σk 1/2 n ∈[C -1 ,C] ,
(5.17) for all C > 0. We then choose a function χ : R + → R continuous and bounded such that, for all

t ∈ R + , t1 t∈[2C -1 ,C/2] ≤ χ(t) ≤ t1 t∈[C -1 ,C] ,
and (5.17) is larger than

σk 1/2 n E χ S kn + L σk 1/2 n 1 S kn ≥-L = σk 1/2 n (1 + o(1)) θR(L) k 1/2 n ∞ 0 χ(t)te -t 2 /2 dt ≥ σθR(L)(1 + o(1)) C/2 2C -1
t 2 e -t 2 /2 dt, by applying (2.6). Coming back to (5.16), we get

lim inf n→∞ n 3βn/2 α 2 n E[Y n (F n )] ≥ 2 π 1 σ 3 E[F (e)]σθR(L) C/2 2C -1 t 2 e -t 2 /2 dt ----→ C→∞ θ σ 2 R(L)E[F (e)], using that ∞ 0 t 2 e -t 2 /2 dt = π 2 . Since E Q (L) [Y n (F n )/D (L) n ] = E[Y n (F n )]/R(L)
, it concludes the proof of Lemma 5.4.

Addition of the second barrier

In this section, we prove Lemma 5.2 with a method similar to the one used by Madaule [START_REF] Madaule | Convergence in law for the branching random walk seen from its tip[END_REF] for his Lemma 4.9 (or Lemma 3.3 of Aïdékon [START_REF] Aïdékon | Convergence in law of the minimum of a branching random walk[END_REF]). The main difference is that we do not only consider particles that are at a distance of order 1 from (3/2) log n, but we can nevertheless apply some of Madaule's results.

Proof of Lemma 5.2. We fix L > 0 and ε, η > 0. For all F ∈ C b (D([0, 1])) and K > 0, we have

|W (L) n,βn (F ) -W (L,K) n,βn (F )| ≤ 2 F |W (L) n,βn -W (L,K) n,βn |, where W (L) n,βn := W (L)
n,βn [START_REF] Aïdékon | Convergence in law of the minimum of a branching random walk[END_REF] and [START_REF] Aïdékon | Convergence in law of the minimum of a branching random walk[END_REF]. Therefore, it is sufficient to show that we have

W (L,K) n,βn := W (L,K) n,βn
lim sup n→∞ Q (L)   n 3βn/2 α 2 n W (L) n,βn -W (L,K) n,βn D (L) n > ε   ≤ η, (5.18)
for K > 0 large enough. Using Proposition A.3 of Aïdékon [START_REF] Aïdékon | Convergence in law of the minimum of a branching random walk[END_REF], we know that

D (L) n converges in L 1 to D (L)
∞ under P, so we can choose M > 0 large enough such that, for all n ∈ N, Q (L) (D

(L) n > M ) = E[D (L) n 1 D (L)
n >M ]/R(L) ≤ η/4. Therefore, the probability in (5.18) is smaller than

Q (L) n 3βn/2 W (L) n,βn -W (L,K) n,βn > εα 2 n D (L) n , η 4 ≤ D (L) n ≤ M + η 4 + Q (L) D (L) n < η 4 = E D (L) n 1 n 3βn/2 |W (L) n,βn -W (L,K) n,βn |>εα 2 n D (L) n , η 4 ≤D (L) n ≤M + η 4 + 1 R(L) E D (L) n 1 D (L) n < η 4 ≤ M P n 3βn/2 W (L) n,βn -W (L,K) n,βn > εα 2 n η 4 + η 4 + η 4 ,
because R(L) ≥ 1. Thus, we now want to prove that, for some K > 0 large enough, we have lim sup

n→∞ P n 3βn/2 W (L) n,βn -W (L,K) n,βn > ε α 2 n ≤ 2η ,
with η := η/4M and ε := εη/4. Moreover, using (1.8), we can fix K ≥ 0 such that P(min |z|=n V (z) < 3 2 log n -K ) ≤ η . Thus, our aim is now to show that we have lim sup

n→∞ P W (L) n,βn -W (L,K) n,βn 1 min |z|=n V (z)≥ 3 2 log n-K > ε α 2 n n 3βn/2 ≤ η , ( 5.19)
for some K ≥ K large enough. Now, following Madaule's [START_REF] Madaule | Convergence in law for the branching random walk seen from its tip[END_REF] proof of his Lemma 4.9, we introduce the intervals

J n (x) := [ 3 2 log n -x -1, 3 2 log n -x)
, for x ∈ R, and the events, for i, ∈ N and n/2

≤ k ≤ n, E i,k, (z) := V (z) ≥ -L, V (z k ) = min n/2 ≤j≤n V (z j ) ∈ J n (K) -, V (z) ∈ J n (K ) + i ,
and, denoting a := e ν( +K) for some fixed ν ∈ (0, 1),

F 1 (z) := i≥1, n/2 ≤k<n-a E i,k, (z) and F 2 (z) := i≥1,n-a ≤k≤n E i,k, (z).
Then, we have

W (L) n,βn -W (L,K) n,βn 1 min |z|=n V (z)≥ 3 2 log n-K ≤ ≥0 |z|=n e -βnV (z) 1 F 1 (z) + 1 F 2 (z) .
(5.20)

On the one hand, by Madaule's [START_REF] Madaule | Convergence in law for the branching random walk seen from its tip[END_REF] proof of his Lemma 4.9, we have the inequality

P |z|=n 1 F 2 (z) ≥ 1 ≤ c 14 (1 + a )(1 + L)e -K-
and, therefore,

P ≥0 |z|=n e -βnV (z) 1 F 2 (z) > 0 ≤ ≥0 P |z|=n 1 F 2 (z) ≥ 1 ≤ c 15 (1 + L)e -(1-ν)K .
(5.21)

On the other hand, by using the many-to-one lemma, we get, for i ≥ 1, n/2 ≤ k < n -a and ≥ 0,

E |z|=n e -βnV (z) 1 E i,k, (z) = E e -Sn/αn 1 E i,k, ≤ e -(i-1-K )/αn n 3/2αn P(E i,k, ), (5.22)
where we set

E i,k, := S n ≥ -L, S k = min n/2 ≤j≤n S j ∈ J n (K) -, S n ∈ J n (K ) + i .
We recall Equation (4.27) of Madaule [START_REF] Madaule | Convergence in law for the branching random walk seen from its tip[END_REF]:

P(E i,k, ) ≤    c 16 (1+L) log n n 3/2 (n-k+1) 3/2 (1 + + i) if n/2 ≤ k < 3n/4 , c 16 (1+L) n 3/2 (n-k+1) 3/2 (1 + + i) if 3n/4 ≤ k ≤ n.
P(E i,k,l ) ≤ c 17 (1 + L) n 3/2 (1 + + i) log n √ n + a -1/2 .
(5.24) Using (5.22), (5.24) and that P(E i,k, ) = 0 for ≥ 3 2 log n -K + L, we have

E ≥0 |z|=n e -βnV (z) 1 F 1 (z) ≤ ≥0 i≥1 n-a -1 k= n/2 E |z|=n e -βnV (z) 1 E i,k, (z) ≤ c 17 e (K +1)/αn (1 + L) n 3βn/2 3 2 log n-K+L =0 log n √ n + a -1/2 i≥1 e -i/αn (1 + + i). (5.25)
We can bound the sum on i in (5.25) by (1 + )α 2 n (1 + o( 1)) uniformly in . Moreover, taking K large enough such that e -νK < 1/2, we have a ≥ e ν( +K) /2 for all ≥ 0. Thus, we get that (5.25) is smaller than

c 18 (1 + o(1)) (1 + L) n 3βn/2 α 2 n (log n) 3 n 1/2 + e -νK/2 = c 18 e -νK/2 (1 + L)(1 + o(1)) α 2 n n 3βn/2 ,
and, thus, with the Markov inequality, we have

P ≥0 |z|=n e -βnV (z) 1 F 1 (z) > ε α 2 n n 3βn/2 ≤ c 18 ε e -νK/2 (1 + L)(1 + o(1)).
(5.26)

Finally, (5.19) follows from (5.20), (5.21) and (5.26) by taking K large enough and it concludes the proof of Lemma 5.2.

From F to F n

We prove here that considering F n instead of F does not change significantly the first moment.

Proof of Lemma 5.3. To control the first moment of W n , but, instead of applying directly Proposition 2.8 as in (5.12), we use that

(L,K) n,βn (|F -F n |)/D (L) n under E Q (L) ,
E |F -F n |(S (n) )1 S n ≥-L,min n/2 ≤j≤n S j ≥ 3 2 log n-K,Sn-( 3 2 log n-K)∈[ih,(i+1)h) = o π 2 θθ - σ R(L) n 3/2 hR -((i + 1)h) , uniformly in i ∈ [0, h( (α + n + K)/h -1)],

The peeling lemma

The aim of this subsection is to prove Lemma 5.5, which shows that introducing the event {z ∈ B n } does not change the first moment. This proof is based on the so-called peeling lemma (see Shi [START_REF] Shi | Branching random walks[END_REF]), which controls that the spine, conditioned to have a specific trajectory, does not have too many and too low children. Such lemmas have been proved in the case where the spine ends up at a distance of constant order from 3 2 log n (see [START_REF] Aïdékon | Convergence in law of the minimum of a branching random walk[END_REF]Lemma C.1], [START_REF] Madaule | Convergence in law for the branching random walk seen from its tip[END_REF]Lemma 7.1] and [START_REF] Shi | Branching random walks[END_REF]Theorem 5.14]) and also when it ends up at a position of order √ n (see [START_REF] Aïdékon | The Seneta-Heyde scaling for the branching random walk[END_REF]Lemma 4.7]). Here we have to deal with the intermediate case where the spine end up far above 3 2 log n and far below √ n. In order to state the peeling lemma in a general setting, we introduce some notation. For b, u, v ∈ R and n ∈ N, we set

A b,u,v n := |z| = n : V (z) ∈ [v + b, v + b + 1), V (z) ≥ -u, min n/2 ≤j≤n V (z j ) ≥ v .
We consider

a (n) i := -u if 0 ≤ i < n/2 , v if n/2 ≤ i ≤ n, and (n) i :
= i 1/7 if 0 ≤ i < n/2 , (n -i) 1/7 if n/2 ≤ i ≤ n,
and the following set

B ρ n := |z| = n : ∀ j ∈ 0, n -1 , y∈Ω(z j+1 ) 1 + [V (y) -a (n) j ] + e -[V (y)-a (n) j ] ≤ ρe -(n) j .
We can now state our version of the peeling lemma, which covers the case where the spine ends up far below √ n and is therefore more general than the peeling lemmas in [START_REF] Aïdékon | Convergence in law of the minimum of a branching random walk[END_REF][START_REF] Madaule | Convergence in law for the branching random walk seen from its tip[END_REF][START_REF] Shi | Branching random walks[END_REF].

Lemma 5.7 (Peeling lemma). For all ε > 0, there exist ρ > 0 and n 0 ∈ N such that, for all

n ≥ n 0 , b ∈ R + and u, v ∈ [0, n 1/8 ], Q w n ∈ A b,u,v n ∩ (B ρ n ) c ≤ ε R(u)R -(b) n 3/2 .
Remark 5.8. We present here the peeling lemma in terms of probability measure Q, because it simplifies somehow the proof (for example, under Q, the reproduction law along the spine does not depend on the position), but it is a direct consequence that, for all ε > 0, there exist ρ > 0 and n 0 ∈ N such that, for all n ≥ n 0 , b ∈ R + and u, v ∈ [0, n 1/8 ],

Q (u) w (u) n ∈ A b,u,v n ∩ (B ρ n ) c ≤ ε R(u + v + b)R -(b) n 3/2 .
Before proving the peeling lemma, we first use it to show Lemma 5.5.

Proof of Lemma 5.5. We set Y n := Y n (1) and Y n := Y n (1) and, since F is bounded, it is sufficient to show that

E Q (L) Y n -Y n D (L) n = o α 2 n n 3βn/2
We first change probabilities from Q (L) to Q: we have

E Q (L) Y n -Y n D (L) n = 1 R(L) E Q Y n -Y n W n = 1 R(L) E Q e -V (wn)/αn 1 wn∈An∩B c n .
(5.27)

Then, setting u = L and v = 3 2 log n and cutting the segment [α - n , α + n ] in pieces of length 1, we get that (5.27)

is smaller than 1 R(L) α + n -1 i= α - n e -i/αn Q w n ∈ A i,u,v n ∩ B c n ≤ o 1 n 3/2 α + n -1 i= α - n e -i/αn R -(i), (5.28)
using Lemma 5.7 uniformly in i, noting that B n ⊂ B ρn n and ρ n → ∞. Using then (2.2) and (5.10), we get that (5.28) is a o(α 2 n /n 3/2 ) and it concludes the proof of Lemma 5.5.

Proof of Lemma 5.7. By Lemma 2.4 and (2.2), it exists µ > 0 such that, for all b, u ∈ R + , v ∈ R and n ∈ N * , we have

Q w n ∈ A b,u,v n , ∃ i ∈ 0, n : V (w i ) < a (n) i + 2 (n) i -µ ≤ ε 2 R(u)R -(b) n 3/2 .
Thus, it is sufficient to show that

Q w n ∈ A b,u,v n ∩ B c n , ∀ j ∈ 0, n , V (w j ) ≥ a (n) j + 2 (n) j -µ ≤ ε 2 R(u)R -(b) n 3/2 , ( 5.29)
for n large enough, b ∈ R + and u, v ∈ [0, n 1/8 ]. Therefore, we now prove (5.29). We first set, for 0

≤ i ≤ n -1, B ρ n,i := |z| = n : y∈Ω(z i+1 ) 1 + [V (y) -a (n) i ] + e -[V (y)-a (n) i ] ≤ ρe -(n) i . Since for all u, v ∈ R, 1 + (u + v) + ≤ (1 + u + )(1 + v + ), we have y∈Ω(z i+1 ) 1 + [V (y) -a (n) i ] + e -[V (y)-a (n) i ] ≤ 1 + [V (w i ) -a (n) i ] + e -[V (w i )-a (n) i ] Θ(w i+1 ),
where we set, for x ∈ T, by noting ←x the parent of x and ∆(x

) := V (x) -V ( ← - x ), Θ(x) := y∈Ω(x)
(1 + ∆(y) + )e -∆(y) .

Thus, we have, on event

{w n / ∈ B ρ n,i } ∩ {V (w i ) ≥ a (n) i + 2 (n) i -µ}, 1 + [V (w i ) -a (n) i ] + e -[V (w i )-a (n) i ] Θ(w i+1 ) ≥ ρe -(n) i ≥ ρe -[V (w i )-a (n) i +µ]/2
and, if we are moreover on event

{w n ∈ A b,u,v n } so that V (w i ) ≥ a (n) i , it implies that Θ(w i+1 ) ≥ ρe -µ/2 e [V (w i )-a (n) i ]/2 1 + [V (w i ) -a (n) i ] + ≥ c 19 ρe -µ/2 e [V (w i )-a (n) i ]/3 ,
where c 19 := inf u≥0 e u/6 /(1 + u) > 0. Therefore, we get

Q w n ∈ A b,u,v n ∩ (B ρ n ) c , ∀ j ∈ 0, n , V (w j ) ≥ a (n) j + 2 (n) j -µ ≤ n-1 i=0 q i , (5.30)
where we set, for 0 ≤ i ≤ n -1,

q i := Q w n ∈ A b,u,v n , ∀ j ∈ 0, n , V (w j ) ≥ a (n) j + 2 (n) j -µ, Θ(w i+1 ) > ρe [V (w i )-a (n) i ]/3
and ρ := c 19 ρe -µ/2 . From now, we choose ρ such that ρ ≥ e. We first consider the case 0 ≤ i < n/2 . Since a

(n) i = -u, we have q i ≤ E Q 1 V (w i )≥-u,Θ(w i+1 )>ρe [V (w i )+u]/3 G i (V (w i+1 )) ,
where we set, for x ∈ R,

G i (x) := Q x V (w n-i-1 ) ≥ -u, V (w n-i-1 ) -v ∈ [b, b + 1), min n/2 -i-1≤j≤n-i-1 V (w j ) ≥ v ≤ P S n-i-1 ≥ -u -x, S n-i-1 -(v -x) ∈ [b, b + 1), min (n-i-1)/2 ≤j≤n-i-1 S j ≥ v -x ≤ c 5 (1 + x + u)2(b + 2) (n -i) 3/2 ≤ c 20 (1 + x + u)R -(b) n 3/2 ,
using successively Proposition 2.2 (ii), (2.8) and (2.2). We thus get

q i ≤ c 20 R -(b) n 3/2 E Q 1 V (w i )≥-u,Θ(w i+1 )>ρe [V (w i )+u]/3 (1 + V (w i+1 ) + u) ≤ c 20 R -(b) n 3/2 E Q 1 V (w i )≥-u,V (w i )+u<3 log Θ(w i+1 ) ρ (1 + V (w i ) + u + ∆(w i+1 )) ≤ c 20 R -(b) n 3/2 E Q 1 V (w i )≥-u,V (w i )+u<3 log Θ(w i+1 ) ρ 1 + 3 log + Θ(w i+1 ) ρ + (∆(w i+1 )) + .
But, under Q, (Θ(w i+1 ), ∆(w i+1 )) is independent of (V (w j )) 0≤j≤i and has moreover the same law as (X + X, V (w 1 )), where X and X are defined in (1.4). Therefore, we get, by integrating first on (V (w j )) 0≤j≤i ,

q i ≤ c 20 R -(b) n 3/2 E Q F i (X + X) 1 + 3 log + X + X ρ + V (w 1 ) + ] , (5.31)
where we set, for x > 0,

F i (x) := Q V (w i ) ≥ -u, V (w i ) + u < 3 log x ρ = 1 0<log x ρ P S i ≥ -u, S i < -u + 3 log + x ,
by using Proposition 2.2 (ii) and that ρ ≥ 1. Then, applying (2.9), we have

n/2 -1 i=kn F i (x) ≤ 1 x>ρ c 6 1 + 3 log + x (1 + u) ≤ 3 c 6 c 1 R(u)(1 + log + x)1 x>ρ
using also (2.2). Coming back to (5.31) and noting that, since X + X > ρ ≥ e, we have log + (X + X) ≥ 1, this gives

n/2 -1 i=kn q i ≤ c 21 R(u)R -(b) n 3/2 E Q 1 X+ X>ρ log + (X + X) log + (X + X) + V (w 1 ) + ] (5.32)
and concludes the case 0 ≤ i < n/2 . We now consider the case n/2 ≤ i < n, so a

(n) i = v. On event {w n ∈ A b,u,v n , Θ(w i+1 ) > ρe [V (w i )-v]/3 }, we have V (w i ) = V (w n ) -(V (w n ) -V (w i )) ∈ [v + b -(V (w n ) -V (w i )), v + b + 1 - (V (w n ) -V (w i ))], Θ(w i+1 ) > ρ ∨ ρe [V (w i )-V (wn)+b]/3 and, for all i + 1 ≤ j ≤ n, V (w n ) -V (w j ) ≤ b + 1. Moreover, we set ã(n) j := -u if 0 ≤ j < n/4 , v if n/4 ≤ j ≤ n,
and note that, on event

{w n ∈ A b,u,v n , ∀ j ∈ 0, n , V (w j ) ≥ a (n) j + 2 (n) j -µ}, for n large enough, we have ∀ j ∈ n/4 , n/2 -1 , V (w j ) ≥ v = ã(n) j , because (n)
j ≥ n/4 1/7 and u + v ≤ 2n 1/8 . Thus, we get

q i ≤ Q ∀ j ∈ 1, i , V (w j ) ≥ ã(n) j , V (w i ) -(v -(V (w n ) -V (w i ))) ∈ [b, b + 1] Θ(w i+1 ) > ρ ∨ ρe [V (w i )-V (wn)+b]/3 , ∀ j ∈ i + 1, n , V (w n ) -V (w j ) ≤ b + 1 .
(5.33)

Under Q, (Θ(w i+1 ), (V (w n ) -V (w j )) i≤j≤n ) is independent of (V (w j )) 0≤j≤i , so the right-hand side of (5.33) is equal to

E Q H i (V (w n ) -V (w i ))1 Θ(w i+1 )>ρ∨ρe [V (w i )-V (wn)+b]/3 ,∀ j∈ i+1,n ,V (wn)-V (w j )≤b+1 , (5.34)
where we set, for x ∈ R,

H i (x) := Q V (w i ) ≥ -u, min n/4 ≤j≤i V (w j ) ≥ v, V (w i ) -(v -x) ∈ [b, b + 1] ≤ P S i ≥ -u, min i/2 ≤j≤i S j ≥ v, S i -v ∈ [b -x, b -x + 1] ≤ c 22 R(u) n 3/2 (1 + b -x),
using successively Proposition 2.2 (ii), (2.8) and (2.2) as before. On the event {Θ(w i+1 ) > ρe [V (w i )-V (wn)+b]/3 }, using that ρ ≥ 1, we have b -

(V (w n ) -V (w i )) ≤ 3 log + Θ(w i+1 ) and also V (w n ) -V (w i+1 ) ≥ b -3 log + Θ(w i+1 ) -∆(w i+1 ) + , therefore, (5.34) is smaller than c 22 R(u) n 3/2 E Q 1 + 3 log + Θ(w i+1 ) 1 Θ(w i+1 )>ρ × 1 V (wn)-V (w i+1 )≥b-3 log + Θ(w i+1 )-∆(w i+1 ) + ,∀ j∈ i+1,n ,V (wn)-V (w j )≤b+1 .
(

Note then that, under Q, (Θ(w i+1 ), ∆(w i+1 )) is independent of (V (w n ) -V (w j )) i+1≤j≤n and has the same law as (X + X, V (w 1 )), so (5.35) is equal to

c 22 R(u) n 3/2 E Q 1 + 3 log + (X + X) 1 X+ X>ρ Γ i 3 log + (X + X) + V (w 1 ) + , (5.36)
where we set, for x ≥ 0,

Γ i (x) := Q(V (w n ) -V (w i+1 ) ≥ b -x, ∀ j ∈ i + 1, n , V (w n ) -V (w j ) ≤ b + 1) = P S - n-i-1 ≤ (x + 1) -(b + 1), S - n-i-1 ≥ -(b + 1) ,
by applying Proposition 2.2 (ii) and then reversing time. Thus, using (2.9) and (2.2), we get

n-1 i= n/2 Γ i (x) ≤ c - 6 (1 + x + 1)(1 + b + 1) ≤ 4 c - 6 c - 1 (1 + x)R -(b).
On event {X + X > ρ}, we have log + (X + X) ≥ 1 so, coming back to (5.36), we get

n-1 i= n/2 q i ≤ c 23 R(u)R -(b) n 3/2 E Q log + (X + X)1 X+ X>ρ log + (X + X) + V (w 1 ) + .
(5.37)

Finally, using (5.32), (5.37) and that

E Q [V (w 1 ) + | F 1 ] = X/X by Proposition 2.2 (i), we get n-1 i=0 q i ≤ (c 21 + c 23 ) R(u)R -(b) n 3/2 E X log 2 + (X + X) + X log + (X + X) 1 X+ X>ρ .
(

Using (1.3), we can choose ρ large enough such that the expectation in (5.38) is smaller than ε/2(c 21 + c 23 ) and, recalling (5.30), it proves (5.29) and concludes the proof of Lemma 5.7.

Second moment of Y n (F n )

For F ∈ C(D([0, 1])), by decomposing along the spine, Y n (F ) is equal to

n-1 i=0 y∈Ω(w (L) i+1 ) |z|=n,z≥y e -βnV (z) F (V(z))1 z∈An∩Bn + e -βnV (w (L) n ) F (V(w (L) n ))1 w (L) n ∈An∩Bn .
We cut this sum in two pieces, depending on whether the lineage of the considered particle z splits off from the spine's lineage before or after time k n :

Y [0,kn) n (F ) := kn-1 i=0 y∈Ω(w (L) i+1 ) |z|=n,z≥y e -βnV (z) F (V(z))1 z∈An∩Bn Y [kn,n] n (F ) := Y n (F ) -Y [0,kn) n (F ).
We define in the same way D

Q (L) D (L),[kn,n] n ≤ n -2 V (w (L) kn ) = u ---→ n→∞ 1.
(5.39)

Proof of Lemma 5.6. First note that, using Proposition 2.3 (i),

Y n (F n ) D (L) n = |z|=n Q (L) w (L) n = z F n e -V (z)/αn R L (V (z)) F n (V(z))1 z∈An∩Bn = E Q (L)   e -V (w (L) n )/αn R L (V (w (L) n )) F n (V(w (L) n ))1 w (L) n ∈An∩Bn F n  
and, thus, we have

E Q (L)   Y n (F n ) D (L) n 2   = E Q (L)   Y n (F n ) D (L) n e -V (w (L) n )/αn R L (V (w (L) n )) F n (V(w (L) n ))1 w (L) n ∈An∩Bn   =: E [0,kn) Q (L) + E [kn,n] Q (L) , by splitting Y n (F n ) = Y [0,kn) n (F n ) + Y [kn,n] n (F n ).
The first part will give the right order and constant and the second part will be negligible. Recall that F is assumed to be nonnegative.

We begin by bounding E [0,kn)

Q (L) . Using D (L) n ≥ D (L),[0,kn) n and 1 w (L) n ∈Bn ≤ 1, we get E [0,kn) Q (L) ≤ E Q (L)   Y [0,kn) n (F n ) D (L),[0,kn) n e -V (w (L) n )/αn R L (V (w (L) n )) F n (V(w (L) n ))1 w (L) n ∈An   = E Q (L) Y [0,kn) n (F n ) D (L),[0,kn) n ϕ V (w (L) kn ) 1 V (w (L) kn )∈[k 1/3 n ,kn] ,
(

where we set, for b ∈ [k

1/3 n , k n ] and with m := n -k n , ϕ(b) := E Q (L) b   e -V (w (L) m )/αn R L (V (w (L) m )) F V(w (L) m ) - b σ √ m 1 ∀ 0≤i≤n-kn,V (w (L) i )∈I n,i+kn   ≤ 1 R L (b) E e -(Sm+b)/αn F (S (m) )1 S m ≥-(L+b),min λm ≤j≤m S j ≥ 3 2 log n-b,Sm-( 3 2 log n-b)∈[α - n ,α + n ] ,
for some fixed λ ∈ (1/2, 1) and n large enough, by applying Proposition 2.3 (ii). Then, proceeding in the same way as for the lower bound of ψ(b) in the proof of Lemma 5.4 (but with a sum on i from α - n /h to α + n /h -1), we get the upper bound

ϕ(b) ≤ 1 R L (b) (L + b) 2 π 1 σ 3 α 2 n n 3βn/2 (E[F (e)] + o(1)) = θ σ 2 α 2 n n 3βn/2 (E[F (e)] + o(1)), ( 5
Q (L) ≤ θ σ 2 α 2 n n 3βn/2 (E[F (e)] + o(1))E Q (L) Y [0,kn) n (F n ) D (L),[0,kn) n 1 V (w (L) kn )∈[k 1/3 n ,kn] ,
(5.42) and thus we now want to bound the expectation in (5.42). We proceed in a way similar to the proof of Lemma 4.5 of Aïdekon et Shi [START_REF] Aïdékon | The Seneta-Heyde scaling for the branching random walk[END_REF], by introducing the event {D

(L),[kn,n] n ≤ n -2 }: E Q (L) Y [0,kn) n (F n ) D (L),[0,kn) n 1 D (L),[kn,n] n ≤n -2 = E Q (L) Y [0,kn) n (F n ) D (L),[0,kn) n Q (L) D (L),[kn,n] n ≤ n -2 V (w (L) kn ) = u ≥ E Q (L) Y [0,kn) n (F n ) D (L),[0,kn) n 1 V (w (L) kn )∈[k 1/3 n ,kn] inf u∈[k 1/3 n ,kn] Q (L) D (L),[kn,n] n ≤ n -2 V (w (L) kn ) = u .
Thus, applying (5.39), we get that the expectation in (5.42) is smaller than

(1 + o(1))E Q (L) Y [0,kn) n (F n ) D (L),[0,kn) n 1 D (L),[kn,n] n ≤n -2 ≤ (1 + o(1)) E Q (L) Y [0,kn) n (F n ) D (L),[0,kn) n 1 D (L) n ≥n -7/4 ,D (L),[kn,n] n ≤n -2 + F Q (L) D (L) n < n -7/4 ≤ (1 + o(1)) E Q (L) Y [0,kn) n (F n ) (1 -n -1/4 )D (L) n + F n -7/4 E Q (L) 1 D (L) n , ( 5
Q (L) ≤ (E[F (e)] + o(1)) θ σ 2 α 2 n n 3βn/2 2 .
(5.44)

We now want to show that E

[kn,n]

Q (L) = o((α 2 n /n 3βn/2 ) 2 )
Q (L) is smaller than n-1 i=kn E Q (L) 1 D (L),[0,kn) n e -V (w (L) n )/αn R L (V (w (L) n )) 1 w (L) n ∈An∩Bn y∈Ω(w (L) i+1 ) |z|=n,z≥y e -βnV (z) 1 z∈An + E Q (L) 1
D (L),[0,kn) n e -V (w (L) n )/αn R L (V (w (L) n )) 1 w (L)
n ∈An∩Bn e -βnV (w (L) n ) .

(5.45)

Noting that, on the event {z ∈ A n }, we have e -βnV (z) ≤ e -V (z) n -3/2αn , the first term in (5.45) is smaller than

n-1 i=kn E Q (L) 1
D (L),[0,kn) n e -V (w (L) n )/αn R L (V (w (L) n )) 1 w (L) n ∈An∩Bn y∈Ω(w (L) i+1 ) |z|=n,z≥y e -V (z) n -3/2αn = n -3/2αn n-1 i=kn E Q (L) 1 D (L),[0,kn) n e -V (w (L) n )/αn R L (V (w (L) n )) 1 w (L) n ∈An∩Bn y∈Ω(w (L) i+1 )
e -V (y) , (5.46) by conditioning with respect to G ∞ := σ(V (w

(L) i ), V (y), y ∈ Ω(w (L) i+1 ), i ∈ N) and noting that, given G ∞ , D (L),[0,kn) n is independent of (V (z), |z| = n, z ≥ y, y ∈ Ω(w (L) i+1 ))
. Noting that we are on the event {w

(L) n ∈ B n }, (5.46) is smaller than n -3/2αn n-1 i=kn E Q (L)   1 D (L),[0,kn) n e -V (w (L) n )/αn R L (V (w (L) n )) 1 w (L) n ∈An∩Bn ρ n e -a (n) i - (n) i   ≤ ρ n n 3/2αn   n/2 -1 i=kn e L-i 1/7 + n-1 i= n/2 e -(n-i) 1/7 n 3/2   E Q (L)   1 D (L),[0,kn) n e -V (w (L) n )/αn R L (V (w (L) n )) 1 w (L) n ∈An   ≤ c 24 e L ρ n n 3βn/2 (1 + o(1))E Q (L)   1 D (L),[0,kn) n e -V (w (L) n )/αn R L (V (w (L) n )) 1 w (L) n ∈An   , (5.47) because k 1/7 n
log n and therefore the sum for i ∈ k n , n/2 -1 is a o(n -3/2 ). For the second term in (5.45), we use that, on the event {w

(L) n ∈ A n }, e -βnV (w (L) n ) ≤ n -3βn/2
, and, thus, combining with (5.47), we get

E [kn,n] Q (L) ≤ c 24 e L ρ n n 3βn/2 (1 + o(1))E Q (L)   1 D (L),[0,kn) n e -V (w (L) n )/αn R L (V (w (L) n )) 1 w (L) n ∈An   = c 24 e L ρ n n 3βn/2 (1 + o(1))E Q (L) 1 D (L),[0,kn) n ϕ V (w (L) kn ) 1 V (w (L) kn )∈[k 1/3 n ,kn] ,
where the function ϕ has been defined previously in the proof. Using (5.41) again, we get

E [kn,n] Q (L) ≤ c 25 e L ρ n n 3βn/2 α 2 n n 3βn/2 (1 + o(1))E Q (L) 1 D (L),[0,kn) n 1 V (w (L) kn )∈[k 1/3 n ,kn] .
(5.48)

Proceeding in the same way as before by using (5.39) to introduce the event {D (L),[kn,n] n

≤ n -2 }, the expectation in (5.48) is smaller than

(1 + o(1))E Q (L) 1
D (L),[0,kn) n 1 D (L),[kn,n] n ≤n -2 ≤ (1 + o(1)) E Q (L) 1
D (L),[0,kn) n 1 D (L) n ≥n -7/4 ,D (L),[kn,n] n ≤n -2 + Q (L) D (L) n < n -7/
Q (L) ≤ c 26 e L R(L) ρ n n 3βn/2 α 2 n n 3βn/2 (1 + o(1)) = o   α 2 n n 3βn/2 2   , because ρ n α 2 n .
This concludes the proof of Lemma 5.6.

A Convergence of random measures

In this section, we present some results concerning convergence of random or deterministic probability measures on a polish space S or more specifically on D([0, 1]). Some of these results are classical, but we state them here with uniformity in some parameter θ ∈ Θ.

B.1 Convergence towards the 3-dimensional Bessel process and the Brownian meander

We first recall a known invariance principle for the random walk conditioned to stay nonnegative for all time. For all n ∈ N, u ∈ R + and B ∈ F n , we set

P + u (B) := 1 R(u) E u R(S n )1 B 1 S n ≥0 . (B.1)
It defines a probability measure P + u , that is called the law of the random walk started at u ∈ R + and conditioned to stay nonnegative for all time. Then we have the following invariance principle, by Theorem 1.1 of Caravenna and Chaumont [START_REF] Caravenna | Invariance principles for random walks conditioned to stay positive[END_REF]:

for any b ∈ R + and (b n ) n∈N such that b n /σ √ n → b as n → ∞ and for any F ∈ C b (D([0, 1])), E + bn F (S (n) ) ---→ n→∞ E b [F (R)], (B.2)
where R denotes the 3-dimensional Bessel process on [0, 1]. Proposition 2.6 follows from (B.2) and from the following link between the 3-dimensional Bessel process and the Brownian meander (see Imhof [START_REF] Imhof | Density factorizations for Brownian motion, meander and the three-dimensional Bessel process, and applications[END_REF]): for all F ∈ C b (D([0, 1])), we have

E[F (M)] = π 2 E 1 R(1) F (R) . (B.3)
Proof of Proposition 2.6. We can assume that F is nonnegative. For K > 0, we consider

χ : R + → [0, 1] continuous such that 1 [K -1 ,K] ≤ χ ≤ 1 [(2K) -1 ,2K
] . On the one hand, we have

E u F (S (n) ) 1 -χ S n σ √ n 1 S n ≥0 ≤ F θR(u) √ n ∞ 0 (1 -χ(t))te -t 2 /2 dt + o(1) , (B.4)
uniformly in u ∈ [0, γ n ], using (2.6). On the other hand, we have

E u F (S (n) )χ S n σ √ n 1 S n ≥0 = R(u) √ n E + u h n S n σ √ n F (S (n) ) , (B.
E + u h n S n σ √ n F (S (n) ) -E + u h S n σ √ n F (S (n) ) = 0.
Moreover, using (B.2), (B.3) and (2.5), we have

E + u h S n σ √ n F (S (n) ) ---→ n→∞ E χ(R(1)) c 0 σR(1) F (R(t), t ∈ [0, 1]) = θE[χ(M(1))F (M)],
uniformly in u ∈ [0, γ n ]. Coming back to (B.4) and (B.5) and using that the density of M(1) is t → te -t 2 /2 1 t>0 , we showed that lim sup

n→∞ sup u∈[0,γn] √ n θR(u) E u F (S (n) )1 S n ≥0 -E[F (M)] ≤ 2 F ∞ 0 (1 -χ(t))te -t 2 /2 dt,
which tends to 0 as K → ∞, so it concludes the proof.

Proof of Corollary 2.7. By Lemma A.2 of Madaule [START_REF] Madaule | First order transition for the branching random walk at the critical parameter[END_REF] (that holds under the assumption of this corollary, see Remark 3.1), it exists c 27 (L, C) > 0 such that for all n large enough and K ≥ 0,

E e CSn/ √ n 1 S n ≥-L,Sn≥K √ n ≤ c 27 (L, C) √ n e -CK/2 . (B.6)
We consider some ε > 0 and fix K large enough such that c 27 (L, C)e -CK/2 ≤ εθR(L)/ F and also E[e CσM (1) 1 M(1)≥K ] ≤ ε/ F . Considering a continuous function χ : R → [0, 1] such that 1 x≤K ≤ χ(x) ≤ 1 x≤K+1 and using the triangle inequality, we get

√ n θR(L) E e CSn/ √ n F (S (n) )1 S n ≥-L -E e CσM(1) F (M) ≤ 2ε + √ n θR(L) E e CSn/ √ n F (S (n) )χ S n √ n 1 S n ≥-L -E e CσM(1) F (M)χ(σM(1)) . (B.7)
Then, note that in Proposition 2.6, we can replace

E u [F (S (n) )1 S n ≥0 ] by E[F (S (n) )1 S n ≥-u ]: indeed, it works when F ∈ C u b (D([0, 1

B.2 Lower envelope for the random walk above two barriers

We prove here Lemma 2.4. Firstly, by Equation (A.9) of Shi [START_REF] Shi | Branching random walks[END_REF], we have: it exists c 28 > 0 such that, for all b > a ≥ 0, u ≥ 0 and n ≥ 1,

P(S n ∈ [b -u, b -u + 1], S n ≥ -u) ≤ c 28 b + 1 n . (B.8)
Secondly, it exists c 29 > 0 such that for all b > a ≥ 0, u ≥ 0, v ∈ R and n ≥ k ≥ 0,

P S n ∈ [b + v, b + v + 1], S k ≥ -u, min k≤j≤n S j ≥ v ≤ c 29 (u + 1)(b + 1) (n -m) √ k + 1 . (B.9)
Indeed, the left-hand side of (B.9) is equal to

E 1 S k ≥-u P S k (S n-k ∈ [b + v, b + v + 1], S n-k ≥ v) ≤ E 1 S k ≥-u c 28 b + 1 n -k ,
by using (B.8), and then we get (B.9) by applying (2.4).

Proof of Lemma 2.4. Recall that, for , i ∈ 0, n , u, µ ≥ 0 and v ∈ R, we have

m (n, ) i := -u + r i -µ if 0 ≤ i < , v + r n-i -µ if ≤ i ≤ n.
Moreover, for J ⊂ 0, n , we set

P J (n, , u, v, b, µ) := P S ≥ -u, min ≤j≤n S j ≥ v, S n ∈ [b + v, b + v + 1], ∃ i ∈ J : S i ≤ m (n, ) i
and write simply P J = P J (n, , u, v, b, µ) when the parameters are obvious. For any ε > 0 and λ ∈ (0, 1/2), we want to prove that for µ large enough, for any b, u ≥

0, v ∈ R, n ∈ N and ∈ [λn, (1 -λ)n], P 0,n ≤ ε(1 + u)(1 + b)n -3/2 .
For this, it is sufficient to prove that

∀ ε > 0, ∀ λ ∈ (0, 1/2), ∃ µ > 0 : ∀ b, u ≥ 0, ∀ v ∈ R, ∀ n ∈ N, ∀ ∈ [λn, (1 -λ)n], P 0, ≤ ε(1 + u)(1 + b)n -3/2 .
(B.10) Indeed, we have P 0,n ≤ P 0, + P +1,n and, setting Sk := S n-k -S n ,

P +1,n ≤ P min n-≤j≤n Sj ≥ -u -(b + v + 1), Sn-≥ -b -1, Sn ∈ [-(b + v + 1), -(b + v)], ∃ i ∈ 0, n --1 : Si ≤ m (n, ) n-i -(b + v) = P 0,n--1 (n, n -, b + 1, -u -b -v -1, u, µ -1),
where PJ is the analogue of P J for the random walk S. Then, using (B.10) for P 0, and for P 0,n--1 , we get the wanted bound for P 0,n .

We now prove (B.10). We first have P 0, ≤ P 0,i 0 + P i 0 , , for some i 0 ≥ 1 that will be chosen afterwards. We take µ = r i 0 + 1, so that m (n, ) i < -u for 0 ≤ i ≤ i 0 , and thus P 0,i 0 = 0.

We set τ := max{i ∈ 0, : S i ≤ m (n, ) i } and get

P i 0 , = i=i 0 P S n ≥ -u, min ≤j≤n S j ≥ v, S n ∈ [b + v, b + v + 1], τ = i = i=i 0 p i ,
where, for i ∈ i 0 , , we set

p i := P S n ≥ -u, min ≤j≤n S j ≥ v, S n ∈ [b + v, b + v + 1], S i ≤ m (n, ) i , ∀k ∈ i + 1, , S k > m (n, ) k .
We first consider the case i ∈ i 0 , -1 and the case i = will be treated after. Applying the Markov property at time i + 1 and noting that for k

∈ i + 1, , m (n, ) k ≥ m (n, ) i , we have p i ≤ E 1 S i ≥-u,S i ≤m (n, ) i 1 S i+1 >m (n, ) i P S i+1 S -i-1 > m (n, ) i , min -i-1≤j≤n-i-1 S j ≥ v, S n-i-1 ∈ [b + v, b + v + 1] . (B.11)
If i ≤ λn/2, we can use (2.8) to bound the probability in (B.11) (because -i -1 ≥ λn/2), but, if i > λn/2, we use instead (B.9). Thus, the probability in (B.11) is smaller than

(1 + S i+1 -m (n, ) i )(1 + b) 2c 5 (n -i -1) 3/2 1 i≤λn/2 + c 29 (n -)( -i) 1/2 1 i>λn/2 .
Setting ξ := S i+1 -S i , we get

p i ≤ c 30 (1 + b) n 3/2 1 + √ n1 i>λn/2 ( -i) 1/2 E 1 S i ≥-u,S i ≤m (n, ) i 1 ξ>m (n, ) i -S i (1 + S i + ξ -m (n, ) i ) . (B.12) Cutting the interval [-u, m (n, ) i
] in pieces of length 1, the expectation in (B.12) is smaller than

m (n, ) i +u k=1 E 1 S i ≥-u,S i ∈[m (n, ) i -k,m (n, ) i -k+1] 1 ξ>k-1 (2 -k + ξ) ≤ m (n, ) i +u k=1 c 4 (1 + u)(1 + u + m (n, ) i -k + 1)2 i 3/2 E[1 ξ>k-1 (2 -k + ξ)],
by noting that ξ is independent of (S j ) 0≤j≤i and applying (2.7). Recalling that m (n, ) i = r iu -µ ≤ r i -u and coming back to (B.12), we get

p i ≤ c 31 (1 + b)(1 + u) n 3/2 (1 + r i ) i 3/2 1 + √ n1 i>λn/2 ( -i) 1/2 k≥1 E[1 ξ>k-1 (2 + ξ)]. (B.13) Moreover, we have k≥1 E[1 ξ>k-1 (2 + ξ)] = i≥0 (i + 1)E 1 ξ∈(i,i+1] (2 + ξ) ≤ E (2 + ξ) 2 = 4 + σ 2 , (B.14)
using that ξ has the same law than S 1 . We now deal with the case i = : we have

p ≤ E 1 S ≥-u,S ≤m (n, ) P S i S n-≥ v, S n-∈ [b + v, b + v + 1] ≤ E 1 S ≥-u,S ≤v+r n-c 5 (1 + S -v)(1 + b)2 (n -) 3/2 ≤ c 32 (1 + r n-)(1 + b) n 3/2 (1 + u) √ n ,
p i ≤ c 33 (1 + b)(1 + u) n 3/2   -1 i=i 0 (1 + r i ) i 3/2 + -1 i= λn/2 +1 (1 + r i ) i 3/2 √ n ( -i) 1/2 + (1 + r n ) √ n   ≤ c 33 (1 + b)(1 + u) n 3/2   ∞ i=i 0 (1 + r i ) i 3/2 + (1 + r n ) √ n ( λn/2 + 1) 3/2 -λn/2 -1 j=1 1 j 1/2 + (1 + r n ) √ n   ≤ c 34 (1 + b)(1 + u) n 3/2   ∞ i=i 0 (1 + r i ) i 3/2 + (1 + r n ) n n j=1 1 j 1/2 + (1 + r n ) √ n   ≤ c 35 (1 + b)(1 + u) n 3/2   ∞ i=i 0 (1 + r i ) i 3/2 + (1 + r n ) √ n   ,
where we used that (r n ) n∈N is an increasing sequence and the constants depend only on λ. Since furthermore n≥1 r n n -3/2 < ∞, we have r n / √ n → 0. Thus, we can choose i 0 such that for all n ≥ i 0 , c 35 (1 + r n )/ √ n ≤ ε/4. Moreover, we can choose i 0 such that c 35

∞ i=i 0 (1+r i ) i 3/2 ≤ ε/4.
This concludes the proof of (B.10) and, therefore, of the lemma.

B.3 Local limit theorems

We first recall the classical Stone's [START_REF] Stone | A local limit theorem for nonlattice multi-dimensional distribution functions[END_REF] local limit theorem: letting h > 0 be any real number if S 1 is nonlattice and be the span of the lattice if S 1 is lattice, we have We set f : t ∈ R → te -t 2 /2 1 t≥0 .

P(S n ∈ [b, b + h)) = h σ √ 2πn e -b 2 /2σ 2 n + o 1 √ n , (B.15) as n → ∞, uniformly in b ∈ R.
(i) If the law of S 1 is nonlattice, then, for all h > 0,

P(S n ∈ [u -b, u -b + h), S n ≥ -b) = θ - σn f b σ √ n u+h u R -(t) dt + o R -(u) n , as n → ∞, uniformly in b ∈ R and in u ∈ [0, γ n ].
(ii) If the law of S 1 is (h, a)-lattice, then,

P(S n = u -b, S n ≥ -b) = θ - σn f b σ √ n hR -(u) + o R -(u) n , as n → ∞, uniformly in b ∈ R and u ∈ [0, γ n ] ∩ (b + an + hZ).
Proof. First note that, for each D > 0, by Propositions 11, 18 and 24 of Doney [START_REF] Doney | Local behaviour of first passage probabilities[END_REF], both estimates of Lemma B.1 holds uniformly in b ∈

[D -1 √ n, D √ n] and u ∈ [0, γ n ) 4 .
Noting also that f tends to 0 at 0 and at infinity, it is sufficient to prove that, for each h > 0 and ε > 0, there exist D and n large enough such that P

(S n ∈ [u -b, u -b + h), S n ≥ -b) ≤ εR -(u)/n for all b / ∈ [D -1 √ n, D √ n] and u ∈ R + .
Reversing time, we can equivalently prove that

P(S n ∈ [b -u, b -u + h), S n ≥ -u) ≤ εR(u)/n. For b ∈ [0, D -1 √ n], using (2.7) and (2.2), we get P(S n ∈ [b -u, b -u + h), S n ≥ -u) ≤ c 37 R(u)(1 + D -1 √ n)/n 3/2 .
For D and n large enough and independent of u, this is smaller than εR(u)/n.

For b > D √ n, we cancel the lower barrier between times n/2 + 1 and n so that we get where g D : R + → [0, 1], obtained by taking the supremum on b > D √ n, is continuous and converges simply to 0 as D → ∞. Thus, using (2.6) and (2.4), we get that (B.17

P(S n ∈ [b -u, b -u + h), S n ≥ -u) ≤ E 1 S n/2 ≥-u h S n/2 + u σ n/2

B.4 Convergence towards the Bessel bridge

In this subsection, we are going to prove the following generalization of Lemma 2.4 of Chen, Madaule and Mallein [START_REF] Chen | On the trajectory of an individual chosen according to supercritical gibbs measure in the branching random walk[END_REF]. It proves that conditioned by the event of Lemma B.1 the trajectory S (n) converges to the Bessel bridge. This is a first step in the proof of Proposition 2.8. Lemma B.2. Let (γ n ) n∈N be a sequence of positive numbers such that γ n √ n as n → ∞.

We set f : t ∈ R → te -t 2 /2 1 t≥0 and we denote by ρ 1 b,0 the 3-dimensional Bessel bridge of length 1 from b ∈ R + to 0. Proof. We will treat only the nonlattice case, because the proof in the lattice case is exactly the same, with hR -(u) instead of where we set, for z ∈ R + ,

h n ε,b (z) := f z √ ε R(bσ √ n) R(zσ √ n) = z √ ε e -z 2 /2ε R(bσ √ n) R(zσ √ n) .
Since e -z 2 /2ε → 0 as z → ∞ and using (2.1), it is clear that We first prove (B.25): we have

E F (ρ 1 b,0 ) -F • ϕ ε (ρ 1 b,0 ) ≤ E ω F ( ρ 1 b,0 -ϕ ε (ρ 1 b,0 ) ∞ ) ≤ E ω F sup t∈[1-ε,1]
ρ 1 b,0 (t) =: E b,ε .

Since the function b → E b,ε is nondecreasing and f ≤ 1, the left-hand side of (B.25) is smaller than E K,ε and, thus, it tends to 0 as ε → 0 by dominated convergence, because ω F is bounded. Now, we prove (B.24): using that d(S (n) , ϕ ε (S (n) )) ≤ max 0≤k≤εn S n-k /σ √ n and reversing time, we get that the expectation in (B. [START_REF] Derrida | Polymers on disordered trees, spin glasses, and traveling waves[END_REF]

B.5 Convergence towards the Brownian excursion

We prove here Proposition 2.8, in a similar way as Lemma 2.5 of Chen, Madaule and Mallein [START_REF] Chen | On the trajectory of an individual chosen according to supercritical gibbs measure in the branching random walk[END_REF], but directly with a first barrier that can be different of 0. Following [START_REF] Chen | On the trajectory of an individual chosen according to supercritical gibbs measure in the branching random walk[END_REF], we fix some λ ∈ (0, 1) and, for

Theorem 1 . 1 .σ 2 π

 112 Assume (1.1), (1.2), (1.3) and that α n → ∞ as n → ∞. Let M denotes the Brownian meander of length 1. (i) If β n := 1 + 1/α n and √ n/α n → ∞ as n → ∞, then we have n in P * -probability.(ii) Ifβ n := 1 + 1/α n and √ n/α n → γ ∈ [0, ∞) as n → ∞, then we have √ nW n,βn ---→ n→∞ 1 E e -σγM(1) D ∞ , in P * -probability.

Theorem 1 . 2 .

 12 Assume (1.1), (1.2), (1.3) and that α n → ∞ as n → ∞. Let e denotes the normalized Brownian excursion, M the Brownian meander of length 1 and B the Brownian motion.

Corollary 1 . 3 .

 13 Assume (1.1), (1.2), (1.3) and that α n → ∞ as n → ∞.

Proposition 5 . 1 .

 51 For all L > 0 and F ∈ C u b (D([0, 1]

  e)]D ∞ , in P * -probability. (5.1) Using (2.5), it proves part (i) of Theorem 1.1 by taking F ≡ 1. Moreover, noting that µ n,βn (F ) = W n,βn (F )/W n,βn (1) and D ∞ > 0 P * -a.s., it proves part (i) of Theorem 1.2 in the case F ∈ C u b (D([0, 1]

Figure 1 -

 1 Figure 1 -Representation of the trajectory of a particle in An. It has to stay above the gray area and to pass through both thick segments.

Lemma 5 . 3 .

 53 For all L, K > 0 and F ∈ C u b (D([0, 1]

3 . 5 . 4 .

 354 (F ), Lemma 5.3 combined with the following lemma shows that the first moments underQ (L) of W (L,K) n,βn (F ), W (L,K) n,βn (F n ), Y n (F ) and Y n (F n ) (divided by D (L)n ) have the same equivalent as n → ∞. It will be proved in Subsection 5.Lemma For all L, K > 0 and F ∈ C b (D([0, 1])) nonnegative, we have

  we follow the proof of Lemma 5.4 for the upper bound of the first moment of W

  by using Proposition 2.8 combined with Lemmas A.3 and A.5, because F ∈ C u b (D([0, 1])). The result follows with the same calculations as in the proof of Lemma 5.4.

3 n

 3 ,kn]

.41) uniformly in b ∈ [k 1 / 3 n

 13 , k n ], using (2.1) and (2.5) for the last equality. Coming back to (5.40), we showed that E [0,kn)

  and, by bounding F ≤ F , it is sufficient to deal with the case F ≡ 1. Using that D (L),[0,kn) n ≤ D (L) n , 1 z∈Bn ≤ 1 and breaking down Y [kn,n] n (1) along the spine, we first have that E [kn,n]

  ])) and we extend to F ∈ C b (D([0, 1])) by Lemma A.1. Thus, applying Proposition 2.6 with the function x ∈ D([0, 1]) → e Cσx 1 F (x)χ(σx 1 ) which belongs to C b (D([0, 1])) we get that the right-hand of (B.7) tends to 2ε as n → ∞ and it concludes the proof of Corollary 2.7.

g 3 R

 3 D (t)te -t 2 /2 dt + o(1) + c b ≥ D √ n and u ∈ [0, γ n ]. By the dominated convergence theorem, the integral in (B.18) tends to 0 as D → ∞ and, thus, (B.17) is smaller than εR(u)/n for D and n large enough and independent of u.

( i )

 i If the law of S 1 is nonlattice, then, for all h > 0 andF ∈ C u b (D([0, 1])), E bσ √ n F (S (n) )1 Sn∈[u,u+h),S n ≥0 = θ - σn u+h u R -(t) dtf (b)E F (ρ 1 b,0 ) + o R -(u) n , as n → ∞, uniformly in b ∈ R and in u ∈ [0, γ n ]. (ii) If the law of S 1 is (h, a)-lattice, then, for all F ∈ C u b (D([0, 1])), E bσ √ n F (S (n) )1 Sn=u,S n ≥0 = θ - σn hR -(u)f (b)E F (ρ 1 b,0 ) + o R -(u) n ,as n → ∞, uniformly in b ∈ R and u ∈ [0, γ n ] ∩ (bσ √ n + an + hZ).

R

  u+h u R -(t) dt. Moreover, since F is bounded, f (b) → 0 as b → ∞ and Lemma B.1 deals with the case F ≡ 1, it is sufficient to show that, for each K > 0, the estimate holds uniformly in b ∈ [0, K] instead of b ∈ R + .We first assume that∀ x ∈ D([0, 1]), F (x) = F (x t , t ∈ [0, 1 -ε]) for some F ∈ C u b (D([0, 1ε])). Thus, F (S (n) ) is F m -measurable with m := (1 -ε)n and we have E bσ √ n F (S (n) )1 Sn∈[u,u+h),S n ≥0 = E bσ √ n F (S (n) )g S m σ √ n 1 S m ≥0 , (B.19)where we set, for z ∈ R + ,g(z) := P zσ √ n (S n-m ≥ 0, S n-m ∈ [u, u + h)) -(t) dt + o R -(u) n ,uniformly in u ∈ [0, γ n ] and z ∈ R + , using Lemma B.1. Therefore, (B.[START_REF] Chauvin | KPP equation and supercritical branching Brownian motion in the subcritical speed area. Application to spatial trees[END_REF]) is equal toθ - σεn u+h u R -(t) dtE bσ √ n F (S (n) )f S m σ √ εn 1 S m ≥0 + E bσ √ n F (S (n) )1 S m ≥0 o R -(u) n , uniformly in u ∈ [0, γ n ] and b ∈ [0, K]. Thus, it is now sufficient to prove that E bσ √ n F (S (n) )f S m σ √ εn 1 S m ≥0 ---→ n→∞ εf (b)E F (ρ 1 b,0 ) , (B.20)uniformly in b ∈ [0, K]. On the one hand, by Equation (2.30) of Chen, Madaule and Mallein[START_REF] Chen | On the trajectory of an individual chosen according to supercritical gibbs measure in the branching random walk[END_REF], we haveεf (b)E F (ρ 1 b,0 ) = b √ ε E b F (R(s), s ∈ [0, 1 -ε])e -R(1-ε) 2 /2ε . (B.21) On the other hand, recalling (B.1), we get

1 S 1 bF

 11 m ≥0 -εf (b)E F (ρ (S (n) )e -S 2 m /2σ 2 εn -E b F (R(s), s ∈ [0, 1 -ε])e -R(1-ε) 2 /2ε ,which is equal to 0 by applying (B.2). It proves (B.20) and so it concludes the case whereF (x) = F (x t , t ∈ [0, 1 -ε]) for some F ∈ C u b (D([0, 1 -ε])). We now want to extend the result to the caseF ∈ C u b (D([0, 1])). For ε > 0 and x ∈ D([0, 1]), we define ϕ ε (x) ∈ D([0, 1]) by ϕ ε (x)| [0,1-ε) = x| [0,1-ε) and ϕ ε (x)| [1-ε,1] ≡ 0, so that F • ϕ ε satisfiesthe assumption of the particular case that is already proved. Thus, it is now sufficient to show that, for each η > 0, it exists ε > 0 such that lim supn→∞ sup u∈[0,γn],b∈[0,K] E bσ √ n F (S (n) ) -F • ϕ ε (S (n) ) 1 Sn∈[u,u+h),S n ≥0 ≤ η R -(u) n , (B.24) sup b∈[0,K] f (b)E F (ρ 1 b,0 ) -F • ϕ ε (ρ 1 b,0 ) ≤ η. (B.25)

  u ∈ [0, γ n ] et b ∈ [0, K],by using Lemma 2.6 to get the last equality. The expectation in the right-hand side of (B.27) does not depend on n, b and u and tends to 0 as ε → 0 by dominated convergence, so it shows (B.24) and concludes the proof.

G 1 :

 1 D([0, λ]) → R, G 2 : D([0, 1 -λ]) → R and x ∈ D([0, 1]), we set G 1 G 2 (x) := G 1 (x s , s ∈ [0, λ])G 2 (x λ+s , s ∈ [0, 1 -λ]).Ψ(pβ) ≤ Ψ(β) < pΨ(β) and, by Theorem 1 of Biggins[START_REF] Biggins | Uniform convergence of martingales in the branching random walk[END_REF], we get that W n,β → W ∞,β in L p . The proof follows the lines of the proof of part (iv) of Theorem 1.2, without having to introduce the barrier at -L.By Lemma A.1, we can reduce the proof to the caseF ∈ C u b (D([0, 1])). Since W ∞,β is positive P * -a.s., it is sufficient to prove that U n (F ) := e -nΨ(β) |z|=n e -βV (z) F V (n) (z) ---→ n→∞ W ∞,β E[F (B)],(C.1) in P * -probability, whereV (n) t (z) := [V (z tn ) + tnΨ (β)]/σ β √ n for t ∈ [0, 1]. We consider a sequence of integers (k n ) n∈N such that 1 k n n and, for each x ∈ D([0, 1]), F n (x) := F (x ((n-kn)t+kn)/n -x kn/n , t ∈ [0, 1]). Then, using (4.2) and setting S(n,β) := (S tn ,β /σ β √ n) t∈[0,1] , we get E * [|U n (F ) -U n (F n )|] ≤ E * [U n (|F -F n |)] = E |F -F n |(S (n,β) ) ---→ n→∞ 0,by Lemma A.5. Thus, in order to prove (C.1), it is now sufficient to prove thatU n (F n ) → W ∞,β E[F (B)] in P * -probability.First, we prove thatζ n := U n (F n ) -E[U n (F n ) | F kn ] → 0 in P * -probability. We set ζ n := E[|ζ n | p |F n ]and, by (4.4), we haveP * (|ζ n | ≥ ε) ≤ εP(S) -1 + P * (ζ n ≥ ε 1+pn). By the branching property at time k n , we haveζ n = |x|=kn e -βV (x)-knΨ(β) Υ (x) n -E[Υ n ] , (C.2)where, conditionally on F kn , the Υ(x)n for |x| = k n are independent variables with the same law as Υ n defined byΥ n := |z|=n-kn e -βV (z)-(n-kn)Ψ(β) F V (z t(n-kn) ) + t(n -k n )Ψ (β) σ β √ n , t ∈ [0, 1] . (C.3) Since p ∈ [1, 2] and c 39 := sup n∈N E[ W p n,β] < ∞, we get, in the same way as for (4.12),ζ n ≤ 2c 39 (4 F ) p W kn,pβ e kn(Ψ(pβ)-pΨ(β)) .Using that Ψ(pβ) -pΨ(β) < 0 and W kn,pβ → W ∞,pβ < ∞ P * -a.s., we get ζ n → 0 P * -a.s. and, therefore,ζ n → 0 in P * -probability. Finally, we prove that E[U n (F n ) | F kn ] → W ∞,β E[F (B)] P * -a.s.Using the branching property in the same way as for (C.2), we haveE[U n (F n ) | F kn ] = W kn,β E[Υ n ],where Υ n is defined in (C.3). By (4.2) and recalling that S(n,β) := (S tn ,β /σ β √ n) t∈[0,1] ,we haveE[Υ n ] = E F S t(n-kn) ,β + (t(n -k n ) -t(n -k n ) )Ψ (β) σ β √ n , t ∈ [0, 1] = E F u n S (n-kn,β) + v n , where (u n ) n∈N ∈ (R * + ) N and (v n ) n∈N ∈ D([0, 1]) N satisfy u n → 1 and v n ∞ → 0.Using that S (n-kn,β) converges in law towards the Brownian motion and applying Lemma A.3, we get that E[Υ n ] → E[F (B)] as n → ∞. On the other hand, we have W kn,β → W ∞,β P * -a.s. Therefore, E[U n (F n ) | F kn ] → W ∞,β E[F (B)] P * -a.s. and it concludes the proof.

  max Sn/αn 1 S n ≥-L = o On the other hand, using Markov property at time k n , the second term in (4.14) is smaller than 2 F E e S kn /αn 1 S kn ≥-L,max 0≤k≤kn S k >M αn E e S n-kn /αn

	and, thus, the first term in (4.14) is smaller than
	ω F	Cα n n	2	∨	3M α n σ √ n	E e e nΨ(βn) α n	,
	using (4.3) to bound the expectation and recalling that α n n. = 2 F √ θR(L) √ k n E e CσM(1) 1 max M>M/(Cσ) + o(1) e (n-kn)Ψ(βn)
	applying Corollary 2.7 and recalling that	√	k n = Cα n . Coming back to (4.13), we finally get
	lim sup
	n→∞
							|x|
							[0,kn/n]

  Corollary 1.3. First note that it is sufficient to prove that, for all sequences (α - n ) n∈N and (α + n ) n∈N such that 1 α -

	n	α n	α + n	√	n, we have
	ν n,βn (3/2) log n + α -n , (3/2) log n + α + n	---→ n→∞	1, in P * -probability.	(5.6)

  this expectation in two pieces depending on whether S n ≤ 3

	2 log n + α + n or S n > 3 2 log n + n : we have, by cutting the interval [α 2 log n + α + n . Let start with the case S n > 3 α +

  Thus we have the following uniform bound: it exists c 36 > 0 such that, for all n ≥ 1 and b ∈ R,

	P(S n ∈ [b, b + 1)) ≤	c 36 √ n	.	(B.16)
	Now, we state a local limit theorem for the random walk staying above a barrier, in the case where the starting point is at distance of order √ n from the barrier and the endpoint at distance o( √ n).

Lemma B.1. Let (γ n ) n∈N be a sequence of positive numbers such that γ n √ n as n → ∞.

  is smaller thanConditioning with respect to F n/2 and applying (B.16), we get that (B.26) is smaller thanc 36 (1 + h) (n -n/2 ) 1/2 E ω F sup

	E ω F	sup 0≤k≤εn	S -k + u + h σ √ n		1 S -n -bσ	√	n∈(-u-h,-u],S -n ≥-u-h .	(B.26)
					0≤k≤εn	S -k + u + h σ √ n	1 S -n/2 ≥-u-h
	=	c 36 (1 + h) n/2	θ -R -(u + h) n/2	E ω F	1 √ 2	sup t∈[0,2ε]

Actually, Madaule[START_REF] Madaule | First order transition for the branching random walk at the critical parameter[END_REF] considers the linear interpolation of the trajectory, instead of V, and the convergence on C([0, 1]), instead of D([0, 1]). But the convergence (1.12) follows from Madaule's result.

To our knowledge, this result has not been proved yet (except when F only depends on the final position, see[START_REF] Biggins | Growth rates in the branching random walk[END_REF]), so a proof is given in Section C of the appendix.

log n-b)∈[α - n ,α + n ],S m ≥-(L+b),min λm ≤j≤m S j

≥ 3 2 log n-b .

Doney states his results in terms of the renewal function for the first weak increasing ladder height process of S and of P(min 1≤k≤n Sn > 0). Our formulation follows from Remark 4.6 of Caravenna and Chaumont[START_REF] Caravenna | An invariance principle for random walk bridges conditioned to stay positive[END_REF] (although they work only in the lattice and absolutely continuous cases, this remark does not rely on these assumptions).

A.1 General space

Let (S, d), (S 1 , d 1 ) and (S 2 , d 2 ) be Polish spaces. We consider some set Θ. In the sequel, for each θ ∈ Θ, (µ θ n ) n∈N will denote a sequence of random probability measures on S and (ξ θ n ) n∈N a sequence of deterministic probability measures on S. Moreover, µ and ξ will be probability measures on S, that are deterministic and do not depend on θ.

Lemma A.1. Assume that for all Lipschitz F ∈ C b (S) and ε > 0, P(|µ θ n (F ) -µ(F )| ≥ ε) → 0 as n → ∞ uniformly in θ ∈ Θ. Then, the same convergence holds for all F ∈ C b (S).

Note that in the case where we consider a deterministic sequence (ξ θ n ) n∈N , it simply means that ξ θ n (F ) → ξ(F ) uniformly in θ ∈ Θ. It is necessary that the limit does not depend on θ.

Proof. We follow the proof of Portmanteau Theorem in Billingsley [START_REF] Billingsley | Convergence of probability measures[END_REF]Theorem 2.1]. Thus, we first consider a closed set A and ε > 0 and we want to show that

We consider, for each η > 0, the function

and such that F η ↓ 1 A . Thus, by dominated convergence, we have µ(F η ) → µ(A) as η → 0. We

by using the assumption of the lemma. From (A.1), we get that, for all set A such that µ(∂A) = 0 P-a.s. and all ε > 0,

We now consider F ∈ C b (S) and we can assume that F is nonnegative. We fix ε > 0 and set M := F . Firstly, we have

and, therefore, for almost every t ∈ [0, M ] (in the sense of the Lebesgue measure), P-a.s., µ(∂{F > t}) = 0. Thus, for all N ∈ N, we can fix a subdivision 0 = t

and the same holds for µ θ n instead of µ, for all n ∈ N and θ ∈ Θ. Since, in (A.3), the left-hand side and right-hand side of (A.3) tend to µ(F ) as N → ∞, we can choose N large enough such that they are at most at distance ε/2 from µ(F ). Then, using (A.3) for µ θ n , it follows that

by noting that, in the sums, the term for k = N is zero, since t N = M = F , and the term for k = 0 is smaller than 2t 1 ≤ 4M/N ≤ ε/4, if we choose N large enough. Then, we have

Lemma A.2. We consider the product space S := S 1 ×S 2 . Assume that, for all

Using again the compacity of K , there exist nonnegative Lipschitz functions χ

We can now construct some functions of the form G 1 G 2 to approach F . For 0

. By the triangle inequality, we have that

The second term in (A.4) tends to 0 as n → ∞ uniformly in θ ∈ Θ by the assumption of the lemma. On the other hand, we have

because of the choice of η. Thus, we get that the first and third terms of (A.4) are smaller than 2ε +

, we can use again the assumption of the lemma to get that

A.2 Weak convergence in D([0, 1])

We keep here the notation of the previous subsection, but we take S = D([0, 1]). Recall the definition of the Skorokhod distance

where we set Λ := {λ : [0, 1] → [0, 1] | λ(0) = 0, λ(1) = 1, λ continuous and increasing}, and that, equipped with this distance, D([0, 1]) is a polish space (see Billingsley [START_REF] Billingsley | Convergence of probability measures[END_REF]).

Lemma A.3. Assume that, for all

Proof. For x ∈ D([0, 1]), we first have

Now, we consider ε > 0 and we fix K > 0 such that ξ({ x ∞ ≥ K}) ≤ ε and some Lipschitz

. Thus, we have

On the one hand, for x ∈ D([0, 1]), we have, using χ(x) ≤ 1 x ∞<K+1 and (A.5),

On the other hand, by using the assumption of the lemma with the function 1 -χ, the first and third terms in the right-hand side of (A.6) tend towards 2 F ξ(1 -χ) uniformly in θ ∈ Θ. Since ξ(1 -χ) ≤ ε, it concludes the proof of Lemma A.3.

Remark A.4. In Lemma A.3, if the limit measure ξ θ depends on θ ∈ Θ, then the result is still true under the additional assumption that sup θ∈Θ ξ({ x ∞ ≥ K}) → 0 as K → ∞, so that in the proof K and χ could be chosen independently of θ.

Proof. The function ϕ n : t ∈ [0, 1] → κ n + (1 -κ n )t is not bijective from [0, 1] to [0, 1], so we consider a function λ n ∈ Λ such that λ n (t) = κ n + (1 -κ n )t for t ≥ κ n and that is linear on [0, κ n ]. Then, for x ∈ D([0, 1]), we have ]). Thus, we have, using the triangle inequality and then (A.7),

for n large enough such that κ n ≤ δ (so independent of θ). Finally, using the assumption of the lemma with the function 1 -χ and recalling that ξ(1 -χ) ≤ ε, it concludes the proof.

B Proofs of the preliminary results concerning random walk

In this section, we prove the results stated in Section 2. Subsection B.

where e is the normalized Brownian excursion, M is the Brownian meander of length λ and (ρ 1-λ z,0 ) z∈R + is a family of Bessel bridges of length 1 -λ from z to 0, independent of M.

Proof of Proposition 2.8. Using Lemma A.2, it is sufficient to consider the case where

. The proof will be only treated in the lattice case: in the nonlattice case, the reasoning is exactly the same with b+h b R -(t) dt instead of hR -(b). We set m := λn and take the conditional expectation according to F m to get that

where we set ϕ(z)

Then, since G 2 is uniformly continuous, using Lemma B.2 combined with Lemma A.3 5 , we get that ϕ(z) is equal to

] by scaling properties of the Bessel bridge, (B.29) is equal to

where (ρ 1-λ z,0 ) z∈R + is independent of (S n ) n∈N . Then, since the function z ∈ R → E[G 2 (ρ 1-λ z,0 )] is continuous, using Lemma 2.6 combined with Lemma A.3, we get that (B.30) is equal to

where M is the Brownian meander of length λ, independent of (ρ 1-λ z,0 ) z∈R + . Finally, recalling the definition of f and using (B.28), it concludes the proof of Proposition 2.8.

C Trajectories in the weak disorder regime

We prove here (1.14). We consider some β < 1 and p > 1 such that E[W p 1,β ] < ∞. Using also (1.2), (1.3) and the convexity of Ψ, we get that Ψ is finite and nonincreasing on [β, 1] and positive on [β, 1). Moreover, we can assume that p ≤ 2 and that pβ < 1. Then, we have 5 Here the limit measure is the law of ρ 1-λ z,0 that depends on the parameter (b, v, z). Thus, using Remark A.4, we should prove that P( ρ 1-λ z,0 ∞ > K) → 0 as K → ∞ uniformly in z ∈ R+, but it is obviously false. However, since f (z/ √ 1 -λ) → 0 as z → ∞, it is sufficient to have, for each M > 0, P( ρ 1-λ z,0 ∞ > K) → 0 as K → ∞ uniformly in z ∈ [0, M ] and this is clearly true.