Pierre Andreoletti

Xinxin Chen

Range and critical generations of a random walk on Galton-Watson trees

Keywords: random walks, range, random environment, branching random walk 1

In this paper we consider a random walk in random environment on a tree and focus on the boundary case for the underlying branching potential. We study the range R n of this walk up to time n and obtain its correct asymptotic in probability which is of order n/ log n. This result is a consequence of the asymptotical behavior of the number of visited sites at generations of order (log n) 2 , which turn out to be the most visited generations. Our proof which involves a quenched analysis gives a description of the typical environments responsible for the behavior of R n .

Introduction

Let us consider a random walk with a random environment given by a branching random walk. This branching random walk is governed by a point process L := {A 1 , A 2 , • • • , A N } on the real line, where N is also random in N ∪ {∞}. The initial ancestor (i.e. the root), denoted by φ, gives birth to N children with displacements A 1 , A 2 , • • • , A N they form the first generation. Then, for any integer n ≥ 1, each individual in the n-th generation gives birth independently of all others to its own children in the (n + 1)-th generation. Their displacements are given by independent copies of L.

We thus obtain a genealogical tree, denoted by T, which is a Galton-Watson tree with offspring N.

For each vertex (individual or site) z ∈ T, A(z) denotes its displacement and V(z) its position with respect to the root. If y is the parent of z, write ←z = y, also if y is an ancestor of z, write y < z. V can then be written as

V(z) = ∑ φ<y≤z A(y),
with V(φ) = 0. In particular L = {V(z), |z| = 1}, with |z| the generation of z. e -V (u) e -V(z) +∑ v: ← -v =z e -V(v) , if u is a child of z, e -V(z) e -V(z) +∑ v: ← -v =z e -V(v) , if u is the parent of z.

(1.1)

For convenience, we add a parent ←φ to the root and assume that (1.1) holds also for z = φ with p E ( ←φ , φ) = 1. Let P be the probability measure of the environment and P * , the probability conditioned on the survival set of the tree T (which is assumed to be supercritical, see (1.2) below). Let P E , the quenched probability measure of this random walk that is P E (•) := P(•|E ) and P(•) := P E (w) (•)P(dw) the annealed probability measure. Similarly we also define P * with respect to P * .

The walk (X n , n ∈ N * , X 0 = φ) belongs to the family of biased random walks on a tree first introduced by R. Lyons ([20] and [START_REF] Lyons | Random walks, capacity and percolation on trees[END_REF]). In our case where the bias is random, the first references go back to R.

Lyons and R. Pemantle [START_REF] Lyons | Random walk in a random environment and first-passage percolation on trees[END_REF] and M.V. Menshikov and D. Petritis [START_REF] Menshikov | On random walks in random environment on trees and their relationship with multiplicative chaos[END_REF]. These works give a classification of these random walks on a regular tree in term of recurrence criteria, their results are extended lately for Galton-Watson trees by G. Faraud [START_REF] Faraud | A central limit theorem for random walk in a random environment on marked galtonwatson trees[END_REF]. This classification which can be determined from the fluctuations of the log-Laplace transform ψ defined below is resumed in Figure 1. Assume that there exists θ > 0, such that ∀s ∈ [-1, 1 + θ] ψ(s) := log E ∑ |z|=1 e -sV(z) < +∞, where ∑ |z|=k with k ∈ N + means sum over all the individuals z of generation k. In this paper we focus on the boundary case for the environment (in the sense of Biggins-Kyprianou [START_REF] Biggins | Measure change in multitype branching[END_REF]), that is :

> 0 Positive recurrent = 0 ψ (1) ψ(1) inf [0,1] ψ > 0 ≤ 0 Transient Positive Recurrent < 0 = 0 Null recurrent < 0 Null recurrent
(1.2)

E [N] > 1, ψ(1) = log E ∑ |z|=1 e -V(z) = 0, ψ (1) = E ∑ |z|=1 V(z)e -V(z) = 0.
2

Notice that the first hypothesis E [N] > 1 implies that we work on a supercritical Galton-Watson tree. In particular (X n ; n ≥ 0) can not be reduced to the one-dimensional random walk in random environment. Also we need additional hypothesis given below : there exists θ > 0 such that

E ∑ |z|=1 e -(1+θ)V(z) + E ∑ |z|=1 e θV(z) <∞ (1.3) E ∑ |z|=1 (1 + |V(u)|)e -V(u) 2
<∞. (1.4) The hypothesis (1.4) will be required in Lemma 4.3. But the hypothesis (1.3) is more elementary which gives finite exponential moments.

It is proved in [START_REF] Faraud | A central limit theorem for random walk in a random environment on marked galtonwatson trees[END_REF], see also Figure 1, that the random walk X is null recurrent under (1.2). Moreover in this case X is very slow, indeed Y. Hu and Z. Shi [START_REF] Hu | Slow movement of recurrent random walk in random environment on a regular tree[END_REF] (see also [START_REF] Faraud | Almost sure convergence for stochastically biased random walks on trees[END_REF] with G. Faraud ) proved that the largest generation visited up to time n, X * n := max k≤n |X k | behaves in (log n) 3 . In fact it is the slowest null recurrent random walk in random environment on the tree, the other cases that is when ψ (1) < 0 being diffusive or sub-diffusive but without logarithmic behavior (see [START_REF] Hu | A subdiffusive behavior of recurrent random walk in random environment on a regular tree[END_REF], [START_REF] Faraud | A central limit theorem for random walk in a random environment on marked galtonwatson trees[END_REF], [3]). One of the questions raised by the authors at this time was : is (log n) 3 the typical fluctuation of this walk, that is of |X n | for example ? If we now look at the largest generation entirely visited M n := max{k ≥ 1 : {|z| = k} ⊂ {X i ; 0 ≤ i ≤ n}}, then it is of order log n as shown in P. Andreoletti and P. Debs [START_REF] Andreoletti | The number of generations entirely visited for recurrent random walks on random environment[END_REF], and we could also ask here the same question. It turns out that neither of the two is the good answer.

A first result in that direction is obtained in the work of [START_REF] Andreoletti | Spread of visited sites of a random walk along the generations of a branching process[END_REF]. For any z ∈ T, define (1.5)

T z = T1 z := inf{m ≥ 1 : X m = z} and T k z := inf{m ≥ T k-1 z : X m = z}, ∀k ≥ 2.

Then for any generation ≥ 1, the number of sites visited at this generation up to time n is given by

N n ( ) := ∑ |z|= 1 T z <n .
We also introduce the same variable stopped at the n-th return to the root:

K n ( ) := N T n φ ( ).
It is proved in [START_REF] Andreoletti | Spread of visited sites of a random walk along the generations of a branching process[END_REF] that the typical generations which maximise the number of distinct visited sites are of the order (log n) 2 :

(1.6) lim

n→+∞ E K n ((log n) 2 ) E (K n ((log n) 1+ζ )) = ∞, ∀ζ = 1 and E K n ((log n) 2 ) n/ log n 1 .
They also notice that only the sites such that the branching potentiel V(•) is high enough (typically larger than log n) are of importance. That is to say produce the main contribution for E K n ((log n) 2 ) , conversely the sites with low potential are mostly visited but there are very few of them (typically of order n/(log n) 2 compared to n/(log n)). More recently, in [START_REF] Hu | The slow regime of randomly biased walks on trees[END_REF], it is proved that (log n) 2 is actually the right normalisation for the generation of X at the instant n, this unexpected behavior makes us think to the one dimensional case of Sinai's walk [START_REF] Ya | The limit behaviour of a one-dimensional random walk in a random medium[END_REF]. However the walk on the tree has its own particularities, for example, contrarily to the one-dimensional case which remains in the site of low potential, it can reach height of potential of order (log n) 2 (see [START_REF] Hu | The potential energy of biased random walks on trees[END_REF]).

Another motivation, as working on the tree, is to understand more precisely the way the walk spread on the tree so we turn back to the number of distinct visited sites. The main lack in the paper [START_REF] Andreoletti | Spread of visited sites of a random walk along the generations of a branching process[END_REF] is first that nothing precise is said on the behavior in probability of N n (neither for K n ), and that their annealed results say few things on the typical behavior of the potentials leading to this critical (log n) 2 -th generation. Our results here bring answers to these points.

We have split our results into two parts, the first subsection below deals with the normalization for the number of distinct visited sites per critical generation as well as for the total number of distinct visited sites up to time n. The second subsection is devoted to a quenched results making a link between the range of X and the behavior of the environment. In a the third subsection we present the key ideas of proofs.

Annealed results

Our first theorem shows that the behavior in probability of the number of distinct visited sites at critical generations is of order n/(log n) 3 .

Theorem 1.1. For any integers = (n) such that lim n→+∞ (log n) 2 = γ > 0, there exists a positive constant

λ(γ) > 0 such that as n → ∞, (1.7) (log n) 3 n N n ( ) in P * ---→ λ(γ)σ 2 4 ,
where

σ 2 := E ∑ |x|=1 V 2 (x)e -V(x) ∈ (0, ∞) by (1.3).
The function λ(γ) can be written explicitly (see below (1.21)), it is related to the convergence of variables depending only on the environment. This theorem is the consequence of the behaviors of K n and of the local time at the root. To be more precise, let us introduce the derivative martingale (D m , m) given by

D m := ∑ |z|=m V(z)e -V(z) , (1.8)
and denote its almost sure limit by D ∞ (see [START_REF] Biggins | Measure change in multitype branching[END_REF] for its existence and [START_REF] Chen | A necessary and sufficient condition for the non-a necessary and sufficient condition for the non-trivial limit of the derivative martingale in a branching random walk[END_REF] for its positivity under P * ). The behavior in probability of K n is given by Theorem 1.2. For any = (n

) such that lim n→+∞ (log n) 2 = γ > 0, (1.9) (log n) 2 n K n ( ) in P * ---→ λ(γ)p E (φ, ← - φ )D ∞ .
If we compare this results with the behavior in mean (see (1.6)), a multiplicative (log n) appears. It comes from the behavior of the branching potential which typically remains positive in probability (see 2.1) reducing the number of possible visited sites.

Also the main difference between N n ( ) and K n ( ) comes essentially from the normalisation. The additional log n which appears above for K n ( ) comes from the local time of X at the root of the tree, it is indeed proved in [START_REF] Hu | The slow regime of randomly biased walks on trees[END_REF]:

Proposition 1.3 ([18]). (1.10) T n φ n log n in P * ---→ 4D ∞ p E (φ, ← - φ )/σ 2 .
Instead of one critical generation, we now turn to consider the total number of visited sites, in other words, the range of the random walk:

R n := ∑ z∈T 1 {T z ≤n} .
Following (1.6) and Theorem 1.1 we can ask wether or not critical generations contribute mainly to R n ? The answer is yes : Proposition 1.4 below states that for non-critical generations, the total number of visited sites contributes to something negligible compared to n/ log n, while the range R n is of order n/ log n in probability, as stated in Theorem 1.5.

Proposition 1.4. For any δ > 0,

lim ε→0 lim sup n→∞ P ∑ |z|≤ε(log n) 2 1 {T z ≤n} + ∑ |z|≥(log n) 2 /ε 1 {T z ≤n} ≥ δn/ log n = 0.
So as the main contribution comes from generations of order (log n) 2 , we have that with high probability,

R n ≈ ∑ ε(log n) 2 ≤ ≤(log n) 2 /ε N n ( ) with ε ↓ 0.
As a consequence we obtain the following result for the range of X : Theorem 1.5. We have

(1.11) log n n R n in P * ---→ σ 2 4 Λ,
where

Λ := ∞ 0 λ(γ)dγ ∈ (0, ∞).
Remark 1.6. In fact, once again by Proposition 1.3, this theorem follows from the following convergence:

R T n φ n in P * ---→ Λp E (φ, ← - φ )D ∞ .
Also, the integrability of λ is stated in Lemma A.1 of the Appendix

These first results give a quantitative description of the number of visited sites and of the generations involved, but no description of the underlying environment is given. In the following section we discuss what we have learnt about the typical behavior of the potential that leads to the above behavior of R n .

A quenched point of view

Like we said in the first part of the introduction, Andreoletti-Debs [START_REF] Andreoletti | Spread of visited sites of a random walk along the generations of a branching process[END_REF] observe that the sites where the potential remains small (always lower than log n) have a negligible contribution for the number of visited sites. One of the reasons for this is the fact that the number of such sites is actually negligible on the tree (see their Proposition 1.3). Intuitively these sites are easily accessible as the potential remains low, but the set of these sites still has a low conductance.

Here we give some more details of the sites that the random walk is inclined to visit, i.e. the sites that contribute importantly to the range.

For sites y, z ∈ T, recall that y ≤ z means that y belongs to the shortest path from the root φ to z. Let V(z) := max φ<y≤z V(y). Define for any a 0 > 1,

A 1 := z ∈ T : log n a 0 ≤ max φ<y≤z V(y) -V(y) ≤ log n + g(n) ,
where {g(n), n} is a positive increasing function such that lim n→+∞ (g(n)log log n) = +∞. Moreover, for any a 1 > 0, let

A 2 := z ∈ T : log n + log log n ≤ V(z) ≤ a 1 log n log log n ,
and

A 3 := z ∈ T : V(z) > max y≤z;|y|≤|z|-|z| 1/3 V(y) .
Let us introduce a notation for truncated versions of K n , R n and their quenched mean : if A is an event depending only on the environment E , then for any ≥ 1,

K A n ( ) := ∑ |z|= 1 {T z <T n φ } 1 {z∈A} , R A T n φ := ∑ m≥0 K A n (m), (1.12) K A n ( ) := E E K A n ( ) , R A T n φ := E E R A T n φ . (1.13)
Notice that the above means are easily computable (see section 2), but we are not interested in their expressions for now. The following result proves tightness of the range up to T n φ minus the truncated quenched mean of R T n φ : R A 1 ∩A 2 ∩A 3 n , this makes appear favorite environments described by potential V.

Proposition 1.7. For any η > 0, there exists a 1 > 0 such that

lim a 0 →+∞ lim sup n→+∞ P * 1 n R T n φ -R A 1 ∩A 2 ∩A 3 T n φ ≥ η = 0.
From this result together with the well known fact in [START_REF] Aidékon | Convergence in law of the minimum of a branching random walk[END_REF] about the potential : P(inf z∈T V(z) ≥ -α) ≥ 1e -α , we are able to draw a typical trajectory of potential that maximises the number of visited sites (see Figure 2).

a1 log n log log n V(z) ≤ 1/3 |z| = ∼ γ(log n) 2 log n/a0 ≤ ≤ log n + g(n) log n + log log n

Sketch of proofs and organization of the paper

As we have already seen, Theorem 1.1, comes from Proposition 1.3 together with Remark 1.6, so only the remark as to be proved. Also thanks to Proposition 1.4 (which is a consequence of Lemma 2.4 for which the proof is posponed in Section 4.2.4) together with Proposition 1.3, only the critical generations of order (log n) 2 have to be considered. For that we first study individually each of these generations which is the purpose of Theorem 1.2 : Skech of proof of Theorem 1.2 : The first step for the study of K n ( ) for ∼ γ(log n) 2 is to compare it with its quenched expectation K n ( )

:= E E [K n ( )].
The main idea here is simple : we would like to apply Tchebychev's inequality to the quenched probability P E (|K n ( ) -K n ( )| ≥ εK n ( )). Unfortunately this gives nothing usable if we do it directly. Indeed it turns out that the quenched variance V E ar(K n ( )) which appears when applying this inequality can not be controlled properly with respect to measure P * . In order to overcome this, we add restrictions to the environment, the first one comes from the reflecting barrier introduced by [START_REF] Hu | The slow regime of randomly biased walks on trees[END_REF]

] : let δ > 0 introduce L δ := {z ∈ T : max φ<y≤z (V(y) -V(y)) ≤ log n -(1 + δ) log log n}.
In other words, we consider the restriction of K n to the sites of L δ , that is to say y) , (1.14) obtained by the strong Markov property, also the last equality in 1.14 comes from Lemma C.1 in [START_REF] Andreoletti | Spread of visited sites of a random walk along the generations of a branching process[END_REF].

K L δ n ( ) = ∑ |z|= 1 {T z <T n φ } 1 {z∈L δ } and its quenched mean K L δ n ( ) := ∑ |z|= P E (T z < T n φ )1 {z∈L δ } = ∑ |z|= (1 -(1 -a z ) n )1 {z∈L δ } , where a z := P E φ (T z < T φ ) = p E (φ, z 1 )P E z 1 (T z < T φ ) = p E (φ, ← - φ ) ∑ φ<y≤z e V(
Then, following the ideas of [START_REF] Andreoletti | Spread of visited sites of a random walk along the generations of a branching process[END_REF], we add a second restriction by defining the set U := {z ∈ T : V(z) ≥ log n + log log n}. This restriction, which comes from the fact that only sites with a high level of potential count, contributes to a simplification of the expression of the quenched mean defined above: for

any z ∈ U, a z ≤ e -V(z) ≤ 1 n log n , thus (1.15) 0 ≤ na z -[1 -(1 -a z ) n ] ≤ n 2 a 2 z ≤ 1 log n na z , so in particular 1 -(1 -a z ) n = (1 + o n (1)
)na z , and for any event D ⊆ {z ∈ T : V(z) ≥ log n + log log n} depending only on the environment

K D n ( ) = E E ∑ |z|= 1 {T z <T n φ } 1 {z∈D} ∼ n ∑ |z|= a z 1 {z∈D} =: K D n ( ). (1.16)
We prove rigorously, in Subsection 2.1, that the cost of these restrictions L δ ∩ U is negligible for the number of distinct visited sites before n return to the origin (see Lemmata 2.1 and 2.2). So we are left to study the restriction K U∩L δ n ( ). For that we apply Tchebychev's inequality (see Section 2.2) and, thanks to the restriction, the expectation with respect to measure P * of the quenched variance (Section 4.1)

V E ar(K U∩L δ n ( )
) is well controlled. Finally we obtain that in probability K n ( ) can be approximated for large n by K U∩L δ n ( ) :

K n ( ) P ∼ K U∩L δ n ( ) = ∑ |z|= na z 1 {z∈L δ ∩U} = np E (φ, ← - φ ) × ∑ |z|= e -V(z) e V(z) ∑ φ<y≤z e V(y) 1 {z∈U∩L δ } . (1.17)
The second step is to obtain the convergence of (log n) 2 K U∩L δ n ( )/n to some non trivial limit under P * .

For that we introduce the following martingale-like variable, for any m ≥ 1 and a, b ≥ 0,

W m (F a,b ) := ∑ |z|=m e -V(z) F a,b (z), where F a,b (z) := √ m e V(z) ∑ φ<y≤z e V(y) 1 {V(z)≥b} 1 {max φ<y≤z (V(y)-V(y))≤a} . (1.18) With this notation K U∩L δ n ( ) can be re-write, (1.19) K U∩L δ n ( ) = np E (φ, ← - φ ) √ W (F log n-(1+δ) log log n,log n+log log n ).
Notice that if F a,b (z) = 1 for any of its arguments, then W m (F a,b ) is exactly the well-known additive martingale W m := ∑ |z|=m e -V(z) . Aidékon and Shi [START_REF] Aidekon | The seneta-heyde scaling for the branching random walk[END_REF] showed that √ mW m converge in P * -probability to the positive martingale D ∞ =: lim m→∞ ∑ |z|=m V(z)e -V(z) . More recently Madaule [23] proved that if one chooses one site z at the m-th generation, according to the measure e -V(z) /W m , the corresponding rescaled trajectory (V(y)1 {|y|= mt ,y≤z} / √ m) 0≤t≤1 is asymptotically a Brownian meander.

Unfortunately in our case F a m ,b m (z) is not simply a functional of this rescaled trajectory, so their results cannot be applied directly. However, our proof of Proposition 1.8 below (see Section 3.2) is mainly inspired by their arguments. We are going to take a = O( √ m), and the factor √ m is used to "balance"

e V(z)
∑ φ<y≤z e V(y) 1 V(z)≥b .

m √ m = b ∈ R + , then as m → ∞, there exists C a,b ∈ (0, ∞) such that (1.20) √ mW m (F a m ,b m ) in P * ---→ m→∞ C a,b D ∞ . see (1.8) for definition of D ∞ . C a,
K U∩L δ n ( ) np E (φ, ← - φ ) in P * --→ C γ -1/2 ,γ -1/2 D ∞ .
Then (1.17)

∈ R 2 + and that W m (F a √ m,b √ m ) is also monotone on (a, b) ∈ R 2 + . It follows that (1.20) holds uniformly for W m (F a √ m,b √ m ) in (a, b) ∈ R 2
+ in the following sense: for any ε > 0,

(1.22) lim m→∞ P * sup a≥0,b≥0 √ mW m (F a √ m,b √ m ) -C a,b D ∞ ≥ ε = 0.
This induces the following corollary which proof can be found Section 3.3.

Corollary 1.9.

(1.23) lim β→∞ ∞ ∑ m=1 W m (F β,β ) √ m = ΛD ∞ , in P * -probability with Λ = ∞ 0 C 1 √ x , 1 √ x dx x .
This corollary still holds if we replace F β,β by F β±O(log β),β in the sum. This result brings out Remark 1.6 and therefore Theorem 1.5.

Remark 1.10. (1.22) suggests that uniformity may also occur in probability for K n ( ), meaning that the "for any " in Theorem 1.1 could actually be placed inside the probability. Unfortunately, this uniformity can not be obtained by the way of our proofs and we believe in fact that this is not true and that the right normalisation for max N n ( ) could be different from n/(log n) 3 .

The rest of the paper is organized as follows :

In Section 2 we use results of Sections 3 and 4 to give the main steps of the proofs of theorems and propositions stated in Section 1.1. In Section 3 we focus on the environment and show Proposition 1.8

and Corollary 1.9. This section is independent of the other sections and uses only the Appendix. In Section 4 we compute the annealed mean of K n and give an upper bound for the mean of the quenched variance. Also we prove lemmata used in section 2 and finish with the proof of Proposition 1.7. In the Appendix we collect and prove many estimations for centered random walk with i.i.d increments.

In this paper, we use c or c for constants which may change from line to line. We write c(x) when that constant depends on some parameter x.

Proof of the theorems

This section is devoted to proving Theorems 1.2 and 1.5, i.e. the convergence in probability of K n ( ). Theorem 1.1 follows immediately from Theorem 1.2 and Proposition 1.3, so we feel free to omit its proof. Recall that for convenience, we fixe some γ ∈ (0, ∞) and always write for the integer sequence

{ (n); n ≥ 1} such that lim n→+∞ (n)/(log n) 2 = γ.
Our arguments are based on the study of truncated versions of K n . This decomposition of K n appears naturally when computing the mean of K n as well as the mean of its quenched variance. We therefore start with this decomposition.

Quenched expectation and truncated versions of K n ( )

For any measurable event C obtained from the environment, the number of visited sites at generation up to the n-th return to φ can be written as

K n ( ) = K C n ( ) + K C c n ( ) = ∑ |z|= 1 {T z <T n φ } 1 {z∈C} + ∑ |z|= 1 {T z <T n φ } 1 {z∈C c } .
To exclude the sites in C that make few contribution to K n , we add restrictions for the potentials on the above sum. First (see [START_REF] Aidékon | Convergence in law of the minimum of a branching random walk[END_REF]) for any ε > 0, we can choose α > 0 such that (2.1)

P inf u∈T V(u) < -α ≤ e -α ≤ ε.
Let V(z) := min φ<y≤z V(y), it is then natural to consider the set

B 1 := {z ∈ T : V(z) ≥ -α}.
Secondly, in [START_REF] Hu | The slow regime of randomly biased walks on trees[END_REF], a reflecting barrier is introduced by

L r := z ∈ T : ∑ φ<u≤z e V(u)-V(z) > r, max φ<y<z ∑ φ<u≤y e V(u)-V(y) ≤ r with r > 0.
This reflecting barrier allows to reduce the number of interesting sites for the walk in the following sense : let f be a positive increasing function such that lim n→+∞ f (n) = +∞, then

lim n→+∞ P ∃k ≤ T n φ , X k ∈ L n f (n) log n = 0. (2.2)
The above result is a direct consequence of Theorem 2.8 (in [START_REF] Hu | The slow regime of randomly biased walks on trees[END_REF]) together with Proposition 1.3. Following this idea, we introduce the following sets

B 2 := z ∈ T : max φ<y≤z ∑ φ<u≤y e V(u)-V(y) ≤ n =: {z ∈ T : z < L n }, then according to (2.2) lim n→+∞ P ∀k ≤ T n φ , X k ∈ B 2 = 1.
Also, for any δ > 0 let s n := n(log n) -1-δ and

B δ 2 := z ∈ T : max φ<y≤z ∑ φ<u≤y e V(u)-V(y) ≤ s n = {z ∈ T : z < L s n }.
We will see that for our specific problem, we can restrict the set B 2 to B δ 2 for well chosen δ. For convenience, denote

B := B 1 ∩ B 2 and B δ := B 1 ∩ B δ 2 .
Because of (2.1) and (2.2), one sees that with high probability, K n ( )

∼ K B n ( ) = ∑ |z|= 1 {T z <T n φ } 1 {z∈B} . Moreover, if z ∈ B δ 2 , we have z ∈ L δ (recall the definition of L δ just above 1.14), and conversely, if z ∈ L δ+2 , then z ∈ B δ
2 . Also we add the last restriction over the values of V: U = {z ∈ T : V(z) ≥ log n + log log n}. The following lemma shows that the cost of this restriction is negligible. Lemma 2.1.

(2.3) E K B\U n ( ) = o E[K B∩U n ( )] = o n (log n) 2 .

Our arguments will show indeed that E[K

B∩U n ( )] = Θ( n (log n) 2
), so that the sites in B ∩ U mainly contribute. We postpone the proof of this lemma to Section 4.2.

Here is our strategy to obtain Theorem 1.1. We first show that for suitable δ > 0, with high probability, K n ( ) ≈ K B∩U n ( ) ≈ K B δ ∩U n ( ), while the last quantity can be approached by its quenched mean by bounding its quenched variance. This observation combined with the fact that the quenched mean K B δ ∩U n ( ) converges in probability because of Proposition 1.8, imply our theorem.

We stress on the fact that replacement of B by B δ helps to correctly bound the quenched variance, it appears that the price of this replacement is negligible, as shown in the following Lemma: Lemma 2.2. For any δ > 0, we have

(2.4) E[K B∩U n ( ) -K B δ ∩U n ( )] = o n (log n) 2 .
The next step is to approach

K B δ ∩U n ( ) by its quenched mean K B δ ∩U n ( ), or more conveniently by K B δ ∩U n ( ) = ∑ |z|= na z 1 {z∈B δ ∩U} . Notice indeed that, in view of (1.15), we have (2.5) 0 ≤ K B δ ∩U n ( ) -K B δ ∩U n ( ) ≤ 1 log n K B δ ∩U n ( ).
Proposition 2.3. For any η > 0 and δ > 3,

lim n→+∞ P |K B δ ∩U n ( ) -K B δ ∩U n ( )| ≥ η n (log n) 2 = 0. (2.6)
The proof of this proposition can be found in Section 4.2, now with these restrictions introduced, we are ready to prove the theorems.

Convergence of K n ( ) and R n : proofs of Theorems 1.2 and 1.5

We are now ready to prove Theorem 1.2: it suffices to show that for any η > 0, (2.7)

lim n→+∞ P (log n) 2 n K n ( ) -λ(γ)p E (φ, ← - φ )D ∞ ≥ η = 0.
Proof of (2.7). Let p n := P

(log n) 2 n K n ( ) -λ(γ)p E (φ, ← - φ )D ∞ ≥ η . We first add the restrictions B 1 and B 2 ( recalling that B = B 1 ∩ B 2 ). For that let us introduce the events B 1 := {inf u∈T V(u) ≥ -α}, B 2 := { n i=1 {X i < L n }} and for any x > 0 and random variable H, B 3 (H, x) := (log n) 2 n H - λ(γ)p E (φ, ← - φ )D ∞ ≥ x . We have p n ≤ P B 1 + P B 2 + P(B 3 (K B n ( ), η)). That is to say using (2.2), lim sup n→∞ p n ≤ P B 1 + lim sup n P(B 3 (K B n ( ), η)).
For the second term on the right hand side of the previous inequality, we involve the restrictions B δ 2 and U, it then follows from (2.3) and (2.4) that lim sup n→∞ p n

≤ P B 1 + lim sup n P(B 3 (K B δ ∩U n ( ), η/2). Then by Proposition 2.3, we can use K B δ ∩U n ( ) to approach K B δ ∩U n ( ) and obtain that lim sup n→∞ p n ≤ P B 1 + lim sup n P(B 3 ( K B δ ∩U n ( ), η/4)).
By releasing the restriction B 1 , one gets that lim sup n→∞ p n ≤ 2P B 1 + lim sup n P(B 3 ( K

B δ 2 ∩U n ( ), η/4).
Recall that by definition (see above (1.14), and below (2.2)

) L δ+2 ⊂ B δ 2 ⊂ L δ so clearly K L δ+2 ∩U n ( ) ≤ K B δ 2 ∩U n ( ) ≤ K L δ ∩U n ( ). So by (1.21) lim sup n→∞ p n ≤ 2P(B 1 ). Letting α ↑ ∞, we deduce (2.7) from (2.

1).

It remains to show the convergence of the range R n , that is Theorem 1.5. As mentioned in Remark 1.6, by Proposition 1.3, we only need to prove that

(2.8) R T n φ n in P * ---→ Λp E (φ, ← - φ )D ∞ , with R T n φ = ∑ ∞ m=0 K n (m)
. First, we claim that only the critical generations really count in this sum, and that the truncated version of (K n (m), m) make the main contribution : Lemma 2.4. We have

(2.9) lim ε↓0 lim sup n→∞ 1 n E ∑ m≤ε(log n) 2 K B n (m) + E ∑ m≥(log n) 2 /ε K B n (m) = 0,
and for any ε > 0,

lim n→∞ 1 n E (log n) 2 /ε ∑ m=ε(log n) 2 K B\U n (m) + E (log n) 2 /ε ∑ m=ε(log n) 2 K (B∩U)\(B δ ∩U) n (m) = 0. (2.10)
The proof of this Lemma is postponed in Section 4.2.4. Notice here that Proposition 1.4 follows from (2.9) and Proposition 1.3. As non-critical generations are negligible, we can borrow the previous arguments for K n ( ) to show the convergence for R T n φ .

Proof of Theorem 1.5 (i.e. (2.8)). For any η > 0, let us consider

P |R T n φ -Λp E (φ, ← - φ )D ∞ n| ≥ ηn .
Considering restrictions B 1 and B 2 , one sees that for any α > 0,

P |R T n φ -Λp E (φ, ← - φ )D ∞ n| ≥ ηn ≤ P(B 1 ) + P(B 2 ) + P ∞ ∑ m=0 K B n (m) -Λp E (φ, ← - φ )D ∞ n ≥ ηn . By (2.1) and (2.2), (2.11) lim sup n→∞ P |R T n φ -Λp E (φ, ← - φ )D ∞ n| ≥ ηn ≤ e -α + lim sup n→∞ P ∞ ∑ m=0 K B n (m) -Λp E (φ, ← - φ )D ∞ n ≥ ηn .
For the K n (m), we only need to consider the generations m of order (log n) 2 . For any ε > 0, define for any x > 0 and random variables (

H(m), m ≥ 0), B 4 (H, x) := | ∑ (log n) 2 /ε m=ε(log n) 2 H(m) -Λp E (φ, ← - φ )D ∞ n ≥ xn , we have P ∞ ∑ m=1 K B n (m) -Λp E (φ, ← - φ )D ∞ n ≥ ηn ≤ P ∑ m≥(log n) 2 /ε, or m≤ε(log n) 2 K B n (m) ≥ ηn/2 + P(B 4 (K B n , η/2)),
where the first probability on the right hand side is negligible because of (2.9). For the second probability, we consider only the sites z ∈ B δ ∩ U and obtain that

P(B 4 (K B n (m), η/2)) ≤ P (log n) 2 /ε ∑ m=ε(log n) 2 K B\U n (m) ≥ ηn/6 + P (log n) 2 /ε ∑ m=ε(log n) 2 K (B∩U)\(B δ ∩U) n (m) ≥ ηn/6 + P B 4 (K B δ ∩U n , η/6) .
In view of (2.10) together with (2.9), we obtain that (2.12) lim sup

n→∞ P ∞ ∑ m=0 K B n (m) -Λp E (φ, ← - φ )D ∞ n ≥ ηn ≤ o ε (1) + lim sup n→∞ P(B 4 (K B δ ∩U n , η/6)).
It remains to bound the second term on the right hand side. Recall that the quenched expectation of

K B δ ∩U n (m) is denoted K B δ ∩U n (m), introducing the variable ∆ n (H, G) := ∑ (log n) 2 /ε m=ε(log n) 2 H(m) -G(m) for any sequences (H(m), G(m), m ≥ 0), we can write P(B 4 (K B δ ∩U n , η/6)) ≤ P(∆ n (K B δ ∩U n , K B δ ∩U n )| ≥ ηn/12) + P(B 4 (K B δ ∩U n , η/12)). (2.13) First, by Markov inequality, P ∆ n (K B δ ∩U n , K B δ ∩U n )| ≥ ηn/12 ≤ 144(η 2 n 2 ) -1 E ∆ n (K B δ ∩U n , K B δ ∩U n ) 2 ,
which by Cauchy-Schwartz inequality is bounded by 144

η 2 n 2 (log n) 2 /ε ∑ m=ε(log n) 2 1 (log n) 2 /ε ∑ m=ε(log n) 2 E Var E (K B δ ∩U n (m)) .
Applying Lemma 4.3 with δ > 5 to this term implies that (2.14) lim sup

n→∞ P ∆ n (K B δ ∩U n , K B δ ∩U n ) ≥ ηn 12 = 0. Second, by replacing K B δ ∩U n by K B δ ∩U n (recall the definition of K in (1. 16
)), one sees that

P(B 4 (K B δ ∩U n , η/12)) ≤P ∆ n (K B δ ∩U n (m), K B δ ∩U n ) ≥ ηn/24 + P(B 4 ( K B δ ∩U n , η/24)) ≤P   (log n) 2 /ε ∑ m=ε(log n) 2 1 log n K B δ ∩U n (m) ≥ ηn 24   + P(B 4 ( K B δ ∩U n , η/24)), =:RH 1 + P(B 4 ( K B δ ∩U n , η/24))
where the last inequality follows from (2.5). Plugging this inequality and (2.14) into (2.13) yields

P(B 4 (K B δ ∩U n (m), η/6)) ≤ o n (1) + RH 1 + P B 4 ( K B δ ∩U n , η/24) .
Now, observe that

RH 1 ≤ P B 4 ( K B δ ∩U n , η/24) + P Λp E (φ, ← - φ )D ∞ n ≥ ηn(log n -1)/24 ,
where the second probability on the right hand side vanishes as n → ∞ because p E (φ, ←φ )D ∞ is finite P-a.s. So moving back to (2.12), we deduce that

(2.15) lim sup n→∞ P ∞ ∑ m=1 K B n (m) -Λp E (φ, ← - φ )D ∞ n ≥ ηn ≤ o ε (1) + 2P B 4 ( K B δ ∩U n , η/24) .
So in view of (2.11) and (2.15), we have lim sup

n→∞ P |R T n φ -Λp E (φ, ← - φ )D ∞ n| ≥ ηn ≤ e -α + o ε (1) + 2P B 4 ( K B δ ∩U n , η/24
V(u) ≥ -α}, K B δ ∩U n (m) = K B δ 2 ∩U n (m) for any m ≥ 0, hence P B 4 ( K B δ ∩U n , η/24) ≤ P(B 1 ) + P B 4 ( K B δ 2 ∩U n
, η/48) where the first term on the right hand side is bounded by e -α . So again by Lemma 2.4, we have

P B 4 ( K B δ ∩U n (m), η/24) ≤ e -α + o n,ε (1) + P ∞ ∑ m=1 K B δ 2 ∩U n (m) -Λp E (φ, ← - φ )D ∞ n ≥ ηn/96 . Recall that L δ+2 ⊂ B δ 2 ⊂ L δ , by (1.19), we have (2.18) W m (F log n-(3+δ) log log n,log n+log log n ) √ m ≤ K B δ 2 ∩U n (m) np E (φ, ← - φ ) ≤ W m (F log n-(1+δ) log log n,log n+log log n ) √ m .
Finally, (2.17) follows immediately from Corollary 1.9 .

Convergence of martingale-like variables

(W m (F a m ,b m ), m ≥ 1)
This section is devoted to proving Proposition 1.8 and Corollary 1.9 which only concern the environment. The main idea is borrowed from [START_REF] Aidekon | The seneta-heyde scaling for the branching random walk[END_REF], on the Seneta-Heyde norm of the additive martingale W m in the boundary case (1.2). To do so, we need to introduce a change of measure and the corresponding spinal decomposition.

Lyons' change of measures and spinal decomposition

We begin with the following Biggins-Kyprianou [START_REF] Biggins | Senata-heyde norming in the branching random walk[END_REF] identity usually called many-to-one Lemma : Lemma 3.1. In the boundary case (1.2), there exists a sequence of i.i.d. real-valued random variables (S i -S i-1 , i ≥ 0) with S 0 = 0 such that for any n ≥ 1 and any Borel function g : R n → R + ,

(3.1) E ∑ |x|=n g V(x i ), 1 ≤ i ≤ m = E e S n g(S i ; 1 ≤ i ≤ n) .
It immediately follows from (1.2) and (1.3) that the sequence (S n , n ≥ 0) is a centered random walk of finite variance z) ]. For notational simplicity, let

σ 2 := E[∑ |z|=1 V(z) 2 e -V(
S n := min 1≤i≤n S i , S n := max 1≤i≤n S i .
Also let R(•) be the renewal function associated with the strict descending ladder heights of (S i , i), it can be expressed as

(3.2) R(u) = ∞ ∑ k=0 P(S k < S k-1 , S k ≥ -u), ∀u ≥ 0. Obviously, R(u) ≥ R(0) = 1.
The renewal theorem implies the existence of c 0 ∈ (0, +∞) such that

c 0 := lim u→+∞ R(u) u . (3.3)
Moreover, there exist two constants 0 < C -< C + < ∞ such that for any u ≥ 0,

(3.4) C -(1 + u) ≤ R(u) ≤ C + (1 + u).
For α > 0, define the truncated variables adapted to {F n := σ((z, V(z)); |z| ≤ n); n ≥ 0}, the natural filtration of the branching random walk, for any n ≥ 0 :

W (α) n (F a n ,b n ) := ∑ |z|=n e -V(z) F a n ,b n (z)1 {V(z)≥-α} , D (α) n := ∑ |z|=n R(α + V(z))e -V(z) 1 {V(z)≥-α} .
See (1.18) for the definition of F a n ,b n (z). For any a ∈ R, let P a be the probability measure such that P a ({V(z), z ∈ T} ∈ •) = P({a + V(z), z ∈ T} ∈ •). For a ≥ -α, we introduce the change of measure as follows:

(3.5) Q (α) a | F n := D (α) n R(α + a)e -a P a | F n . The fact that D (α)
n is a non-negative martingale which converges a.s. to some limit D a is well define. Following their ideas, we present a spinal decomposition of the branching random walk under Q (α) a : we start with one individual w 0 (i.e., the root φ ), located at V(w 0 ) = a. Then for any n ≥ 0, 1. in the n-th generation, each individual u except w n , gives birth independently of all others to its children of the n + 1-th generation whose positions constitute a point process distributed as (V(z), |z| = 1) under P V(u) ; 2. w n produces, independently, its children in the n + 1-th generation, whose positions are given by a point process distributed as (V(z), |z| = 1) under

Q (α) V(w n ) ;
3. Among the children of w n , w n+1 is chosen to be z with probability proportional to

R(α + V(z))e -V(z) 1 {V(z)≥-α} .
In this description, the infinite ray (w n , n ≥ 0) is called the spine under

Q (α)
a . For simplicity, we write

Q (α) for Q (α) 0 .
The following fact makes explicit the distribution of ω n and (V(w k ),

1 ≤ k ≤ n) under Q (α) . Fact 3.2 ([9]). Assume (1.2). Let α ≥ 0, for any n ≥ 1 and |z| = n, (3.6) Q (α) (w n = z|F n ) = R(α + V(z))e -V(z) 1 {V(z)≥-α} D (α) n .
The spine process (V(w n ), m ≥ 0) under Q (α) is distributed as the random walk (S n , n ≥ 0) under P conditioned to stay above -α. In other words, for any n ≥ 1 and any measurable function g : R n → R + ,

(3.7) E Q (α) g(V(w k ), 1 ≤ k ≤ n) = 1 R(α) E g(S k , 1 ≤ k ≤ n)R(α + S n ); S n ≥ -α .

Convergence in probability of

W (α) n (F)/D (α)
n under Q (α) In this section we prove that if

a n = a √ n + o( √ n) and b n = b √ n + o( √ n) for some a, b > 0, then there exists some constant C a,b ∈ (0, ∞) such that under Q (α) , (3.8) √ n W (α) n (F a n ,b n ) D (α) n -→ C a,b , in probability.
This convergence also holds for b = 0. When a = 0, C a,b is trivially zero by first moment estimation.

It is known that lim n→∞ min |z|=n V(z) = ∞, P-a.s. As a consequence of (3.3), D The proof of (3.8) is based on the computations of the first and second moments of

∞ = c 0 D ∞ on {inf z∈T V(z) ≥ -α}. (α)
W (α) n (F an,bn ) D (α) n
. By (3.6), for any measurable function F : R n → R + of the trajectory of V that is that F(z) = F(V(y); φ < y ≤ z), we have

(3.9) E Q (α) F(w n ) R(α + V(w n )) F n = ∑ |z|=n e -V(z) F(z)1 {V(z)≥-α} D (α) n =: W (α) n (F) D (α) n .
Taking expectation under Q (α) then applying (3.7) yields that

(3.10) E Q (α) W (α) n (F) D (α) n = E Q (α) F(w n ) R(α + V(w n )) = 1 R(α) E F(S k ; 1 ≤ k ≤ n); S n ≥ -α
Recall that for |z| = n (see (1.18))

F a n ,b n (z) = √ n e V(z) ∑ φ<y≤z e V(y) 1 {V(z)≥b n } 1 {max φ<y≤z (V(y)-V(y))≤a n } .
In order to deal with the factor e V(z)

∑ φ<y≤z e V(y) , we have to add some restrictions to the sites. Observe that if

V(z) V(z), then e V(z) ∑ φ<y≤z e V(y) ≤ e V(z)-V(z)
1.

So it is reasonable to count only the sites |z| = n such that V(z) ≈ V(z). And this choice gives an extra factor 1

Υ z := inf{k : V(z k ) = V(z) = max 0≤m≤n V(z m )}.
Similarly, we also define Υ S = inf{k : S k = S(k) := max 0≤m≤n S m } and S [m,n] := min m≤k≤n S k for one-dimensional random walk (S k , k). Instead of F a n ,b n (z), it is more convenient to consider

G(z) := √ n e V(z) ∑ φ<y≤z e V(y) 1 {V(z)≥b n , max y≤z (V(y)-V(y))≤a n } 1 {Υ z >n 0 } , (3.11)
with n 0 := nn 1/3 . Moreover, following [START_REF] Aidekon | The seneta-heyde scaling for the branching random walk[END_REF], let us introduce the events E z n for |z| = n as follows. Let Ω(y)

:= {u ∈ T : u = y, ← - u = ← -y } be the collection of brothers of y. If (k n , n) is a positive sequence such that k n = o(n 1/2 ) and (log n) 6 = o(k n ), let E z n := E z n,1 ∩ E z n,2 ∩ E z n,3
, where

E z n,1 = {k 1/3 n ≤ V(z k n ) ≤ k n } ∩ n i=k n {V(z i ) ≥ k 1/6 n }; E z n,2 = n i=k n { ∑ y∈Ω(z i+1 ) [1 + (V(y) -V(z i )) + ]e -[V(y)-V(z i )] ≤ e V(z i )/2 }; E z n,3 = { n ∑ i=k n ∑ y∈Ω(z i+1 ) ∑ |u|=n,u≥y R(α + V(u))e -V(u) 1 {V(u)≥-α} ≤ 1 n 2 }, (3.12
) with x + := max(x, 0). In particular, for w n , write E n (resp. E n,i ) instead of E w n n (resp.

E w n n,i ). Let H(z) := G(z)1 E z n .
Here we choose k n = o(n 1/2 ) so that E n,1 happens with high probability and ( V(w i )

√ n ; k n ≤ i ≤ n)
is still asymptotically Brownian meander. At the same time, we take (log n) 6 = o(k n ) to make sure that the probability in (3.30) is o n [START_REF] Afanasyev | A limit theorem for a critical branching process in a random environment[END_REF]. Moreover, it is proved in Lemma 4.7 of [START_REF] Aidekon | The seneta-heyde scaling for the branching random walk[END_REF] that for (k n , n) chosen as stated above,

lim n→∞ Q (α) (E n ) =1, lim n→∞ inf u∈[k 1/3 n ,k n ] Q (α) (E n |V(w k n ) = u) =1. (3.13)
One will see later that involving E n helps us to control the second moment of

W (α) n (F) D (α) n
without influencing its first moment. Let us now state the main lemma of this section.

Lemma 3.3. Let α ≥ 0, we have lim n √ nE Q (α) W (α) n (H) D (α) n = lim n √ nE Q (α) W (α) n (F a n ,b n ) D (α) n = C a,b , (3.14) lim n E Q (α) √ n W (α) n (H) D (α) n 2 = C 2 a,b . (3.15)
This lemma shows immediately that under [START_REF] Afanasyev | A limit theorem for a critical branching process in a random environment[END_REF] in probability. We hence conclude the convergence (3.8).

Q (α) , √ n W (α) n (H) D (α) n converges in probability towards C a,b while √ n W (α) n (F an,bn -H) D (α) n = o n
Moreover, by the change of measures (3.5), this means that

(3.16) √ nE[W (α) n (F a n ,b n )] → C a,b R(α).
Before starting the proof of Lemma 3.3, let us state a useful result on the random walk {S k ; k ≥ 0} and the definition of constants C a,b , C a,b and λ(•).

It is proved in [START_REF] Afanasyev | A limit theorem for a critical branching process in a random environment[END_REF] that the following joint convergence in law holds (3.17)

S nt √ σ 2 n , t ∈ [0, 1] ; n ∑ i=0 e -S i S n > 0 =⇒ {(m t , t ∈ [0, 1]), H ∞ }, where (m t , t ∈ [0, 1]) is a Brownian meander independent of H ∞ ∈ [1, ∞).
In fact, in the sense of [START_REF] Bertoin | On conditioning a random walk to stay non-negative[END_REF], the associated random walk conditioned to stay positive, denoted (ζ n , n ≥ 0), is a Markov chain with probabilities of transition p(x, dy) :=

R(y) R(x) 1 {y>0} P x (S 1 ∈ dy), with P(ζ 0 = 0) = 1. Consequently H ∞ can be defined as H ∞ := ∞ ∑ j=0 e -ζ j .
Also we denote

c + 1 := lim n→∞ √ nP(S n ≥ 0), c + 2 := lim n→∞ √ nP(S n > 0), (3.18)
where the existence and positivity of c + 1 and c + 2 have been proved in [14, Th.1 in XII. 7 & Th.1 in XVIII.5]. We also introduce two functions which appears in the definition of λ(•). The first one involves the discrete random walk (S j , j). For any j ≥ 1 and x ≥ 1, define

G j (x) := E e S j x + ∑ 1≤i≤j e S i ; S j ≤ 0 , with G 0 (x) := 1 x . (3.19)
The second function depends on Brownian meander (m s , 0 ≤ s ≤ 1). Let m s := sup 0≤t≤s m t and m [s,1] := inf s≤t≤1 m t for any s ∈ [0, 1]. Take a > 0 and b ≥ 0, for any (x, h) ∈ R 2 + , let

Ψ a,b (x, h) := c + 2 P σm 1 > ( √ 2b -x) ∨ h, σ(m 1 -m 1 ) ≤ ( √ 2a -h) + ∧ x, max 0≤s≤1 σ(m s -m [s,1] ) ≤ √ 2a .
Finally, let

C a,b := 2c + 1 c + 2 E Ψ a,b (σm 1 , σ(m 1 -m 1 )); max 0≤s≤1 σ(m s -m s ) ≤ √ 2a , and (3.20) C a,b := C a,b +∞ ∑ j=0 E G j (H ∞ ) . (3.21)
C a,b is well defined positive and finite [see Lemma A.1 and its proof in Appendix A.1], also we set C 0,b = 0. Note also that G j (x) ≤ G j := E[e S j 1 S j ≤0 ] for any j ≥ 1 and x ≥ 1 so C a,b is finite [see (A.13)]. This implies that for any γ > 0,

λ(γ) := C γ -1/2 ,γ -1/2 γ = c 0 C γ -1/2 ,γ -1/2 γ ∈ (0, ∞).
The integrability of λ is stated in Lemma A.1 of Appendix, so Λ in Theorem (1.5) is well defined, i.e.

(3.22) Λ = +∞ 0 λ(x)dx ∈ (0, ∞).

First moment estimate: proof of (3.14)

Let us turn to the proof of Lemma 3.3. First of all, note that 0 ≤ H ≤ G ≤ F a n ,b n . (3.14) follows from the following lemma. √ n e V(z) ∑ φ<y≤z e V(y) 1 {V(z)≥b n , max y≤z (V(y)-V(y))≤a n } 1 {Υ z ≤n 0 } , with recall n 0 = nn 1/3 , also it is clear that 0 ≤ F a n ,b n -G ≤ r. So to obtain (3.24), it suffices to show that

Lemma 3.4. If lim n→∞ a n √ n = a ∈ (0, ∞) and lim n→∞ b n √ n = b ∈ (0, ∞), then lim n √ nE Q (α) W (α) n (G) D (α) n =C a,b , (3.23) lim n √ nE Q (α) W (α) n (F a n ,b n -G) D (α) n =0, (3.24) lim n √ nE Q (α) W (α) n (G -H) D (α) n =0. (3.25)
(3.26) E Q (α) W (α) n (r) D (α) n = o n (1) √ n .
Applying (3.10) for r yields that

E Q (α) W (α) n (r) D (α) n = 1 R(α) E √ ne S n ∑ 1≤j≤n e S j ; S n ≥ b n , max j≤n (S j -S j ) ≤ a n , Υ S ≤ n 0 , S n ≥ -α .
Partitioning on the values of Υ S gives that

E Q (α) W (α) n (r) D (α) n = n 0 ∑ k=0 1 R(α) E √ ne S n ∑ 1≤j≤n e S j ; Υ S = k, S n ≥ b n , max j≤n (S j -S j ) ≤ a n , S n ≥ -α ≤ n 0 ∑ k=0 √ n R(α) E e S n -S k 1 {Υ S =k,S k ≥-α} .
Notice that {Υ S = k} = {S k = S k > S k-1 } ∩ {max k<j≤n S j -S k ≤ 0}. By Markov property at time k,

E Q (α) W (α) n (r) D (α) n ≤ n 0 ∑ k=0 √ n R(α) P S k ≥ -α, S k = S k E e S n-k 1 {S n-k ≤0} ,
which by (3.4), (A.4) and (A.13) implies that

E Q (α) W (α) n (r) D (α) n ≤ n 0 ∑ k=0 √ n R(α) c(1 + α) (k + 1)(n -k) 3/2 ≤ c √ n n-n 1/3 ∑ k=0 1 (k + 1)(n -k) 3/2 = O( 1 n 2/3 ). Observe that ∑ 0≤k≤n/2 1 (k+1)(n-k) 3/2 = O(1) n 3/2 ∑ 0≤k≤n/2 1 k+1 = O( log n n 3/2 ). And observe also that ∑ n/2≤k≤n-n 1/3 1 (k+1)(n-k) 3/2 = O(1) n ∑ n/2≤k≤n-n 1/3 1 (n-k) 3/2 = O( (n 1/3 ) -1/2 n
). Thus (3.26) holds.

Proof of (3.23): It follows from (3.10) that

E Q (α) W (α) n (G) D (α) n = √ n R(α) E e S n ∑ 1≤j≤n e S j ; Υ S > n 0 , S n ≥ b n , max j≤n (S j -S j ) ≤ a n , S n ≥ -α .
Partioning over the values of Υ S implies that E Q (α)

W (α) n (G) D (α) n = √ n R(α) ∑ n k=n 0 +1 σ k where σ k := E e S n ∑ 1≤j≤n e S j ; Υ S = k, S n ≥ b n , max j≤n (S j -S j ) ≤ a n , S n ≥ -α .
Let T i = S i+k -S k , and notice that {S j ; 0 ≤ j ≤ k} and {T i ; 0 ≤ i ≤ n -k} are independent, we have

σ k = E e T n-k ∑ 1≤j≤k e S j -S k + ∑ 1≤j≤n-k e T j ; S k-1 < S k , S k ≥ b n , S k ≥ -α, max i≤k (S i -S i ) ≤ a n , T n-k ≥ (-α -S k ) ∨ (-a n ), T n-k ≤ 0 . Note that {(-α -S k ) ∨ (-a n ) = -O( √ n)} as S k ≥ b n , while with high probability, T n-k = O(n 1/6 )
for nk ≤ n 1/3 . The next step is to approximate σ k by σ k which is defined as follows

σ k := E e T n-k ∑ 1≤j≤k e S j -S k + ∑ 1≤j≤n-k e T j ; S k-1 < S k , S k ≥ b n , S k ≥ -α, max i≤k (S i -S i ) ≤ a n , T n-k ≤ 0 . Observe that 0 ≤ σ k -σ k ≤ P(S k-1 < S k , S k ≥ -α, T n-k ≤ (-a n ) ∨ (-α -b n ))
. By independence of S and T, then using (A.4) and (A.17), one sees that

σ k -σ k ≤P S k ≥ -α, S k = S k P T n-k ≤ (-a n ) ∨ (-α -b n ) ≤c(1 + α)k -1 e -c √ n .
Hence,

√ n R(α) ∑ n-1 k=n 0 +1 (σ k -σ k ) = o n (1)
√ n . This implies that [START_REF] Afanasyev | A limit theorem for a critical branching process in a random environment[END_REF].

(3.27) √ nE Q (α) W (α) n (G) D (α) n = 1 R(α) n ∑ k=n 0 +1 nσ k + o n
We now turn to consider σ k . By independence of S and T again,

σ k = E G n-k ( ∑ 1≤j≤k e S j -S k ); S k-1 < S k , S k ≥ b n , S k ≥ -α, max i≤k (S i -S i ) ≤ a n
where G . (x) is defined in (3.19). Observe that for k = ni with i ∈ N fixed,

σ k = σ n-i = E G i ( ∑ 1≤j≤n-i e S j -S n-i ); S n-i-1 < S n-i , S n-i ≥ b n , S n-i ≥ -α, max i≤n-i (S i -S i ) ≤ a n ,
which by (A.12), is asymptotically,

C a,b R(α)E[G i (H ∞ )] n + o n (1) n . Moreover, as sup x≥1 G i (x) ≤ E[e S i ; S i ≤ 0] ≤ ci -3/2 , by (A.4), one sees that for n 0 ≤ n -i ≤ n, σ n-i ≤ c i 3/2 P(S n-i-1 < S n-i , S n-i ≥ b n , S n-i ≥ -α) ≤ c(1 + α) i 3/2 (n -i) .
As a result, for any integer K ≥ 1 fixed,

n ∑ k=n 0 +1 nσ k = K ∑ i=0 nσ n-i + ∑ K<i≤n 1/3 nσ n-i = C a,b R(α) K ∑ i=0 E[G i (H ∞ )] + o n (1) + o K (1), where ∑ K k=0 E[G i (H ∞ )] = ∑ ∞ i=0 E[G i (H ∞ )] + o K (1). Plugging this into (3.27), letting n → ∞ then K → ∞ implies that, lim n→∞ √ nE Q (α) W (α) n (G) D (α) n = C a,b ∞ ∑ j=0 E[G j (H ∞ )] = C a,b ,
which ends the proof of (3.23).

Proof of (3.25): by (3.10), we only need to prove that

(3.28) √ nE Q (α) W (α) n (G -H) D (α) n = E Q (α) √ nG(w n ) R(α + V(w n )) 1 E c n =: LHS = o n (1)
.

First, we have

LHS ≤nE Q (α) e V(w n )-V(w n ) 1 E c n R(α + V(w n )) ; Υ w n > n 0 ≤ LHS 1 + LHS 2 + LHS 3
where

LHS 1 :=nE Q (α) e V(w n )-V(w n ) 1 E c n,1 R(α + V(w n )) ; Υ w n > n 0 , LHS 2 := nE Q (α) e V(w n )-V(w n ) 1 E n,1 ∩E c n,2 R(α + V(w n )) ; Υ w n > n 0 , LHS 3 :=nE Q (α) e V(w n )-V(w n ) 1 E n,1 ∩E n,2 ∩E c n,3 R(α + V(w n )) .
Each term LHS i , i = 1, 2, 3, are treated separately.

For LHS 1 , by (3.7), we have

LHS 1 ≤ n R(α) E[e S n -S n ; S k n ∈ [k 1/3 n , k n ], S [k n ,Υ S ] ≤ k 1/6 n , Υ S > n 0 , S n ≥ -α] + n R(α) E[e S n -S n ; S k n / ∈ [k 1/3 n , k n ], Υ S > n 0 , S n ≥ -α] =: ξ 1 + ξ 1 .
Arguing over the values of Υ S then using Markov property at Υ S = k,

ξ 1 ≤ n R(α) n ∑ k=n 0 +1 E e S n-k 1 {S n-k ≤0} P S k n ∈ [k 1/3 n , k n ], S [k n ,k] ≤ k 1/6 n , S k > S k-1 , S k ≥ -α ≤ n R(α) n ∑ k=n 0 +1 c (n -k + 1) 3/2 P S k n ∈ [k 1/3 n , k n ], S [k n ,k] ≤ k 1/6 n , S k > S k-1 , S k ≥ -α , (3.29)
where the second inequality holds because of (A.13). Moreover, by (A.20), uniformly on k ∈ [n 0 , n] ∩ Z,

P S k n ∈ [k 1/3 n , k n ], S [k n ,k] ≤ k 1/6 n , S k > S k-1 , S k ≥ -α = o n (1) n .
We hence deduce that

ξ 1 = o n (1) since ∑ n k=n 0 +1 c (n-k+1) 3/2 is finite.
For ξ 1 , similarly, applying Markov property at time Υ S = k then (A.13), we have

ξ 1 ≤ n R(α) n ∑ k=n 0 +1 E[e S n-k ; S n-k ≤ 0]P S k n / ∈ [k 1/3 n , k n ], S k ≥ -α, S k = S k ≤ cn R(α) n ∑ k=n 0 +1 1 (n -k + 1) 3/2 P S k n / ∈ [k 1/3 n , k n ], S k ≥ -α, S k = S k
which by (A.21) yields that

ξ 1 ≤ c n R(α) n ∑ k=n 0 +1 1 (n -k + 1) 3/2 nk 1/2 n = o n (1).
For LHS 3 , let G ∞ be the sigma-field generated by the spine and all siblings of the spine. We know from ([4] eq. (4.9)) that (3.30)

Q (α) E n,1 ∩ E n,2 ∩ E c n,3 G ∞ ≤ O(n 3 e -k 1/6 n /3 ),
wich implies that

LHS 3 ≤nE Q (α) e V(w n )-V(w n ) R(α + V(w n )) × Q (α) E n,1 ∩ E n,2 ∩ E c n,3 G ∞ ≤ O(n 4 e -k 1/6 n /3 ) = o n (1).
For LHS 2 , we follow the same lines as in ([4] page 18, below (4.8)) using the same notations. For

any 1 ≤ i ≤ n, Q (α) (E c n,i |V(w k ); 0 ≤ k ≤ n) ≤ c h(V(w i )),
where for any u ≥ -α, h(u

) := E[X1 {X+ X>e u/2 } + X1 X+ X>e u/2
u+α+1 ], with X := ∑ |z|=1 e -V(z) and X := ∑ |z|=1 V + (z)e -V(z) . Note that E[(X + X) 2 ] < ∞ because of (1.4). Markov inequality gives that h(u) ≤ e -u/2 . Recall that V(w i ) ≥ k 1/6 n on E n,i . Therefore,

LHS 2 ≤c n ∑ i=k n nE Q (α) e V(w n )-V(w n ) R(α + V(w n )) h(V(w i ))1 E n,1 ; Υ w n > n 0 ≤c n(n -k n )e -k 1/6 n /2 E Q (α) e V(w n )-V(w n ) R(α + V(w n )) ; Υ w n > n 0
Applying (3.7) then partitioning on the values of Υ S yields

LHS 2 ≤c n(n -k n )e -k 1/6 n /2 n ∑ k=n 0 +1 1 R(α) E e S n -S k ; Υ S = k, S n ≥ -α ≤c n 2 e -k 1/6 n /2 n ∑ k=n 0 +1 1 R(α) E e S n-k 1 S n-k ≤0 P S k ≥ -α, S k = S k ,
by Markov property. By (3.4), (A.13) and (A.4),

LHS 2 ≤ cn 2 e -k 1/6 n /2 n ∑ k=n 0 +1 1 k(n -k + 1) 3/2 = o n (1), since (log n) 6 = o(k n ).
Collecting all the estimations for the LHS i , this ends the proof of (3.25).

Second moment estimate: proof of (3.15)

Recall the definitions of G in (3.11) and H below (3.12). In view of (3.14), it suffices to show that

(3.31) lim sup n→∞ E Q (α) √ nW (α) n (H) D (α) n 2 ≤ C 2 a,b .
By (3.6), n . We thus approximate

LHS (3.31) :=E Q (α) √ nW (α) n (H) D (α) n 2 = E Q (α) √ nW (α) n (H) D (α) n × √ nH(w n ) R(α + V(w n )) ≤E Q (α) √ nW (α) n (G) D (α) n × √ nG(w n )1 E n R(α + V(w n )) . (3.32) For convenience, let W (α),[k n ,n] n (G) : = e -V(w n ) G(w n )1 {V(w n )≥-α} + n-1 ∑ i=k n ∑ y∈Ω(w i+1 ) ∑ |z|=n,z≥y e -V(z) G(z)1 {V(z)≥-α} , W (α),[0,k n ) n (G) : = k n -1 ∑ i=0 ∑ y∈Ω(w i+1 ) ∑ |z|=n,z≥y e -V(z) G(z)1 {V(z)≥-α} , with Ω(ω i+1 ) = {|x| = i + 1 : ← -x = ω i , x = ω i+1 }.
√ nW (α) n (G) D (α) n by √ nW (α)[0,kn ) n (G) D (α),[0,kn ) n
on the right hand side of (3.32). Then Markov property at k n makes it possible to deal with these two terms in the product separately. Clearly

LHS (3.31) ≤ E Q (α) √ nW (α),[k n ,n] n (G)(D (α) n ) -1 × G n + E Q (α) W n × G n , with W n := √ nW (α),[0,k n ) n (G)/D (α),[0,k n ) n and G n := √ nG(w n )1 E n /R(α + V(w n )). For the first expecta- tion above, as G ≤ √ n1 {V(w n )≥b n /2} , it is clear that given E n , W (α),[k n ,n] n (G) ≤ √ nW (α),[k n ,n] n ≤ √ nD (α),[k n ,n] n ≤ n -3/2 .
In view of (3.4), it follows that

E Q (α) √ nW (α),[k n ,n] n (G)(D (α) n ) -1 × G n ≤ E Q (α) n -1 D (α) n × n R(α + b n /2) 1 E n ≤ c 1 + α + b n /2 E Q (α) 1 D (α) n ≤ c n -1/2 , since E Q (α) [(D (α) n ) -1 ] = R(α) -1 ≤ 1. As a consequence, LHS (3.31) ≤ c √ n + E Q (α) W n G n ≤ c √ n + E Q (α) W n × 1 {V(w kn )∈[k 1/3 n ,k n ]} × sup u∈[k 1/3 n ,k n ] E Q (α) G n V(w k n ) = u ,
where the second inequality follows from Markov property at k n . Let

RHS 1 := E Q (α) W n × 1 {V(w kn )∈[k 1/3 n ,k n ]} , RHS 2 (u) := E Q (α) G n V(w k n ) = u .
Next we are going to show that lim sup n→∞ RHS 1 ≤ C a,b , and

(3.33) lim sup n→∞ sup u∈[k 1/3 n ,k n ] RHS 2 (u) ≤ C a,b . (3.34)
For RHS 1 , note that by Markov property

RHS 1 × inf u∈[k 1/3 n ,k n ] Q (α) (E n |V(w k n ) = u) ≤ E Q (α) W n × 1 E n . By (3.13), inf u∈[k 1/3 n ,k n ] Q (α) (E n |V(w k n ) = u) = 1 + o n (1), therefore, RHS 1 ≤(1 + o n (1))E Q (α) W n × 1 E n ≤(1 + o n (1))E Q (α) G n × 1 E n 1 {D (α) n ≥n -3/2 } + 2nQ (α) (D (α) n ) -1 > n 3/2 , since W (α),[0,k n ) n (G) ≤ √ nD (α),[0,k n ) n . Again by Markov inequality with E Q (α) [(D (α) n ) -1 ] = R(α) -1 ≤ 1, 2nQ (α) (D (α) n ) -1 > n 3/2 ≤ 2n -1/2 .
On the other hand, given

E n ∩ {D (α) n ≥ n -3/2 }, D (α),[k n ,n] n ≤ n -2 ≤ D (α) n / √ n. So, D (α),[0,k n ) n = D (α) n -D (α),[k n ,n] n ≥ (1 -1/ √ n)D (α) n .
Consequently,

RHS 1 ≤ (1 + o n (1))E Q (α) W n × 1 E n 1 {D (α) n ≥n -3/2 } + 2 √ n ≤ (1 + o n (1))E Q (α) √ nW (α) n (F a n ,b n ) D (α) n + 2 √ n .
So (3.33) follows from (3.14).

It remains to prove (3.34). Let m := nk n and m 0 :

= n 0 -k n , for any u ∈ [k 1/3 n , k n ], RHS 2 (u) is bounded by E Q (α) n R(α + V(w m )) e V(w m ) ∑ 0<j≤m e V(w m ) 1 {Υ wn >m 0 ,V(w m )≥b n ,max k≤n (V(w k )-V(w k ))≤a n } V(w 0 ) = u which by Markov property and (3.7) is less than n R(α + u) E e S m ∑ 1≤j≤m e S m ; max i≤m (S i -S i ) ≤ a n , Υ S > m 0 , S m ≥ b n -u, S m ≥ -α -u .
Following the same arguments used for (3.23), one obtains that for all u ∈ [k [START_REF] Afanasyev | A limit theorem for a critical branching process in a random environment[END_REF], which completes the proof of (3.34) and conclude (3.31).

1/3 n , k n ], RHS 2 (u) ≤ C a,b + o n

Proof of Corollary 1.9

In this subsection, we show that as β → ∞,

(3.35) ∞ ∑ m=1 ∑ |z|=m 1 ∑ φ<y≤z e V(y) 1 {max φ<y≤z(V(y)-V(y))≤β,V(z)≥β±O(log β) } in P * probability --------→ ΛD ∞ . Proof. Denote W * m (β) := ∑ |z|=m 1 ∑ φ<y≤z e V(y) 1 {max φ<y≤z(V(y)-V(y))≤β,V(z)≥β } = W m (F β,β )/ √ m.
In fact, only those m that are comparable to β 2 really contribute to the sum. First, for m ≤ εβ 2 and m ≥ β 2 /ε with ε ↓ 0, we claim that for any η > 0,

lim ε→0 lim sup β→∞ P   ∑ m≤εβ 2 W * m (β) ≥ η   = 0 (3.36) lim ε→0 lim sup β→∞ P   ∑ m≥β 2 /ε W * m (β) ≥ η   = 0. (3.37)
We postpone the proof of the above facts to Subsection 4.2.4 as the arguments are similar to the proof of (2.9).

For any ε > 0 fixed, by (1.22), as β → ∞, mW * m (a

√ m) converges in probability to D ∞ C a,a uniformly on a ∈ R + . Moreover, ∑ εβ 2 ≤m≤β 2 /ε 1 m < ∞ for any ε > 0. Therefore, ∑ εβ 2 ≤m≤β 2 /ε W * m (β) =D ∞ ∑ εβ 2 ≤m≤β 2 /ε 1 m C β/ √ m,β/ √ m + o P * (1) =D ∞ ∑ εβ 2 ≤m≤β 2 /ε 1 m C β/ √ m,β/ √ m + o P * (1).
where o P * (1) denotes a term such that lim β→∞ o P * (1) = 0 in P * -probability. On the other hand, by change of variables m = γβ 2 ,

β 2 /ε εβ 2 C β/ √ m,β/ √ m dm m = 1/ε ε C γ -1/2 ,γ -1/2 dγ γ .
As C a,b is continuous and monotone, we get that

∑ εβ 2 ≤m≤β 2 /ε 1 m C β/ √ m,β/ √ m = 1/ε ε C γ -1/2 ,γ -1/2 dγ γ + o β (1)
.

When ε → 0, 1/ε ε C γ -1/2 ,γ -1/2 dγ γ → Λ because of Lemma A.1.
In view of (3.36) and (3.37), we conclude that in P * -probability,

lim β→∞ ∞ ∑ m=1 W * m (β) = ΛD ∞ . Lemma 4.2.
Recall the definition of B δ ∩ U in Section 2.1, we have :

(4.1) V E ar K B δ ∩U n ( ) ≤ ∑ |z|=|v|= , z =v na z P E v∧z (T v < T φ )1 {z,v∈B δ ∩U} + na v P E v∧z (T z < T φ )1 {z,v∈B δ ∩U} + ∑ |z|= na z 1 {z∈B δ ∩U} ,
where v ∧ z is the latest common ancestor of v and z in the tree T, and P E y is the quenched probability of the random walk started from y.

Proof. This upper bound is actually true for every truncated version of K n ( ), however it is optimized here for events included in U, in particular for B δ ∩ U. For a v,z one sees that

a v,z = P E (T v < T z ∧ T φ ) + P E (T z < T v ∧ T φ ) =: d v,z + d z,v . We have, (1 -a v,z ) n -(1 -a z ) n (1 -a v ) n ≤ (1 -d v,z -d z,v ) n -(1 -a z -a v ) n ≤ n(a z -d z,v + a v -d v,z ).
Observe that

a z -d z,v + a v -d v,z =P E (T v ∨ T z < T φ ) ≤ P E (T z < T φ )P E z∧v (T v < T φ ) + P E (T v < T φ )P E z∧v (T z < T φ ) =a z P E z∧v (T v < T φ ) + a v P E z∧v (T z < T φ ).
This together with Lemma 4.1 yields that

∑ |z|= ,|v|= , z =v [(1 -a v,z ) n -(1 -a z ) n (1 -a v ) n ] 1 {z∈B δ ∩U} 1 {v∈B δ ∩U} ≤ ∑ |z|=|v|= , z =v na z P E v∧z (T v < T φ )1 {z∈B δ ∩U,v∈B δ ∩U} + ∑ |z|=|v|= , z =v na v P E v∧z (T z < T φ )1 {z∈B δ ∩U,v∈B δ ∩U} Moreover, we have (1 -a z ) n -(1 -a z ) 2n ≤ na z .
This leads to (4.1).

Upper bound for the mean of the quenched variance

In this section we obtain an upper bound of the mean

E V E ar K B δ ∩U n ( ) .
Lemma 4.3. For ∼ γ(log n) 2 , every δ > 0 and n large enough,

E V E ar K B δ ∩U n ( ) ≤ cn 2 (log n) -δ+1/2 .
Proof. Because of (4.1), we only have to bound the means of

t n := ∑ |z|=|v|= , z =v a z P E v∧z (T v < T φ )1 {z,v∈B δ ∩U} , K B δ ∩U n ( ) = ∑ |z|= na z 1 {z∈B δ ∩U}
since the second term on the RHS of (4.1) is its symmetric. We begin with

K B δ ∩U n ( ). As B δ 2 ⊂ L δ , recall- ing (1.14) and (1.19), one sees that K B δ ∩U n ( ) = n √ p E (φ, ← - φ )W (α) (F log s n ,log n+log log n ) since the truncated
martingale-like variable is obtained by adding the restriction B 1 . By (3.16), one gets that

(4.2) E[ K B δ ∩U n ( )] = Θ( n ) = Θ( n (log n) 2 ).
The main idea of the rest proof, is to decompose the double sum ∑ |z|=|v|= according to the latest common ancestor z ∧ v.

Define ∑ 1 := ∑ φ<s≤v∧z e V(s)-V(z∧v) , ∑ 2 := ∑ v∧z<s≤z e V(s)-V(z∧v) and ∑ 3 := ∑ v∧z<s≤v e V(s)-V(z∧v) .

We then have

a z = p E (φ, ← - φ )e -V(v∧z) ∑ 1 + ∑ 2 and P E z∧v (T v < T φ ) = ∑ 1 ∑ 1 + ∑ 3 . By comparing Σ 1 , Σ 2 , Σ 3 , we get t n ≤ t 1 n + t 2 n + t 3 n + t 4 n , with t 1 n := ∑ |z|=|v|= z =v e -V(z∧v) ∑ 1 1 {z∈B δ ∩U,v∈B δ ∩U,∑ 1 ≥∑ 2 ∨ ∑ 3 } , t 2 n := ∑ |z|=|v|= z =v e -V(z∧v) ∑ 3 1 {z∈B δ ∩U,v∈B δ ∩U,∑ 2 ≤∑ 1 ≤∑ 3 } , t 3 n := ∑ |z|=|v|= z =v e -V(z∧v) ∑ 2 1 {z∈B δ ∩U,v∈B δ ∩U,∑ 3 ≤∑ 1 ≤∑ 2 } , t 4
n := ∑ |z|=|v|= z =v e -V(z∧v) ∑ 1 ∑ 2 * ∑ 3 1 {z∈B δ ∩U,v∈B δ ∩U,∑ 1 ≤∑ 2 ∧ ∑ 3 } .
We treat each term separately. Notice that by symmetry

E(t 2 n ) = E(t 3 n ), so we only estimate E(t 1 n ), E(t 2 n ) and E(t 4 n ). Recall that for every z ∈ U, V(z) ≥ log n + log log n and a z ≤ (n log n) -1 . Clearly, {Σ 1 + Σ 2 ≥ e V(z)-V(z∧v) ≥ n log ne -V(z∧v) }. In addition, if {Σ 1 ≥ Σ 2 }, we have V(z ∧ v) ≥ log Σ 1 e V(z∧v) |z ∧ v| ≥ log n log n 2 .
* Upper bound for E(t 1 n ), as Σ 1 is the largest term here, using the above remark we have {z,

v ∈ U, Σ 1 ≥ Σ 2 ∨ Σ 3 } ⊂ {V(z ∧ v) > log n + log log n -log 2 =: m n }, also as z ∈ B δ 2 , Σ 1 ≤ s n = n/(log n) 1+δ , so t 1 n ≤ ∑ |z|=|v|= z =v e -V(z∧v) ∑ 1 1 {V(z∧v)>m n ,Σ 1 ≤s n ,∑ 1 ≥∑ 2 ∨ ∑ 3 , V(z)∧V(v)≥-α} ≤ -1 ∑ j=0 ∑ |u|=j e -V(u) ∑ u 1 1 {V(u)>m n ,Σ u 1 ≤s n ,V(u)≥-α} ∑ ← -x =u= ← -y x =y ∑ |z|= ,z≥x 1 {∑ x,z 2 e V(x)-V(u) ≤∑ u 1 } ∑ |v|= ,v≥y 1 {∑ y,v 2 e V(y)-V(u) ≤∑ u 1 } ,
where Σ u 1 := ∑ φ<s≤u e V(s)-V(u) and Σ x,z 2 := ∑ x≤s≤z e V(s)-V(x) and recall that ←x is the parent of x. Applying Markov property at time |u| + 1 and then Many-to-one equation (3.1) yields

E[t 1 n ] ≤ -1 ∑ j=0 E ∑ |u|=j e -V(u) ∑ u 1 1 {V(u)>m n ,Σ u 1 ≤s n ,V(u)≥-α} ∑ ← -x =u= ← -y x =y f j, (Σ u 1 e V(u)-V(x) ) f j, (Σ u 1 e V(u)-V(y) ) ,
where f j, (t

) := E e S -1-j ; ∑ -1-j i=0 e S i ≤ t . By (A.14), f j, (t) ≤ E(e S -1-j ; S -1-j ≤ log + t) ≤ c(log + t + 1)t/( -j) 3/2 .
Plugging this into the previous inequality yields

E[t 1 n ] ≤ -1 ∑ j=0 E ∑ |u|=j e -V(u) Σ u 1 (1 + log Σ u 1 ) 2 1 {V(u)>m n ,Σ u 1 ≤s n ,V(u)≥-α} × c ( -j) 3 ∑ ← -x =u= ← -y x =y [1 + (V(u) -V(x)) + ]e V(x)-V(u) [1 + (V(u) -V(y)) + ]e V(y)-V(u) ≤ -1 ∑ j=0 c ( -j) 3 E ∑ |u|=j e -V(u) Σ u 1 (1 + log Σ u 1 ) 2 1 {V(u)>m n ,Σ u 1 ≤s n ,V(u)≥-α} E ∑ |x|=1 [1 + (-V(x)) + ]e -V(x) 2 .
By (3.1) and hypothesis (1.4), we get

E(t 1 n ) ≤ ∑ -1 j=0 c ( -j) 3 s n (1 + log + s n ) 2 P S j > m n , Σ S 1 ≤ s n , S j ≥ -α , with ∑ S 1 := ∑ j i=1
e S i -S j . Also by (A.2), P S j ≥ -α ≤ c(1 + α)j -1/2 , so

E(t 1 n ) ≤ -1 ∑ j=0 c ( -j) 3 s n (1 + log + s n ) 2 (1 + α) (j + 1) 1/2 ≤ c(1 + α)n (log n) δ-1/2
. * Upper bound for E(t 2 n ), with the same ideas as for the upper bound of t 1 n , we have

t 2 n ≤ ∑ |z|=|v|= z =v e -V(z∧v) ∑ 3 1 {V(z∧v)>m n ,Σ 1 ≤s n ,Σ 2 ≤Σ 1 ,V(z∧v)≥-α} ≤ -1 ∑ j=0 ∑ |u|=j e -V(u) 1 {V(u)≥m n ,Σ u 1 ≤s n ,V(u)≥-α} ∑ ← -x =u= ← -y x =y ∑ |z|= ,z≥x 1 {Σ x,z 2 e V(x)-V(u) ≤s n } ∑ |v|= ,v≥y 1 Σ y,v 2 e V(y)-V(u) .
By Markov property then by (3.1), it follows that

E(t 2 n ) ≤ -1 ∑ j=0 E ∑ |u|=j e -V(u) 1 {V(u)≥-α} ∑ ← -x =u= ← -y x =y e V(u)-V(y) E e S -j-1 ∑ -j-1 i=0 e S i f j, s n e V(u)-V(x) ,
which by (A.14) and (A.15), is less than

-1 ∑ j=0 cs n (1 + log + s n ) ( -j) 2 E ∑ |u|=j e -V(u) 1 {V(u)≥-α} ∑ ← -x =u= ← -y x =y e V(u)-V(y) [1 + (V(u) -V(x)) + ]e V(u)-V(x) ≤ -1 ∑ j=0 cs n (1 + log + s n ) ( -j) 2 E ∑ |u|=j e -V(u) 1 {V(u)≥-α} E ∑ |x|=1 [1 + (-V(x)) + ]e -V(x) 2 .
Applying again (3.1), (1.4), and then (A.2) we have,

E(t 2 n ) ≤ -1 ∑ j=0 cs n (1 + log + s n ) ( -j) 2 P S j ≥ -α ≤ c(1 + α)n (log n) 1+δ
. * Upper bound for E(t 4 n ), we have :

t 4 n ≤ ∑ |z|=|v|= z =v e -V(z∧v) ∑ 1 ∑ 2 * ∑ 3 1 {V(z∧v)≥-α,∑ 1 ≤s n } ≤ -1 ∑ j=0 ∑ |u|=j e -V(u) s n 1 {V(u)≥-α} ∑ ← -x =u= ← -y x =y ∑ |z|= ,z≥x 1 Σ x,z 2 e V(x)-V(u) ∑ |v|= ,v≥y 1 Σ y,v 2 e V(y)-V(u) .
With the same arguments as above, one sees that

E(t 4 n ) ≤ -1 ∑ j=0 E ∑ |u|=j e -V(u) s n 1 {V(u)≥-α} ∑ ← -x =u= ← -y
x =y e 2V(u)-V(x)-V(y) E e S -j-1

∑ -j-1 i=0 e S i
E(t 4 n ) ≤ -1 ∑ j=0 cs n ( -j) P(S j ≥ -α) ≤ -1 ∑ j=0 cs n (1 + α) ( -j)(j + 1) 1/2 ≤ c(1 + α)n(log log n) (log n) 2+δ .
Consequently, we have t n ≤ cn/(log n) δ , which concludes the proof. ( )] = o(n/(log n) 2 ).

Note that P E (T z < T n φ ) ≤ na z ∧ 1. We have

E[K B\U n ( )] ≤ E ∑ |z|= (na z ∧ 1)1 {V(z)≤log n+log log n,V (
E ∑ |z|= na z 1 {log n-3 log log n≤V(z)≤log n+log log n,V(z)≥-α} ≤ c n log log n (log n) 3 = o( n (log n) 2 ),
which completes the proof.

proof of Lemma 2.2

The quenched mean K

(B∩U)\(B δ ∩U) n ( ) of K B∩U n ( ) -K B δ ∩U n ( ) satisfies that 0 ≤ K (B∩U)\(B δ ∩U) n ( ) ≤ K (B∩U)\(B δ ∩U) n ( ) = ∑ |z|= na z 1 {z∈(B\B δ )∩U} .
As {z ∈ B \ B δ } implies that log s nlog < max φ<y≤z (V(y) -V(y)) ≤ log n, similarly to (1.19), we have

K (B∩U)\(B δ ∩U) n ( ) ≤ n √ W (α) (F log n,log n+log log n ) -W (α) (F log s n -log ,log n+log log n ) ,
with s n = n/(log n) 1+δ . Taking expectation and using change of measures (3.5) yields that

E K (B\B δ )∩U n ( ) ≤ n √ R(α) E Q (α) W (α) (F log n,log n+log log n ) D (α) -E Q (α) W (α) (F log s n -log ,log n+log log n ) D (α) (4.3)
In view of (3.14), as ∼ γ(log n) 2 , we have

E Q (α) √ W (α) (F log n,log n+log log n ) D (α) -E Q (α) √ W (α) (F
∑ ≥(log n) 2 /ε ∑ |z|= K B n ( ) = 0, (4.5) lim ε↓0 lim sup n→∞ 1 n E ∑ ≤ε(log n) 2 ∑ |z|= K B n ( ) = 0. (4.6)
• Proof of (4.5) and (3.37). Recall that a z ≤ e -V(z) . One sees that

LHS (4.5) :=E ∑ ≥(log n) 2 /ε ∑ |z|= K B n ( ) ≤ E ∑ ≥(log n) 2 /ε ∑ |z|= (na z ∧ 1)1 z∈B ≤ ∑ ≥(log n) 2 /ε nE ∑ |z|= e -V(z) 1 {V(z)≥log n,z∈B} + ∑ ≥(log n) 2 /ε E ∑ |z|= 1 {V(z)≤log n,V (
∑ ≥(log n) 2 /ε cn(1 + α)(1 + log n + α) 3/2 ≤ c n(1 + α)(1 + log n + α) (log n) 2 /ε .
For any α > 0 fixed, letting ε ↓ 0 implies that lim ε↓0 lim sup n→∞ n -1 R I I = 0. Also by (3.1),

R I = ∑ ≥(log n) 2 /ε nE ∑ |z|= e -V(z) 1 {V(z)≥log n,z<L n ,V(z)≥-α} equals to ∑ ≥(log n) 2 /ε nE e S -S ; S ≥ log n, max 1≤k≤ k ∑ i=1 e S i -S k ≤ n, S ≥ -α .
Observe that e S k -S k ≤ ∑ k i=1 e S i -S k . It then follows that

R I ≤ ∑ ≥(log n) 2 /ε nE e S -S ; max 1≤k≤ (S k -S k ) ≤ log n, S ≥ -α ,
which by (A.23) is less than

cn(1 + α) ∑ ≥(log n) 2 /ε 1 7/6 + 1 e -c (log n) 2 .
Clearly, ∑ ≥(log n) 2 /ε 1 7/6 ≤ cε 1/6 (log n) -1/3 , so by monotonicity and change of variables, one sees that

∑ ≥(log n) 2 /ε 1 e -c (log n) 2 ≤ ∞ (log n) 2 /ε e -c t (log n) 2 dt t = ∞ 1/ε e -c s ds s ≤ ε/c . Consequently, R I ≤ (1 + α)εn/c + c(1 + α)ε 1/6 (log n) -1/3 n.
We hence deduce that lim ε→0 lim sup n→∞ R I /n = 0. Collecting estimates for R I and R I I together with (4.7), (4.5) follows immediately.

Moreover, observe that W * m (β) ≤ ∑ |z|=m e -V(z) 1 {max φ<y≤z(V(y)-V(y))≤β,V(z)≥β } . So, for any ε, η > 0,

P   ∑ m≥β 2 /ε W * m (β) ≥ η   ≤ P( inf z∈T V(z) ≤ -α) + P   ∑ m≥β 2 /ε ∑ |z|=m e -V(z) 1 {max φ<y≤z(V(y)-V(y))≤β,V(z)≥β,V(z)≥-α } ≥ η  
where the second probability on the right hand side vanishes as β → ∞ then ε → 0 because of the convergence of R I /n by replacing log n by β. The first probability on the right hand side is negligible in view of (2.1).

• Proof of (4.6) and (3.36). Similarly as above,

LHS (4.6) := E ∑ ≤ε(log n) 2 ∑ |z|= K B n ( ) ≤ E ∑ ≤ε(log n) 2 ∑ |z|= (ne -V(z) ∧ 1)1 {z<L n ,V(z)≥-α} ≤ R I + R I I ,
where

R I := ∑ ≤ε(log n) 2 E ∑ |z|= ne -V(z) 1 {V(z)≥log n/2,V(z)≥-α} , R I I := ∑ ≤ε(log n) 2 E ∑ |z|= 1 {V(z)≤log n/2,V(z)≥-α} .
Again by (3.1),

R I =n ∑ ≤ε(log n) 2
E e S -S ; S ≥ log n/2, S ≥ -α which by (A.24) is bounded by

n ∑ ≤ε(log n) 2 c(1 + α) 1/2 log n ≤ c (1 + α) √ εn.
Therefore, lim ε↓0 lim sup n→∞ R I n -1 = 0 . It remains to bound

R I I = ∑ ≤ε(log n) 2 E ∑ |z|= 1 {V(z)≤log n/2,V(z)≥-α} . By (3.1), R I I = ∑ ≤ε(log n) 2 E e S ; S ≤ log n/2, S ≥ -α ≤ ∑ ≤ε(log n) 2 e log n/2 ≤ ε(log n) 2 √ n,
so lim ε↓0 lim sup n→∞ R I I n -1 = 0. This completes the proof of (4.6).

Similarly to the proof of (3.37), the convergence (3.36) follows from (2.1) and the convergence of

R I /n.
Proof of (2.10). We now prove that for any ε > 0,

(log n) 2 /ε ∑ m=ε(log n) 2 E K B\U n (m) = o(n), (4.8) (log n) 2 /ε ∑ m=ε(log n) 2 E K (B∩U)\(B δ ∩U) n (m) = o(n). (4.9)
As shown in the proof of Lemma 2.1, for any m ≥ ε(log n) 2 , there exists some constant c(ε) ∈ R + such that

E K B\U n (m) ≤ c(ε)n log log n (log n) 3 ,
so (4.8) follows. It remains to show (4.9). Observe that

E K (B∩U)\(B δ ∩U) n (m) ≤E K (B∩U)\(B δ ∩U) n (m) = E ∑ |z|=m na z 1 {z∈(B\B δ )∩U} .
Take β = log n + log log n. Because of (4.3), for any ε > 0 fixed, there exists c 1 > 0 such that when n ≥ 10, for any m ∈ [εβ 2 /2,

β 2 /ε] ∩ Z, E K (B∩U)\(B δ ∩U) n (m) ≤ n √ m R(α) E Q (α) W (α) m (F β,β ) D (α) m -E Q (α) W (α) m (F β-c 1 log β,β ) D (α) m .
It follows immediately that (4.10)

(log n) 2 /ε ∑ m=ε(log n) 2 E K (B∩U)\(B δ ∩U) n (m) ≤ nR(α) ×   β 2 /ε ∑ m=εβ 2 /2 1 √ m E Q (α) W (α) m (F β,β ) D (α) m - β 2 /ε ∑ m=εβ 2 /2 E Q (α) W (α) m (F β-c 1 log β,β ) D (α) m   .
Similarly to (1.22), the convergence (3.14) holds uniformly. So following the arguments used to prove Corollary 1.9, we deduce that for any ε > 0, as β → ∞, ∑

εβ 2 /2≤m≤β 2 /ε 1 √ m E Q (α) W (α) m (F β,β ) D (α) m = 1/ε ε/2 C γ -1/2 ,γ -1/2 dγ γ + o β (1)
.

Similarly, we also have

∑ εβ 2 /2≤m≤β 2 /ε E Q (α) 1 √ m W (α) m (F β-c 1 log β,β ) D (α) m = 1/ε ε/2 C γ -1/2 ,γ -1/2 dγ γ + o β (1).
As a consequence, (4.10) becomes

(log n) 2 /ε ∑ m=ε(log n) 2 E K (B∩U)\(B δ ∩U) n (m) ≤ o n (1)nR(α),
which ends the proof.

Proof of Proposition 1.7

Most of the arguments are already present in the proof of Theorem 1.5 in section 2.2. Indeed we have stressed on the fact that the main contribution of visited sites comes from the set of individuals of the tree truncated by B δ ∩ U.

Similarly to the proof of (2.10), the restriction A 3 := {z ∈ T : V(z) > max |y|≤|z|-|z| 1/3 , y≤z V(y)} follows easily from (3.26). So it remains to consider D := z ∈ T : max φ<y≤z (V(y) -V(y)) ≤ log n a 0 , and

F := z ∈ T : V(z) ≥ a 1 log n log log n . We only need to show that lim a 0 →+∞ lim n→+∞ E n -1 (log n) 2 /ε ∑ m=ε(log n) 2 K B∩D∩A 3 n (m) = 0, (4.11) and lim n→+∞ E n -1 (log n) 2 /ε ∑ m=ε(log n) 2 K B∩F n (m) = 0. (4.12)
For (4.11) we do as usual and get that the expectation is smaller than

(log n) 2 /ε ∑ m=ε(log n) 2 E e S m -S m ; max 1≤i≤m (S i -S i ) ≤ log n/a 0 , S m ≥ -α, Υ S > m -m 1/3 ≤ (log n) 2 /ε ∑ m=ε(log n) 2 m ∑ j=m-m 1/3
E e S m-j 1 S m-j ≤0 P max 1≤i≤j (S i -S i ) ≤ log n/a 0 , S j-1 < S j , S j ≥ -α .

Similarly to (A.30) and the lines that follow, the above sum is bounded by

∑ (log n) 2 /ε m=ε(log n) 2 ∑ m j=m-m 1/3 c(1+α)
(m-j+1) 3/2 m e -c ma 0 /(log n) 2 ≤ -2(log ε)e -c εa 0 which goes to zero as a 0 → ∞. Also for the expectation in (4.12) we have that it is smaller than

(log n) 2 /ε ∑ m=ε(log n) 2 E e S m -S m ; S m ≥ a 1 log n log log n, S m ≥ -α ≤ (log n) 2 /ε ∑ m=ε(log n) 2 m ∑ j=1 E e S m-j ; S m-j ≤ 0 P S j ≥ -α, S j ≥ a 1 log n log log n which by (A.13) and (A.18) is bounded by c ∑ (log n) 2 /ε m=ε(log n) 2 ∑ m j=1 (m -j + 1) -3/2 m -c * a 2 1 = o n (
1) by choosing a 1 properly.

A Appendix

A.1 Finiteness of Λ [see (3.22)]

Lemma A.1. The function λ : (0, ∞) → (0, ∞) is well defined and integrable, i.e.,

(A.1) Λ = ∞ 0 λ(γ)dγ = c 0 ∞ 0 C γ -1/2 ,γ -1/2 γ dγ < ∞.
Further, for any a, b > 0, C aγ -1/2 ,bγ -1/2 /γ is integrable.

Proof. Recalling (3.21), it suffices to show that C a,b ∈ (0, ∞) and that C γ -1/2 ,γ -1/2 /γ is integrable. Recall that for any a, b > 0,

C a,b = 2c + 1 c + 2 E Ψ a,b (σm 1 , σ(m 1 -m 1 )); max 0≤s≤1 σ(m s -m s ) ≤ √ 2a with Ψ a,b (x, h) := c + 2 P σm 1 > ( √ 2b -x) ∨ h, σ(m 1 -m 1 ) ≤ ( √ 2a -h) + ∧ x, max 0≤s≤1 σ(m s -m [s,1] ) ≤ √ 2a . Obviously, C a,b ≤ 2c + 1 c + 2 c + 2 < ∞. Moreover, C a,b ≥ 2c + 1 c + 2 c + 2 P σm 1 > √ 2b, σ(m 1 -m 1 ) ≤ √ 2 2 a, max 0≤s≤1 σ(m s -m [s,1] ) ≤ √ 2a × P σm 1 ≥ √ 2 2 a, max 0≤s≤1 σ(m s -m s ) ≤ √ 2 2 (a ∧ b) .
On the one hand, Biane and Yor [START_REF] Biane | Quelques précisions sur le méandre brownien[END_REF] showed that

(m s , m [s,1] ); 0 ≤ s ≤ 1 = d (|b s | + λ 0 s , λ 0 s ); 0 ≤ s ≤ 1 ,
where (b s , s ∈ [0, 1]) is a Brownian bridge and (λ 0 s , 0 ≤ s ≤ 1) its local time at 0. On the other hand, if (R s ; 0 ≤ s ≤ 1) a Bessel(3) process, Imhof [START_REF] Imhof | Density factorizations for brownian motion, meander and the three-dimensional bessel process, and applications[END_REF] showed that for any x > 0,

(m s , s ∈ [0, 1]) given {m 1 = x} = d (R s , s ∈ [0, 1]) given {R 1 = x},
where P(m 1 ∈ dx) = xe -x 2 /2 1 {x≥0} dx. By the continuity of the distribution of (b, λ 0 ) and R, one sees that C a,b is continuous and strictly positive.

Let (m s , s ∈ [0, 1]) and ( m s , s ∈ [0, 1]) be two independent Brownian meanders. Then C a,b ≤    cP σ(m 1 + m 1 ) ≥ √ 2b cP max s∈[0,1] σ(m s -m [s,1] ) ≤ √ 2a .
It follows from the first inequality that

C a,b ≤ cP(m 1 ≥ √ 2b 2σ ) = ce -b 2 /4σ 2 ,
On the other hand, according to [START_REF] Biane | Quelques précisions sur le méandre brownien[END_REF],

P max s∈[0,1] σ(m s -m [s,1] ) ≤ √ 2a = P max s∈[0,1] |b s | ≤ √ 2a/σ ≤ P max s∈[0,1] b s ≤ √ 2a/σ . This shows that C a,b ≤cP max s∈[0,1] b s ≤ √ 2a/σ = c 1 -exp -2 √ 2a σ 2 ≤ 4c σ 2 a 2 .
We are now ready to prove the integrability. Observe that

∞ 0 C γ -1/2 ,γ -1/2 γ dγ = 1 0 C γ -1/2 ,γ -1/2 γ dγ + ∞ 1 C γ -1/2 ,γ -1/2 γ dγ ≤ 1 0 ce -1 4σ 2 γ dγ γ + ∞ 1 4c σ 2 dγ γ 2 = 1 0 ce -1 4σ 2 γ dγ γ + 4c σ 2 .
By change of variables t = 1/γ,

1 0 ce -1 4σ 2 γ dγ γ = ∞ 1 ce -t 4σ 2 dt t < ∞.
We hence conclude the integrability of

C γ -1/2 ,γ -1/2 γ
, as well as

C aγ -1/2 ,bγ -1/2 γ
for any a, b > 0.

By Lemma 2.3 in [START_REF] Aidekon | The seneta-heyde scaling for the branching random walk[END_REF], there exists a constant c such that for any α ≥ 0,

sup n≥1 E [|S n |; S n ≥ -α] ≤ c(α + 1).
It follows from this lemma and Markov's inequality that for any α ≥ 0,

P S n/2 ≥ -α, S n/2 ≥ A/2 ≤ 2E |S n/2 | A ; S n/2 ≥ -α ≤ c(1 + α) A .
As a consequence,

P S n ≥ -α, S n = S n ≥ A ≤ c(1 + α) A √ n .
Proof of (A.7). To obtain (A.7), we consider the two independent random walks (S k , 0

≤ k ≤ m) and (R k , 0 ≤ k ≤ n -m). As S n = R n-m + S m , one immediately sees that P S n ≥ -α, S n = S n , S m -S n ≥ -A, S m -S m ≤ a ≤P S m ≥ -α, S m -S m ≤ a, R n-m ≥ 0, R n-m ∈ [S m -S m , S m -S m + A] ≤E P R n-m ≥ 0, R n-m ∈ [S m -S m , S m -S m + A] (S k , 0 ≤ k ≤ m) ; S m ≥ -α, S m -S m ≤ a .
Applying (A.3) to this conditional probability implies that

P S n ≥ -α, S n = S n , S m -S n ≥ -A, S m -S m ≤ a ≤E c (1 + A)(1 + A + S m -S m ) (n -m) 3/2 ; S m ≥ -α, S m -S m ≤ a ≤c (1 + A)(1 + A + a) (n -m) 3/2 P (S m ≥ -α) , which by (A.2) is bounded by c (1 + α)(1 + A)(1 + A + a) m 1/2 (n -m) 3/2 .
This ends the proof of (A.7).

Proof of (A.8). Let

T k := S n-k -S n = -R k . Then (T k , 0 ≤ k ≤ n) is a random walk distributed as (-S k , 0 ≤ k ≤ n). It follows from (A.3) that P S n = S n ≥ -α ≤ P T n ≥ 0, T n ≤ α ≤ c(1 + α) 2 n 3/2 .
Proof of (A.9). Observe that e S n -S n ≤ ∑ 1≤i≤n e S i -S n ≤ ne S n -S n , then where the last inequality follows from (A.7). Now, given S n/2 ≤ S nn -1/3 , ∑ n j=1 e S j -S n can be replaced by ∑ n/2≤j≤n e S j -S n which is independent of (S k ; 0 ≤ k ≤ n/2). Note that on {S n/2 ≤ S nn -1/3 }, Now we use (R k , 0 ≤ k ≤ n/2) in replace of (S n -S n-k , 0 ≤ k ≤ n/2) and recount on the same arguments as in the proof of (A.11). Thanks to (3.17), (A.12) follows immediately.

Proof of (A. Proof of (A.17). For θ > 0 such that ϕ(θ) := log E[e θS 1 ] ∈ (-∞, ∞), {e θS n -nϕ(θ) ; n ≥ 0} is a non-negative martingale. The existence of θ comes from (1.3). Therefore, by Doob's inequality, P S n ≥ n 1+δ ≤P max 0≤k≤n e θS k -kϕ(θ) ≥ e θn 1+δ -nϕ(θ) ≤e -θn 1+δ +nϕ(θ) E e θS n -nϕ(θ) = e -θn 1+δ +nϕ(θ) .

For n large enough, θn 1+δnϕ(θ) ≥ θn 1+δ /2. Hence, for any n ≥

Figure 1 :

 1 Figure 1: Recurrence criteria for (X n , n)

Figure 2 :

 2 Figure 2: Typical accessible environments within time n

Proposition 1 . 8 .

 18 If (a m ; m ≥ 0) and (b m ; m ≥ 0) are positive sequences such that lim m→∞ a m √ m = a ∈ R * + and lim m→∞ b

∞

  has been proved by Biggins and Kyprianou [9, Th 5.1]. So Q (α)

  As it is shown in[START_REF] Biggins | Measure change in multitype branching[END_REF] Th 5.1] and[START_REF] Chen | A necessary and sufficient condition for the non-a necessary and sufficient condition for the non-trivial limit of the derivative martingale in a branching random walk[END_REF], D (α)n converges P-a.s and in L 1 to D (α)∞ which is positive under P * . So Q(α) is absolutely continuous with respect to P. We thus deduce Proposition 1.8 from (3.8) with C a,b = c 0 C a,b (one can refer to [4, Section 5] for more details).

√n.

  That is why we multiply √ n in the definition of F a n ,b n (z). Let us introduce the following notations. For any |z| = n and 0 ≤ m ≤ n, let z m be the ancestor of z in the m-th generation and define

  Proof of(3.24): For any |z| = n, comparing (1.18) and (3.11), we define r(z) :=

  In the similar way, we define D (α),[0,k n ) n and D (α),[k n ,n] n . Recall (3.12), the event E n,3 means that D (α),[k n ,n] n ≤ n -2 . So under Q (α) , the descendants of the (ω i ; k n ≤ i ≤ n) make little contribution to D (α) n . The same thing happens to W (α)

2 ,

 2 which by(A.15) and(1.4) is less than ∑ -1 j=0 cs n ( -j) E ∑ |u|=j e -V(u) 1 {V(u)≥-α} . Once again by (3.1) and (A.2), we end up with

  z)≥-α} ≤ E ∑ |z|= na z 1 {log n-3 log log n≤V(z)≤log n+log log n,V(z)≥-α} + E ∑ |z|= 1 {V(z)≤log n-3 log log n} . 32 It follows immediately from (3.1) thatE ∑ |z|= 1 {V(z)≤log n-3 log log n} = E e S 1 {S ≤log n-3 log log n} ≤ e log n-3 log log n = n (log n) 3 = o( n (log n) 2 ).On the other hand, for the second term, as a z ≤ e -V(z) ,E ∑ |z|= na z 1 {log n-3 log log n≤V(z)≤log n+log log n,V(z)≥-α} ≤nE ∑ |z|= e -V(z) e V(z)-V(z) 1 {log n-3 log log n≤V(z)≤log n+log log n,V(z)≥-α} ,which by (3.1) is equal to nE e S -S , log n -3 log log n ≤ S ≤ log n + log log n, S ≥ -α .By Markov property at the first hitting time S , one sees thatE ∑ |z|= na z 1 {log n-3 log log n≤V(z)≤log n+log log n,S ≥-α} ≤n ∑j=1 E e S -S j ; S j-1 < S j , S j = S ∈ [log n -3 log log n, log n + log log n], S j ≥ -α ≤n ∑ j=1 P S j ≥ -α, S j = S j ∈ [log n -3 log log n, log n + log log n] E e S -j ; S -j ≤ 0 . By (A.5) and (A.13), one obtains that

  z)≥-α} =:R I + R I I . (4.7) Applying (3.1) to R I I yields that R I I = ∑ ≥(log n) 2 /ε E e S ; S ≤ log n, S ≥ -α , which by (A.22) is bounded by

  -S n ≤ ne -n 1/3 + ∑ n/2≤j≤ne S j -S n . and that g is uniformly continuous. Hence, -S n = o n (1).

  13). Let S n := -S n . Observe that E[e S n ; S n ≤ 0] = E[e -S n ; S n ≥ 0] ≤ ∞ ∑ k=0 e -k P [S n ≥ 0, S n ∈ [k, k + 1)] .Applying (A.3) to S implies thatE[e S n ; S n ≤ 0] ≤ ∞ ∑ k=0 e -k c(1 + k) n 3/2 ≤ c n 3/2 , since ∑ k≥0 (1 + k)e -k < ∞.Proof of (A.14). By applying Markov property at the first hitting time S n , one sees thatE[e S n ; S n ≤ A] = n ∑ k=0 E[e S n ; S k-1 < S k ≤ A, S k ≥ S [k,n] ] = n ∑ k=0 E[e S k ; S k-1 < S k ≤ A]E[e S n-k ; S n-k ≤ 0] S k ; S k > 0, S k ≤ A]E[e S n-k ; S n-k ≤ 0] (A.28)where the last equality follows from time-reversing. Next, one observes that for any k ≥ 1, by (A.3),E[e S k ; S k > 0, S k ≤ A] ≤ ∑ j∈[0,A)∩Z e j+1 P S k > 0, S k ∈ [j, j + 1] ≤ c k 3/2 ∑ j∈[0,A)∩Z e j+1 (1 + j) ≤ c(1 + A)e A k 3/2 , since ∑ j∈[0,A √ n)∩Z e j+1 (1 + j) ≤ c(A + 1)e A .Plugging this inequality and (A.13) into (A.28) yields thatE[e S n ; S n ≤ A] ≤ n ∑ k=0 c(1 + A)e A (k + 1) 3/2 (nk + 1) 3/2 ≤ c(1 + A)e A n 3/2 ,which is what we need.Proof of (A.15). We have,E e S n ∑ 1≤i≤n e S i ≤ E e S n -S n = n ∑ k=0 E e S n -S k ; S k > S k-1 , S k ≥ S [k,n] ,then by Markov property and a time reversal for (S j , 0 ≤ j ≤ k), one gets thatE e S n ∑ 1≤i≤n e S i ≤ n ∑ k=0 P S k > S k-1 E e S n-k ; S n-k ≤ 0 ≤ n ∑ k=0 P (S k > 0) E e S n-k ; S n-k ≤ 0 .By (A.2) and (A.13) 1) 1/2 (nk + 1) 3/2 ≤ c √ n .(A.16) follows immediately from Lemma 3 in[START_REF] Ritter | Growth of random walks conditioned to stay positive[END_REF].

Final ideas for the proof of Remark 1.6 As

  C a,b is continuous and monotone on (a, b)

	and (1.21) yield Theorem 1.2, with λ(γ) := γ -1 C γ -1/2 ,γ -1/2 . More details about the proper-
	ties of λ are given in Lemma A.1.

4.2 Complementary arguments: Proofs of Lemmas 2.1, 2.2, 2.4 and Proposition 2.3 4.2.1 Proof of Lemma 2.1

	In fact, as in (2.5), E[K B∩U n	( )] = Θ(E[ K B∩U n	( )]). Then similar to (4.2), E[K B∩U n	( )] = Θ(n/(log n) 2 ).
	Let us show that E[K	B\U n

  log s n -log ,log n+log log n )

	4.2.4 Proofs of Lemma 2.4, (3.36) and (3.37)
	Proof of (2.9). Let us show that
	lim ε↓0	lim sup n→∞	1 n	E

  (R k -R [k,n/2] ) ≤ a √ n + a n R n/2 > 0 .Again by invariance principle and (3.18), as n → ∞,(A.27) √ n/2Ψ a,b n (x, h) -→ Ψ a,b (x, h) = (x, h) is monotone for x ≥ 0 and h ≥ 0 and Ψ a,b is continuous, by Dini's theorem, we have uniformly for (x, h) ∈ R 2 + , Ψ a,b n (x, h) = Ψ a,b (x, h) + o n (1) √ (S i -S i ) ≤ a √ n + a n S n/2 ≥ -α P S n/2 ≥ -α .Once again by invariance principle and the fact that lim n→∞√ nP(S n ≥ -α) = R(α)c + -S n ; S n ≥ -α, S n > S n-1 , max 1≤i≤n (S i -S i ) ≤ a √ n + a n , S n ≥ b √ n + b n .First, we show that in this case with high probability, S n/2 ≤ S nn -1/3 . In fact, -S n ; S n ≥ -α,S n/2 ≥ S nn -1/3 , S n > S n-1 , max 1≤i≤n (S i -S i ) ≤ a √ n + a n , S n ≥ b √ n + b n≤ ||g|| ∞ P S n ≥ -α, S n/2 ≥ S nn -1/3 , S n = S n , S n/2 -S n/2 ≤ a √

	which equals to
						E Ψ a,b n (	S n/2 √ n/2	,	S n/2 -S n/2 √ n/2	); S n/2 ≥ -α, max 1≤i≤n/2	(S i -S i ) ≤ a	√	n + a n
	where
		Ψ a,b n (x, h) := P R n/2 > 0 × P	R n/2 √ n/2	> (	√	2b +	b n √ n/2	) ∨ (x + h) -x,
			R n/2 -R n/2 √ n/2	≤ (	√	2a -h +	a n √ n/2	) ∧ (x +	√	α n/2	), max 1≤i≤n/2
											√
											2a .
	Because Ψ a,b n n/2	.
	As a consequence,
		P (A.11) = E	 	Ψ a,b (	S n/2 √ n/2 ,	S n/2 -S n/2 √ n/2 ) + o n (1) √ n/2	; S n/2 ≥ -α, max 1≤i≤n/2	(S i -S i ) ≤ a	√	n + a n	 
	=	√	1 n/2	E Ψ a,b (	S n/2 √ n/2	,	S n/2 -S n/2 √ n/2	) + o n (1); max 1≤i≤n/2
											1 ,
											P (A.11) =	R(α)C a,b n	+	o n (1) n	,
	with C a,b defined in (3.20).
	Proof of (A.12). We turn to consider
	nE g e S j E g n ∑ j=1 n ∑ e S j
			j=1

∑ 1≤i≤n e S i -S n ∈ [a, b] ⊂ {log alog n ≤ S n -S n ≤ log b}. c + 2 P σm 1 > ( √ 2bx) ∨ h, σm 1σm 1 ≤ ( √ 2ah) ∧ x, max 0≤s≤1 σ(m sm [s,1] ) ≤ n + a n ≤ c||g|| ∞ a(1 + α) n 1+1/6 ,

  -S n ; S n ≥ -α, S n > S n-1 , max 1≤i≤n (S i -S i ) ≤ a √ n + a n , S n ≥ b √ n + b n S j -S n ; S n ≥ -α, S n > S n-1 , max 1≤i≤n (S i -S i ) ≤ a √ n + a n , S n ≥ b √ n + b n + o n (1) n

	Therefore, we deduce that
	E g e S j =E g n ∑ j=1 ∑
	n/2≤j≤n

e

  1, P S n ≥ n 1+δ ≤ c(δ, θ)e -θn 1+δ /2 .(A.18) can be treated similarly choosing θ properly as a, decreasing to zero, function of n.Proof of (A.[START_REF] Imhof | Density factorizations for brownian motion, meander and the three-dimensional bessel process, and applications[END_REF]). LetP (A.19) := P S [n/2,n] ≤ b n , S n ≥ -α, S n > S n-1Use again the notation R k = S n -S n-k , we observe thatP (A.19) = P S n/2 ≥ -α, R n/2 -R n/2 ∈ [(S n/2b n ) + , S n/2 + α], R n/2 > 0, R n/2 > S n/2 -S n/2 ≤ E S n/2 ≥ -α, fn ( S n/2 √ n/2 ) ,where for the random walk from the time j to 2k/3, we use the fact that {S[k n -j,k/2-j] ≤ k 1/6 n + α, S 2k/3-j ≥ 0} ⊂ {S 7k/12 ≥ 0, S [k n /2,k/2] ≤ k 1/6n + α} as j ≤ k n /2. By time reversal together with (A.2), P S k/3-1 < S k/3 ≤ c/ √ k/3. Also, in view of (A.8) and (A.[START_REF] Hu | A subdiffusive behavior of recurrent random walk in random environment on a regular tree[END_REF]), for anyn 0 ≤ k ≤ n, ∑ j≤k n /2 P S [k n ,k/2] ≤ k 1/6 n , S k > S k-1 , S j-1 > S j = S k ≥ -α ≤ ∑Proof of (A.21). Applying Markov property at time k n yields thatP (A.21) = E P S kn S k-k n ≥ -α, S k-k n = S k-k n , S k n ≥ -α, S k n / ∈ [k 1/3 n , k n ] ,and recall that P u is for the distribution of the random walk starting from u. By (A.4),P S kn S k-k n ≥ -α, S k-k n = S k-k n ≤ c(1 + α + S k n )/(kk n ). This yieldsP (A.21) ≤ E (1 + α + S k n ) kk n ; S k n ≥ -α, S k n / ∈ [k 1/3 n , k n ] .We now split the above expectation into two terms, first by Markov's inequality,E (1 + α + S k n ) kk n ; S k n ≥ -α, S k n ≥ k n ≤ c n(1 + α + k n ) 3 E (1 + α + S k n ) 4 ≤ k n ; S k n ≥ -α, S k n ≤ k 1/3 k n ; S k n ≥ -α, S k n ∈ [l, l + 1] + α + l) n P S k n ≥ -α, S k n ∈ [l, l + 1] ≤ Proof of (A.[START_REF] Lyons | Random walk in a random environment and first-passage percolation on trees[END_REF]. Arguing over the first time hitting S then by Markov property, we haveE (A.22) ≤ ∑j=1 E e S j ; S j ≤ A, S j ≥ -α E e S -j 1 {S -j ≤0} S j ∈ [k, k + 1], S j ≥ -α E e S -j 1 {S -j ≤0} .

						j≤k n /2	c(1 + α) 2 (1 + α + k 1/6 n ) (j + 1) 3/2 nk 1/2 n	=	o n (1) n	.
						c nk n	,
	and also by (A.3)
						k 1/3 n
					n	≤	∑
						l=-α
	≤	k 1/3 n ∑ l=-α	c(1 c n nk 1/2	.
	These two inequalities conclude (A.21).
			≤ ∑	∑	e k+1 P
			j=1	-α≤k≤A

E

(1

+ α + S k n ) k E (1 + α + S k n ) k

By (A.13) and (A.3), we have

E (A.22) ≤ ∑ j=1 ∑ -α≤k≤A e k+1 c(1 + α)(1 + k + α) j 3/2 (j + 1) 3/2 ≤ ce A (1 + α)(1 + A + α) 3/2

.

In[START_REF] Andreoletti | Spread of visited sites of a random walk along the generations of a branching process[END_REF] the lower bound obtained is actually a little smaller than n/ log n.

Acknowledgment : We are grateful to an anonymous referee for comments that were very useful to help improve the presentation of the paper.

Note that if we replace W m (F β,β ) by W m (F β,(1±ε)β ) with β ∈ (0, 1), these arguments still work. By monotonicity of F, we have

∑ φ<y≤z e V(y) 1 {max φ<y≤z (V(y)-V(y))≤β,V(z)≥β±O(log β)} ≤ ∞ ∑ m=1 W m (F β,(1+ε)β ).

By integrability and continuity of

. Consequently, the convergence (3.35) holds.

Variance of K n and secondary results

In this section, we complete the proof of the main theorems by proving Lemmata 2.1, 2.2, 2.4 and Proposition 2.3.

Variance of K B δ ∩U n ( ) and Proof of Proposition 2.3

Recall the definition of B δ ∩ U in Section 2.1, in this section we focus on the mean of the quenched variance of K B δ ∩U n ( ) which is a key step in the proof of Proposition 2.3.

Quenched expression for the variance

Lemma 4.1. Recall that a z = P E (T z < T φ ) and let a v,z := P E (T v ∧ T z < T φ ). For every event A measurable with respect to E , denote the quenched variance of K A n ( ) as follows:

then

Proof. Note that

So the lemma comes directly.

A corollary of this Lemma is the following result, which gives a simple upper bound of the quenched variance when A = B δ ∩ U :

,

). This implies that

This ends the proof of Lemma 2.2.

Proof of Proposition 2.3

Observe that

For the second term on the right hand side, by Markov inequality and (2.5), we have

where the last equalities come from the change of measures (3.5) and (3.14).

For the first term on the right hand side of (4.4), using Tchebychev inequality on the quenched probability yields that

A.2 Results on one-dimensional random walks

In this section we state technical inequalities that are used all along the paper. The sequence (S k , k) which appears here is the one defined in (3.1). The proofs are postpone Section A.3.

We start with two well know inequalities (see [START_REF] Aidekon | The seneta-heyde scaling for the branching random walk[END_REF] for instance) and some basic Facts. There exists constant c > 0 such that for any n ≥ 1 and u ≥ 0 (A.2)

By Lemma 2.2 in [START_REF] Aidekon | The seneta-heyde scaling for the branching random walk[END_REF], there exists some constant c > 0 such that for any u ≥ 0, b ≥ a ≥ -u and any n ≥ 1,

Fact A.2. 1. For any u, α ≥ 0 and ∀n ≥ 1,

2. (a) For any n ≥ 1, B > 0 fixed, there exists c(B) > 0 such that for any u ≥

3. For any a, A, α > 0 and ∀n > m ≥ 1, (A.7)

We now state the following Lemma which is mostly a consequence of the above facts.

Lemma A.3. For any α ≥ 0, 0 < a ≤ b and n ≥ 1, we have

Following Lemma focus on asymptotic results.

We have the following convergences.

1. Moreover, for any α ≥ 0 fixed,

where C a,b ∈ (0, ∞) is a constant depending on a and b, defined in (3.20).

2. Let g : [1, ∞) → R + be a uniformly continuous and bounded function. Then,

Below we collect some more basic facts.

Fact A.5. 1. For any n ≥ 1,

For any

3. For any n ≥ 1,

For any

3)], then for any δ > 0 there exists c(δ, θ) > 0 such that for any n ≥ 1, (A.17)

6. Let a > 0. With the same hypothesis as above there exists c 2 > 0 such that

The following corollaries follow from above lemmas.

Lemma A.7. For any α > 0 fixed and k n = o(n 1/2 ), the following estimates hold uniformly for n/2 ≤ k ≤ n,

2. For A > 0 sufficiently large and any ≥ 1,

3. For any A ≥ 1 and ≥ 1,

A.3 Proofs of (A.4)-(A.19)

We show these results one by one.

Proof of (A.4). Let R k := S n -S n-k for 0 ≤ k ≤ n. Clearly, (R k , 0 ≤ k ≤ n/2) is an independent copy of (S k , 0 ≤ k ≤ n/2). Hence,

Applying (A.2) to both (S • ) and (R • ) yields that (A.25)

which is exactly (A.4).

Proof of (A.5)-(A.6). Using the same arguments as above, as S n = S n/2 + R n/2 , we get that

where ψ(x)

It follows that

This completes the proof of (A.5).

Similarly for (A.6), we have

which by independence between (S i , i ≤ n/2) and (R i , i ≤ n/2) and (A.2), is bounded by

We thus bound the left hand side of (A.9) as follows

By Markov property at the first hitting time S n ,

By (A.4) and (A.3), we deduce that

which ends the proof.

Proof of (A.10). By (A.3), one sees that

Proof of (A.11). Consider the two independent random walks (S k , 0

One observes that max 1≤i≤n

and that

It is immediate that

By invariance principle, P(

Consequently, fn (x) = o n (1) √ n , uniformly for x ∈ R + , so

Proof of (A.20). We need to obtain an upper bound for

By (3.29), to conclude that ξ 1 = o n (1), it suffices to show that uniformly on k ∈ [n 0 , n] ∩ Z,

Considering the first hitting time of S k which should be before k/2, one has

which by (A.8) and (A.4) is bounded by 1) n . Also when j ≤ k n /2, applying Markov property at time 2k/3 then at time j implies that

Proof of (A.23). Arguing over the value of Υ S implies that

which by Markov property at time j, is bounded by

By (A.13),

We split this sum into two parts: ∑ -1/3 j=1 and ∑ -1/3 ≤j≤ . For the first sum, by (A.4), one sees that For the second sum, by Markov property at j/3 and 2j/3 then by reversing time, r ,j ≤P S j/3 ≥ -α P max k≤j/3

(S k -S k ) ≤ A P S j/3 = S j/3 c (j + 1) 3/2 =P S j/3 ≥ -α P max k≤j/3 (S k -S k ) ≤ A P S j/3 ≥ 0 c (j + 1) 3/2 .

It is known by [START_REF] Hu | The slow regime of randomly biased walks on trees[END_REF] that for sufficiently large λ > 0, P(max 1≤k≤j (S k -S k ) ≤ λ) ≤ e -c j Proof of (A.24) By Markov property at time Υ S = j, E (A.24) :=E e S -S ; S ≥ A, S ≥ -α = ∑ j=1 P S j ≥ -α, S j = S j ≥ A E e S -j 1 {S -j ≤0} , which by (A.13), is bounded by ∑ j=1 P S j ≥ -α, S j = S j ≥ A c(j + 1) -3/2 . Then by (A.6), E (A.24) ≤ ∑ j=1 c(1+α) j 1/2 ( -j+1) 3/2 A ≤ c(1+α) 1/2 A .