Gabriel Faraud

Yueyun Hu

Zhan Shi

Almost sure convergence for stochastically biased random walks on trees

Keywords: Biased random walk on a Galton-Watson tree, branching random walk, slow movement, random walk in a random environment. 2010 Mathematics Subject Classification. 60J80, 60G50, 60K37

, and study a phenomenon of slow movement. In order to observe such a slow movement, the bias needs to be random; the resulting random walk is then a tree-valued random walk in random environment. We investigate the recurrent case, and prove, under suitable general integrability assumptions, that upon the system's non-extinction, the maximal displacement of the walk in the first n steps, divided by (log n) 3 , converges almost surely to a known positive constant.

Introduction 1.Stochastically biased random walks on Galton-Watson trees

Let T be a supercritical Galton-Watson tree rooted at ∅. Two vertices x and y are said to be connected, and denoted by x ∼ y, if x is either the parent or a child of y. For a vertex

x ∈ T, we denote by |x| the distance between x and the root ∅, and ∅ = x 0 , x 1 , . . . , x |x| the shortest path between the root and x. Let ω = (ω(x), x ∈ T\{∅}) be a sequence of vectors defined by ω(x) = (ω(x, y), y ∼ x) such that ω(x, y) > 0, ∀y ∼ x and that y∼x ω(x, y) = 1. Given the sequence ω, we define a random walk (X n , n ≥ 0) on T whose transition probabilities are P ω (X n+1 = y | X n = x) = ω(x, y).

For each vertex x ∈ T\{∅}, we denote its parent by ← x, and its children by (x (1) , • • • , x (N (x)) ), where N(x) stands for the number of children of x. Instead of looking at ω(x, y) (for y ∼ x and x ∈ T), it is often more convenient to study A(x) := (A i (x), 1 ≤ i ≤ N(x)) defined by (1.1)

A i (x) := ω(x, x (i) ) ω(x, ← x) , 1 ≤ i ≤ N(x).
Example (Biased random walk on a Galton-Watson tree). When A i (x) ≡ λ, ∀x, ∀i (where 0 < λ < ∞ is a constant), the random walk (X n ) is the λ-biased random walk on T studied by Lyons [START_REF] Lyons | Random walks and percolation on trees[END_REF], [START_REF] Lyons | Random walks, capacity and percolation on trees[END_REF], Lyons, Pemantle and Peres [START_REF] Lyons | Ergodic theory on Galton-Watson trees: speed of random walk and dimension of harmonic measure[END_REF], [START_REF] Lyons | Biased random walks on Galton-Watson trees[END_REF], Peres and Zeitouni [START_REF] Peres | A central limit theorem for biased random walks on Galton-Watson trees[END_REF], and Ben Arous et al. [START_REF] Ben Arous | Biased random walks on a Galton-Watson tree with leaves[END_REF]. More particularly, if A i (x) ≡ 1, ∀x, ∀i, we get the simple random walk on T.

Ben Arous and Hammond [START_REF] Ben Arous | Stable limit laws for randomly biased random walks on trees[END_REF] considered the case that A i (x) does not depend on x nor on i, but can be random. They called the resulting walk (X n ) randomly biased walk on T, and proved that the walk is more regular in some sense than the biased random walk.

We focus, in this paper, on a phenomenon of slow movement of the walk in the recurrent case. In order to exhibit the slow movement, the transition probabilities need to be random (which was already the case in the aforementioned work of Ben Arous and Hammond [START_REF] Ben Arous | Stable limit laws for randomly biased random walks on trees[END_REF]): the resulting random walk (X n ) is a so-called random walk in random environment. In dimension 1 (which, informally, corresponds to the case N(x) = 1 for all x), a celebrated theorem of Sinai [START_REF] Sinai | The limiting behavior of a one-dimensional random walk in a random medium[END_REF] says that Xn (log n) 2 converges in distribution to a non-degenerate law. From now on, we assume (A(x), x ∈ T\{∅}) to be i.i.d., and write A = (A 1 , • • • , A N ) for a random vector having the distribution of any of A(x). Note that here N itself is random and follows the law of reproduction of T. We always use P to denote the probability with respect to the environment, and P := P ⊗ P ω the annealed measure. It is convenient to consider (ω, T) as a marked tree, see (1.6) below in terms of a branching random walk. As such, when we say "for almost all environment ω", we mean, in fact, for almost all (ω, T).

Let us introduce the logarithmic moment-generating function of A:

ψ(t) := log E N i=1 A t i ∈ (-∞, ∞], t ≥ 0.
In particular, ψ(0) = log E(N). We always assume ψ(0) > 0, so that the Galton-Watson tree is supercritical. Furthermore, we assume that there exists some δ > 0 such that (1.2) ψ(t) < ∞, ∀ t ∈ (-δ, 1 + δ), and that E(N 1+δ ) < ∞.

We first recall the following recurrence/transience criterion:

Theorem A (Lyons and Pemantle [START_REF] Lyons | Random walk in a random environment and firstpassage percolation on trees[END_REF]). (i) If inf t∈[0, 1] ψ(t) ≤ 0, then for almost all ω, (X n ) is recurrent.

(ii) If inf t∈[0, 1] ψ(t) > 0, then for almost all ω, (X n ) is transient on the set of nonextinction.

Theorem A was proved in [START_REF] Lyons | Random walk in a random environment and firstpassage percolation on trees[END_REF] under the additional condition that the distribution of A i does not depend on i; this condition was removed in Faraud [START_REF] Faraud | A central limit theorem for random walk in a random environment on marked Galton-Watson trees[END_REF]. See also Menshikov and Petritis [START_REF] Menshikov | On random walks in random environment on trees and their relationship with multiplicative chaos[END_REF] for a proof of this criterion (under the additional assumptions that N > 1 is deterministic and that the law of A i does not depend on i) via Mandelbrot's multiplicative cascades.

The transient case (i.e., if inf t∈[0, 1] ψ(t) > 0) has received much research attention recently ( [START_REF] Aidékon | Transient random walks in random environment on a Galton-Watson tree[END_REF], [START_REF] Aidékon | Large deviations for transient random walks in random environment on a Galton-Watson tree[END_REF], [START_REF] Ben Arous | Biased random walks on a Galton-Watson tree with leaves[END_REF], [START_REF] Ben Arous | Stable limit laws for randomly biased random walks on trees[END_REF]).

If inf t∈[0, 1] ψ(t) < 0, the walk (X n ) is positive recurrent for almost all ω; in this case, it is not hard (see [START_REF] Hu | A subdiffusive behaviour of recurrent random walk in random environment on a regular tree[END_REF], under the additional assumptions that N is deterministic and that the law of A i does not depend on i) to prove that 1 log n max 0≤k≤n |X k | converges almost surely to a positive constant.

We assume inf t∈[0, 1] ψ(t) = 0 from now on. There are two different regimes in this case, depending on the sign of ψ ′ (1) = e -ψ(1) E{ N i=1 A i log A i }. If ψ ′ (1) < 0 (and inf t∈[0, 1] ψ(t) = 0), then by defining κ := inf{t > 1 : ψ(t) = 0} ∈ (1, ∞] (with inf ∅ := ∞) and κ 1 := 1 -1 min{κ, 2} ∈ (0, 1 2 ], the order of magnitude of |X n | is, loosely speaking, n κ 1 . [That κ > 1 is a consequence of the convexity of ψ.] More precisely, as far as strong convergence is concerned, Hu and Shi [START_REF] Hu | A subdiffusive behaviour of recurrent random walk in random environment on a regular tree[END_REF] proved (assuming N is deterministic and that the law of A i does not depend on i) that max 0≤k≤n |X k | = n κ 1 +o (1) , P-almost surely. For (functional) weak convergence, Peres and Zeitouni [START_REF] Peres | A central limit theorem for biased random walks on Galton-Watson trees[END_REF] established a quenched functional central limit theorem for biased random walk on T (corresponding to the case A i = 1 E(N ) , ∀i, and thus κ = ∞; assuming moreover N > 1 a.s.). The latter was extended by Faraud [START_REF] Faraud | A central limit theorem for random walk in a random environment on marked Galton-Watson trees[END_REF] for walks satisfying κ ∈ (8, ∞] for the quenched case and κ ∈ (5, ∞] for the annealed case. The problem of whether |Xn| n κ 1 converges weakly (in either quenched or annealed setting) for the whole region κ ∈ (2, ∞] remains open, to the best of our knowledge. This paper is devoted to the study of the second situation: ψ ′ (1) ≥ 0 (and inf t∈[0, 1] ψ(t) = 0). It was known that the walk is then extremely slow, at least under the additional conditions that N is deterministic and that the law of A i does not depend on i: there exist constants 0 < c 1 ≤ c 2 < ∞ such that (see [START_REF] Hu | Slow movement of recurrent random walk in random environment on a regular tree[END_REF])

(1.3) c 1 ≤ lim inf n→∞ max 0≤k≤n |X k | (log n) 3 ≤ lim sup n→∞ max 0≤k≤n |X k | (log n) 3 ≤ c 2 , P-a.s.
One of the main goals of this paper is to prove that almost sure convergence holds in

(1.3) if ψ ′ (1) ≥ 0 (and inf t∈[0, 1] ψ(t) = 0). The limiting constant in (1.
3), however, will have different natures depending on whether ψ ′ (1) = 0 or ψ ′ (1) > 0. If ψ ′ (1) ≥ 0 (and

inf t∈[0, 1] ψ(t) = 0), there exists 0 < θ ≤ 1 such that (1.4) ψ ′ (θ) = 0.
The case θ = 1 corresponds to ψ ′ (1) = 0; in this case, the condition inf t∈[0, 1] ψ(t) = 0 is equivalent to ψ(1) = 0, and the walk is null recurrent. The case θ < 1, on the other hand, corresponds to ψ ′ (1) > 0, and the walk is positive recurrent (see [START_REF] Faraud | A central limit theorem for random walk in a random environment on marked Galton-Watson trees[END_REF]).

To give the limiting constant in (1.3), we define

(1.5) α θ := 3π 2 2 1 θ E N i=1 A θ i (log A i ) 2 .
We write simply α for α 1 .

Theorem 1.1 Assume ψ(1) = ψ ′ (1) = 0. On the set of non-extinction,

lim n→∞ max 0≤k≤n |X k | (log n) 3 = 4 α , P-a.s.,
where the constant α = α 1 is defined in (1.5).

Theorem 1.2 Assume inf t∈[0, 1] ψ(t) = 0 and ψ ′ (1) > 0. On the set of non-extinction,

lim n→∞ max 0≤k≤n |X k | (log n) 3 = 1 α θ , P-a.s.,
where θ ∈ (0, 1) is as in (1.4) and α θ is defined in (1.5).

Theorem 1.2 is proved in Section 4, whereas the proofs of the upper and lower bounds in Theorem 1.1 are in Sections 5 and 6, respectively. In the next paragraph, we explain how it is that the (log n) 3 rate in Theorem 1.1 arises, by means of an associated branching random walk which plays the role of potential for our walk (X n ).

Branching random walks and maxima along rays

The influence of the random environment on the behaviour of (X n ) is best formulated in terms of an associated potential process. To make the presentation easier, we artificially add a special vertex, ← ∅, which is to be thought of as the parent of ∅. Since the values of the transition probabilities at a finite number of vertices have no influence on any of the results of the paper, we feel free to modify the value of ω(∅, •), the transition probability at ∅, in such a way that (A i (x), 1 ≤ i ≤ N(x)), for x ∈ T (including x = ∅ now), form an i.i.d. collection of random variables. Let ω( ← ∅, ∅) = 1, and define consistently

ω(∅, ← ∅) := 1 1 + N (∅) i=1 A i (∅)
.

The potential process associated with the random environment is defined by V (∅) := 0 and

(1.6) V (x) := - y∈ ]]∅, x]] log ω( ← y , y) ω( ← y , ⇐ y ) , x ∈ T\{∅}, where ⇐ y is the parent of ← y , [[∅, x]
] the set of vertices on the shortest path connecting ∅ to x, and ]]∅

, x]] := [[∅, x]]\{∅}.
Clearly, (V (x), x ∈ T) is a branching random walk, in the usual sense of Biggins [START_REF] Biggins | Chernoff's theorem in the branching random walk[END_REF]. It can be described as follows: Initially, a single particle is located at 0, which is the ancestor of the system. At time 1, the ancestor dies, giving birth to new particles who form the first generation, and who are positioned according to the distribution of (-log A i (∅), 1 ≤ i ≤ N(∅)). At time 2, each of the particles in the first generation dies, giving birth to new particles that are positioned (with respect to their birth places) according to the same distribution of (-log A i (∅), 1 ≤ i ≤ N(∅)); these new particles form the second generation.

The system goes on according to the same mechanism. We assume that for any n, each particle at generation n produces new particles independently of each other and of everything up to the n-th generation. The positions of the particles in the n-th generation are denoted by (

V (x), |x| = n). The condition inf t∈[0, 1] ψ(t) = 0 is equivalent to inf t∈[0, 1] E( |x|=1 e -tV (x) ) = 1, whereas ψ ′ (1) ≥ 0 means E( |x|=1 V (x)e -V (x) ) ≤ 0.
In the recurrent case, there is a simple relationship between the potential (V (x), x ∈ T)

and the walk (X n ). For any k ≥ 0, let

τ k := inf{j ≥ 1 : |X j | = k}, inf ∅ := ∞.
So τ 0 is the first return time to the root if the walk starts from ∅. It turns out that there exists 0 < c(ω) < ∞ possibly depending on the environment, such that for any n ≥ 1,

(1.7)

̺ n := P ω {τ n < τ 0 } ≥ c(ω) n exp -min |x|=n V (x) ,
where, for any vertex x, we write

(1.8) V (x) := max y∈ ]]∅, x]]
V (y).

Inequality (1.7) was proved in [START_REF] Hu | Slow movement of recurrent random walk in random environment on a regular tree[END_REF] under the additional conditions that N is deterministic and that the law of A i does not depend on i. Since the proof is simple, we reproduce it here: For any x ∈ T, let T (x) := inf{i ≥ 0 : X i = x} be the first hitting time of the walk at vertex

x. By definition, for any n ≥ 1, τ n = min |x|=n T (x), so that (1.9)

P ω {τ n < τ 0 } ≥ max |x|=n P ω {T (x) < τ 0 }.
We fix a vertex x with |x| = n. To compute P ω {T (x) < τ 0 }, we define a random sequence (σ j ) j≥0 (depending on x) by σ 0 := 0 and

σ j := inf k > σ j-1 : X k ∈ [[∅, x]]\{X σ j-1 } , j ≥ 1. If the walk (X n ) is recurrent, then (σ j ) is well-defined. Let Z k := X σ k , k ≥ 0, which is the restriction of (X j ) on the path [[∅, x]]. For i ≤ n, let x i be the unique vertex in [[∅, x]] with |x i | = i (in particular, x 0 = ∅, x n = x). Then for 1 ≤ i < n, P ω Z k+1 = x i+1 Z k = x i = ω(x i , x i+1 ) ω(x i , x i+1 ) + ω(x i , x i-1 ) = 1 -P ω Z k+1 = x i-1 Z k = x i ,
which yields

P ω {T (x) < τ 0 } = ω(∅, x 1 ) P ω (Z k ) hits x before hitting ∅ Z 0 = x 1 = ω(∅, x 1 ) e V (x 1 ) z∈ ]]∅, x]] e V (z) , (1.10)
the second identity following from a general formula (Zeitouni [START_REF] Zeitouni | Random Walks in Random Environment, XXXI summer school in probability[END_REF], formula (2.1.4)) for the exit problem of one-dimensional random walk in random environment. x) , this implies (1.11)

Since z∈ ]]∅, x]] e V (z) ≥ max z∈ ]]∅, x]] e V (z) = e V (
P ω {T (x) < τ 0 } ≤ ω(∅, x 1 ) e V (x 1 )-V (x) .
Going back to (1.9), we immediately obtain (1.7) with c(ω

) := min |x|=1 [ω(∅, x) e V (x) ] > 0.
The probability ̺ n is closely related to the maximal displacement of the branching random walk. The following simple observation was implicitly stated in [START_REF] Hu | Slow movement of recurrent random walk in random environment on a regular tree[END_REF] (pp. 1993-1996): 1))n 1/3 for all sufficiently large n, then

Fact 1.3 Assume inf t∈[0,1] ψ(t) = 0 and ψ ′ (1) ≥ 0. Let 0 < c < ∞ be a constant. Almost surely on the set of non-extinction, (i) if ̺ n ≥ e -(c+o(
lim inf n→∞ 1 (log n) 3 max 0≤k≤n |X k | ≥ 1 c 3 ; (ii) if ̺ n ≤ e -(c+o(1)
)n 1/3 for all sufficiently large n, then

lim sup n→∞ 1 (log n) 3 max 0≤k≤n |X k | ≤ 1 c 3 .
As such, an upper bound for min |x|=n V (x) yields, via inequality (1.7), a lower bound for ̺ n , which, in turn, will lead to a lower bound for the maximal displacement of the walk (X j ).

Theorem 1.4 Assume inf t∈[0, 1] E{ |x|=1 e -tV (x) } = 1 and E{ |x|=1 V (x)e -V (x) } ≤ 0. Let θ ∈ (0, 1] be such that E{ |x|=1 V (x)e -θV (x) } = 0.
We have, on the set of non-extinction,

lim n→∞ 1 n 1/3 min |x|=n V (x) = 3π 2 σ 2 θ 2 1/3 , P-a.s.,
where

σ 2 θ := 1 θ E |x|=1 V (x) 2 e -θV (x) .
We mention that Fang and Zeitouni [START_REF] Fang | Consistent minimal displacement of branching random walks[END_REF] have independently obtained Theorem 1.4, under the condition that N is non-random and A i (∅), for 1 ≤ i ≤ N, are i.i.d.

Comparing Theorem 1.4 with Theorems 1.1 and 1.2, we observe that (1.7) is optimal in the case ψ ′ (1) > 0 (or, equivalently, E{ |x|=1 V (x)e -V (x) } < 0), but not in the case ψ ′ (1) = 0 (or, equivalently, E{ |x|=1 V (x)e -V (x) } = 0).

The proofs of the theorems are organized as follows.

• Section 2: Theorem 1.4.

• Section 4: Theorem 1.2.

• Section 5: Theorem 1.1, upper bound.

• Section 6: Theorem 1.1, lower bound. [This is the heart of the paper.] Section 3 is devoted to a probability estimate for one-dimensional random walks, which will be exploited in the proofs of Theorems 1.1 and 1.2 later on.

Throughout the paper, we use the convention ∅ := 0, max ∅ := 0 and min ∅ := ∞. The letter c, with or without subscript, denotes a finite and positive constant, whose value may vary from line to line. Furthermore,

a n ∼ b n , n → ∞, means lim n→∞ an bn = 1.
2 Proof of Theorem 1.4

Assume ψ(1) = 0, i.e., E{ |x|=1 e -V (x) } = 1.

The condition E(N 1+δ ) < ∞ in (1.2) guarantees that P{N(x) < ∞, ∀x} = 1 (N(x) being the number of children of x). Recall that given a vertex x ∈ T, x 0 := ∅, x 1 , • • • ,

F : R n × R n → [0, ∞), (2.1) E |x|=n e -V (x) F [V (x i ), N(x i-1 ), 1 ≤ i ≤ n] = E F [S i , ν i-1 , 1 ≤ i ≤ n] ,
where (S i -S i-1 , ν i-1 ), for i ≥ 1, are i.i.d. random vectors, whose common distribution is determined by

(2.2) E[f (S 1 , ν 0 )] = E |x|=1 e -V (x) f (V (x), N(∅)) = E N i=1 A i f (-log A i , N) ,
for any measurable function f : R 2 → [0, ∞). Considering only the first argument, (2.1) says that for any n ≥ 1 and any measurable function F : R n → [0, ∞),

(2.3) E |x|=n e -V (x) F (V (x i ), 1 ≤ i ≤ n) = E[F (S i , 1 ≤ i ≤ n)],
with the distribution of S 1 determined by

E(f (S 1 )) = E |x|=1 e -V (x) f (V (x)) ,
for any measurable function f : R → [0, ∞). Formula (2.3) is well-known, and can be proved by means of a simple argument by induction in n. See, for example, Biggins and Kyprianou [START_REF] Biggins | Seneta-Heyde norming in the branching random walk[END_REF]. The proof of (2.1) follows exactly from the same argument. In Section 6, we will see an extension of (2.1), which, in particular, gives a probabilistic interpretation of the new random walk (S i ).

[The distribution of S 1 is well-defined upon the assumption ψ(1) = 0. If furthermore ψ ′ (1) = 0, then E(S 1 ) = 0; in words, (S n ) is a mean-zero random walk, with σ 2 = E(S 2 1 ).] Formula (2.3) naturally leads to studying the one-dimensional random walk (S n ). However, we sometimes need to work in a slightly more general setting: For each n ≥ 1, let

X (n) i , 1 ≤ i ≤ n, be i.i.d. real-valued variables; define S (n) 0 := 0 and S (n) j := j i=1 X (n) i for 1 ≤ j ≤ n. Let (a n ) be positive numbers such that a n → ∞ and a 2 n n → 0, n → ∞.
Assume that there exists some η > 0 and a constant σ 2 > 0 such that, as n → ∞,

(2.4) E(X (n) 1 ) = o a n n , sup n≥1 E(|X (n) 1 | 2+η ) < ∞, Var(X (n) 1 ) → σ 2 .
The following estimate is essentially due to Mogulskii [START_REF] Mogulskii | Small deviations in the space of trajectories[END_REF]:

Proposition 2.1 (A triangular version of Mogulskii [START_REF] Mogulskii | Small deviations in the space of trajectories[END_REF]) Assume (2.4). Let g 1 < g 2 be continuous functions on [0, 1] with g 1 (0) < 0 < g 2 (0). Consider the measurable event

F n := g 1 ( i n ) ≤ S (n) i a n ≤ g 2 ( i n ), for 1 ≤ i ≤ n .
We have

(2.5) lim n→∞ a 2 n n log P(F n ) = - π 2 σ 2 2 1 0 dt [g 2 (t) -g 1 (t)] 2 .
Moreover, for any b > 0,

(2.6) lim n→∞ a 2 n n log P F n , S (n) n a n ≥ g 2 (1) -b = - π 2 σ 2 2 1 0 dt [g 2 (t) -g 1 (t)] 2 . If the law of X (n) 1
does not depend on n (in which case we can even take η = 0), Proposition 2.1 is Mogulskii [START_REF] Mogulskii | Small deviations in the space of trajectories[END_REF]'s theorem. For a detailed proof of Proposition 2.1, see [START_REF] Gantert | Asymptotics for the survival probability in a killed branching random walk[END_REF].

A useful consequence of Proposition 2.1 is as follows. Again, if the law of X (n) 1

does not depend on n, we only need X to have a finite second moment in order to have (2.4).

Corollary 2.2 Assume that (2.4) is satisfied with a n := n 1/3 .

(i) Let f : [0, 1] → (0, ∞) be a continuous function, and let (f n ) be a sequence of continuous functions converging uniformly to f on [0, 1]. Then for any b > 0, when n → ∞,

sup 0≤u≤b n 1/3 P u ≥ S (n) i ≥ u -n 1/3 f n ( i n ), 1 ≤ i ≤ n = e -π 2 σ 2 2 (1+o(1))n 1/3 1 0 dt f 2 (t) .
(ii) For any b > a > 0, we have, as n → ∞,

n j=1 e -b(n-j) 1/3 P an 1/3 ≥ S (n) i > an 1/3 -b(n -i) 1/3 , ∀ 1 ≤ i ≤ j = e -min{b, 3π 2 σ 2 2b 2 }(1+o(1))n 1/3 . Proof of Corollary 2.2. We first prove (ii). Let ε > 0. Define k := ⌊ 1 ε ⌋, n ℓ := ℓ⌊εn⌋ for ℓ = 0, • • • , k -1 and n k := n. By (2.5
), the sum in (ii) is, for all large n and some constant c,

≤ ⌊εn⌋ k ℓ=1 e -b(n-n ℓ ) 1/3 P an 1/3 ≥ S (n) i > an 1/3 -b(n -i) 1/3 , ∀ 1 ≤ i ≤ n ℓ-1 ≤ ⌊εn⌋ k ℓ=1 e -b(n-n ℓ ) 1/3 e -( 3π 2 σ 2 2b 2 -ε)(n 1/3 -(n-n ℓ-1 ) 1/3 ) ≤ e -min{b, 3π 2 σ 2 2b 2 }(1-cε)n 1/3 .
This proves the upper bound in (ii) as ε can be arbitrarily small. The lower bound is easier:

we only need to consider two terms: j = ⌊εn⌋ and j = n, and apply again (2.5).

The proof of (i) goes along similar lines by cutting the interval {0 ≤ u ≤ b n 1/3 } into smaller intervals of length of order εn with small ε > 0, using monotonicity and applying Proposition 2.1. The details are omitted.

We now proceed to the proof of Theorem 1.4: if inf t∈[0, 1] E{ |x|=1 e -tV (x) } = 1 and

E{ |x|=1 V (x)e -V (x) } ≤ 0, then on the set of non-extinction, lim n→∞ 1 n 1/3 min |x|=n V (x) = 3π 2 σ 2 θ 2 1/3 , P-a.s.,
where

σ 2 θ := 1 θ E{ |x|=1 V (x) 2 e -θV (x) } and θ ∈ (0, 1] is such that E{ |x|=1 V (x)e -θV (x) } = 0. Without loss of generality, we can assume θ = 1. Indeed, if 0 < θ < 1, then by considering V (x) := θV (x), we have inf t∈[0, 1] E( |x|=1 e -t V (x) ) = 1 and E( |x|=1 V (x)e -V (x) ) = 0, so that by the case θ = 1, 1 n 1/3 min |x|=n max y∈ ]]∅, x]] V (y) → ( 3π 2 σ 2
2 ) 1/3 P-almost surely on the set of non-extinction, where

σ 2 := E{ |x|=1 V (x) 2 e -V (x) }.
So we only need to prove Theorem 1.4 in the case θ = 1. In the rest of the section, we assume E( |x|=1 e -V (x) ) = 1 and E( |x|=1 V (x)e -V (x) ) = 0, and prove that, on the set of non-extinction,

(2.7) lim n→∞ 1 n 1/3 min |x|=n V (x) = 3π 2 σ 2 2 1/3 , P-a.s., with σ 2 := σ 2 1 = E{ |x|=1 V (x) 2 e -V (x) }.
For the sake of clarity, we prove the upper and lower bounds in distinct parts.

Proof of (2.7): lower bound. We assume

E( |x|=1 e -V (x) ) = 1 and E( |x|=1 V (x)e -V (x) ) = 0. Let 0 < a < ( 3π 2 σ 2 2 ) 1/3 and b := ( 3π 2 σ 2 2 ) 1/3 . Let n ≥ 1. For all |x| = n, let H x := inf{j ∈ [1, n] : V (x j ) ≤ an 1/3 -b(n -j) 1/3 }, inf ∅ := ∞. n V (x j ) an 1/3 (a -b)n 1/3 H x Figure 2: H x
Assume there exists a vertex x with |x| = n such that V (x) ≤ an 1/3 . Then H x ≤ n;

writing j := H x and y := x j , we have, for all i < j, an

1/3 ≥ V (y i ) > an 1/3 -b(n -i) 1/3 and V (y) ≤ an 1/3 -b(n -j) 1/3
. Therefore, by writing

U j := |y|=j 1 {V (y)≤an 1/3 -b(n-j) 1/3 , an 1/3 ≥V (y i )>an 1/3 -b(n-i) 1/3 , ∀i<j} ,
we obtain:

P min |x|=n V (x) ≤ an 1/3 ≤ P n j=1 {U j ≥ 1} ≤ n j=1 E(U j ). By (2.3), we have E(U j ) = E[e S j 1 {S j ≤an 1/3 -b(n-j) 1/3 , an 1/3 ≥S i >an 1/3 -b(n-i) 1/3 , ∀i<j} ]. Hence P min |x|=n V (x) ≤ an 1/3 ≤ n j=1 e an 1/3 -b(n-j) 1/3 P an 1/3 ≥ S i > an 1/3 -b(n -i) 1/3 , ∀i < j .
Applying Corollary 2.2 (ii) and noting that min{b,

3π 2 σ 2 2b 2 } = ( 3π 2 σ 2 2 ) 1/3 , we get that, for any 0 < a < ( 3π 2 σ 2 2 ) 1/3 , (2.8) lim sup n→∞ 1 n 1/3 log P min |x|=n V (x) ≤ an 1/3 ≤ a - 3π 2 σ 2 2 1/3 , which implies n P{min |x|=n V (x) ≤ an 1/3 } < ∞.
The lower bound in (2.7) follows from the Borel-Cantelli lemma, as a can be as close to ( 3π 2 σ 2 2 ) 1/3 as possible.

Proof of (2.7): upper bound. Assume E{ |x|=1 e -V (x) } = 1 and

E{ |x|=1 V (x)e -V (x) } = 0. Let n ≥ 1 and b > a > ε > 0.
The key step in the proof of the upper bound in (2.7) is the following estimate, which is a consequence of the Paley-Zygmund inequality (see [START_REF] Gantert | Asymptotics for the survival probability in a killed branching random walk[END_REF] for a proof): For any Borel sets I i,n ⊂ R, 1 ≤ i ≤ n, and any integer r n ≥ 1, we have (2.9)

P ∃|x| = n : V (x i ) ∈ I i,n , ∀1 ≤ i ≤ n ≥ E[e Sn 1 {S i ∈I i,n , ν i-1 ≤rn, ∀ 1≤i≤n} ] 1 + (r n -1) n j=1 h j,n
, where h j,n := sup

u∈I j,n E e S n-j 1 {S i ∈I i+j,n -u, ∀0≤i≤n-j} ,
and

I i+j,n -u := {v -u : v ∈ I i+j,n }. [We recall that (S i -S i-1 , ν i-1 ), i ≥ 1, are i.i.d.
random vectors (with S 0 := 0) whose common distribution is given by (2.

2).]

We choose r n := ⌊e n 1/4 ⌋ and

I i,n := [(a -ε)n 1/3 -b(n -i) 1/3 , an 1/3 ]. In particular, {∃|x| = n : V (x i ) ∈ I i,n , ∀1 ≤ i ≤ n} ⊂ {min |x|=n V (x) ≤ an 1/3 }. It follows from (2.9) that P min |x|=n V (x) ≤ an 1/3 ≥ e (a-ε)n 1/3 P{S i ∈ I i,n , ν i-1 ≤ e n 1/4 , ∀ 1 ≤ i ≤ n} 1 + e n 1/4 n j=1 h j,n . Let X (n) j , 1 ≤ j ≤ n, be i.i.d. random variables such that X (n) 1 has the same distribution as S 1 conditioned on {ν 0 ≤ e n 1/4 }. Let S (n) 0 := 0 and S (n) j := X (n) 1 + ... + X (n) j for 1 ≤ j ≤ n. Then P{S i ∈ I i,n , ν i-1 ≤ e n 1/4 , ∀ 1 ≤ i ≤ n} = [ P(ν 0 ≤ e n 1/4 )] n P max 0≤k≤n S (n) k ≤ an 1/3 , S (n) i ≥ (a -ε)n 1/3 -b(n -i) 1/3 , ∀1 ≤ i ≤ n .

12

The second probability expression on the right-hand side is, according to Proposition 2.1 (we easily check that condition (2.4)

is satisfied), = exp{-(1 + o(1)) π 2 σ 2 2 n 1/3 1 0 dt (ε+b(1-t) 1/3 ) 2 }, which is bounded by exp{-( 3π 2 σ 2 2b 2 -c 1 (ε))n 1/3
} for all sufficiently large n, with c 1 (ε) denoting a constant such that lim ε→0 c 1 (ε) = 0. On the other hand, by (2.2), we have, for any η > 0,

E[(ν 0 ) η ] = E(N η N i=1 A i )
, which is finite (by Hölder's inequality and (1.2)) if η > 0 is chosen to be sufficiently small; thus [P(ν 0 ≤ e n 1/4 )] n → 1 as n → ∞. Accordingly, for all sufficiently large n and some constant c 2 (ε) satisfying lim ε→0 c 2 (ε) = 0, (2.10)

P min |x|=n V (x) ≤ an 1/3 ≥ exp{n 1/3 [a -3π 2 σ 2 2b 2 -c 2 (ε)]} 1 + e n 1/4 n j=1 h j,n
.

We now estimate n j=1 h j,n . By definition,

h j,n = sup 0≤u≤εn 1/3 +b(n-j) 1/3 E e S n-j 1 {u≥S i ≥u-εn 1/3 -b(n-j-i) 1/3 , ∀ i≤n-j} ≤ sup 0≤u≤εn 1/3 +b(n-j) 1/3 e u P u ≥ S i ≥ u -εn 1/3 -b(n -j -i) 1/3 , ∀ i ≤ n -j .
Let A be an integer such that

A ≥ 1 ε 2 . Let n ℓ := ℓ⌊ n A ⌋ for ℓ = 0, 1, • • • , A -1 and n A := n. If j ∈ [n ℓ , n ℓ+1 ] ∩ Z (for some 0 ≤ ℓ ≤ A -1), then h j,n ≤ e εn 1/3 +b(n-n ℓ ) 1/3 sup 0≤u≤(b+ε)n 1/3 P u ≥ S i ≥ u -εn 1/3 -b(n -n ℓ -i) 1/3 , ∀i ≤ n -n ℓ+1 .
We now bound the supremum on the right-hand side. If ℓ is such that 1 -ℓ+1 A ≤ ε, then we simply say that the supremum is bounded by 1, so that max n ℓ ≤j≤n ℓ+1 h j,n ≤ e εn 1/3 +b(n-n ℓ ) 1/3 . If 1 -ℓ+1 A > ε, we bound the supremum by applying Corollary 2.

2 (i) to f (t) := ε (1-ℓ+1 A ) 1/3 + b( A-ℓ A-(ℓ+1) -t) 1/3 : since f (t) ≤ ε 2/3 + b(1 + 1 εA -t) 1/3 ≤ ε 2/3 + b(1 + ε -t) 1/3 (using A ≥ 1 ε 2 for the second inequality), we have 1 0 dt f 2 (t) ≥ 3 b 2 -c 3 (ε), with c 3 (ε) denoting a constant satisfying lim ε→0 c 3 (ε) = 0; hence by Corollary 2.2 (i), max n ℓ ≤j≤n ℓ+1 h j,n ≤ e εn 1/3 +b(n-n ℓ ) 1/3 -( 3π 2 σ 2 2b 2 -c 3 (ε))(n-n ℓ+1 ) 1/3 .
Therefore, for all sufficiently large n and a constant c(ε) satisfying lim ε→0 c(ε) = 0, we have,

uniformly in all ℓ ∈ [0, A -1] ∩ Z, max n ℓ ≤j≤n ℓ+1 h j,n ≤ e n 1/3 [(b-3π 2 σ 2 2b 2 ) + +c(ε)] ,
where u + := max{u, 0}. As a consequence, max 0≤j≤n h j,n = max 0≤ℓ≤A-1 max n ℓ ≤j≤n ℓ+1 h j,n ≤ e n 1/3 [(b-3π 2 σ 2 2b 2 ) + +c(ε)] for all sufficiently large n. In view of (2.10), we obtain that, for any b > a > 0, (2.11) lim inf

n→∞ 1 n 1/3 log P min |x|=n V (x) ≤ an 1/3 ≥ -b - 3π 2 σ 2 2b 2 + + a - 3π 2 σ 2 2b 2 .
We now fix a > ( 3π 2 σ 2 2 ) 1/3 and η > 0. We can choose b > a sufficiently close to a such that (b -3π 2 σ 2 2b 2 ) +a + 3π 2 σ 2 2b 2 < η; accordingly, for all sufficiently large n, (2.12)

P min |x|=n V (x) ≤ an 1/3 ≥ e -η n 1/3 .
From here, it is routine (McDiarmid [START_REF] Mcdiarmid | Minimal positions in a branching random walk[END_REF]) to obtain the upper bound in (2.7); we produce the details for the sake of completeness. Let R n := inf{k : #{x :

|x| = k} ≥ e 2η n 1/3 }. For all large n, P R n < ∞, max k∈[ n 2 , n] min |x|=k+Rn V (x) > max |y|=Rn V (y) + an 1/3 ≤ k∈[ n 2 , n] P R n < ∞, min |x|=k+Rn V (x) > max |y|=Rn V (y) + an 1/3 ≤ k∈[ n 2 , n] P min |x|=k V (x) > an 1/3 ⌊e 2ηn 1/3 ⌋ ,
which, according to (2.12), is summable in n. By the Borel-Cantelli lemma, P-a.s. for all large n, we have either R n = ∞, or max k∈[ n 2 , n] min |x|=k+Rn V (x) ≤ max |y|=Rn V (y) + an 1/3 . By the law of large numbers for the branching random walk (Biggins [START_REF] Biggins | The first-and last-birth problems for a multitype age-dependent branching process[END_REF]), there exists a constant c ∈ (0, ∞) such that 1 n max |y|=n V (y) → c, P-almost surely upon the system's survival. In particular, upon survival, max |y|=n V (y) ≤ 2cn, P-almost surely for all large n. Consequently, upon the system's survival, P-almost surely for all large n, we have either

R n = ∞, or max k∈[ n 2 , n] min |x|=k+Rn V (x) ≤ 2cR n + an 1/3 .
Recall that the number of particles in each generation forms a supercritical Galton-Watson process. In particular, conditionally on the system's survival, #{u: |u|=k}

(EN ) k
min |x|=n V (x) ≤ 5cη log(EN) n 1/3 + an 1/3 .
Since a (resp. η) can be as close to ( 3π 2 σ 2 2 ) 1/3 (resp. 0) as possible, this yields the upper bound in (2.7), and completes the proof of Theorem 1.4.

Our proof of Theorem 1.4 gives the following deviation probability of min |x|=n V (x), which may be of independent interest.

Proposition 2.3 Assume ψ(1) = ψ ′ (1) = 0. For any 0 < a ≤ ( 3π 2 σ 2
2 ) 1/3 , we have

(2.13) lim n→∞ 1 n 1/3 log P min |x|=n V (x) ≤ an 1/3 = a - 3π 2 σ 2 2 1/3 . Proof. If 0 < a < ( 3π 2 σ 2 2 ) 1/3 ,
:= ( 3π 2 σ 2 2 ) 1/3 solves b = 3π 2 σ 2 2b 2 .] If a = ( 3π 2 σ 2
2 ) 1/3 , only the lower bound in (2.13) requires a proof, which follows immediately from (2.11).

Remark 2.4 Assume ψ(1) = ψ ′ (1) = 0. Theorem 1.4 says that, on the set of nonextinction, P-almost surely for n → ∞, there exists

x n with |x n | = n such that V (x n ) = (1 + o(1))( 3π 2 σ 2
2 ) 1/3 n 1/3 . One may wonder whether the vertices (x n ) can be chosen to form an infinite ray (i.e., each x n is a child of x n-1 ). The answer is no: Jaffuel [START_REF] Jaffuel | The critical barrier for the survival of the branching random walk with absorption[END_REF] proves that this is possible only if we increase the function ( 3π 2 σ 2

2 ) 1/3 n 1/3 to ( 81π 2 σ 2

8

) 1/3 n 1/3 .

An estimate for one-dimensional random walks

We present in this section a probability estimate for one-dimensional random walks. It will be used in the proofs of Theorems 1.1 and 1.2 in the forthcoming sections. For each

n ≥ 1, let X (n) i , 1 ≤ i ≤ n, be i.i.d. real-valued variables; let S (n) 0 := 0 and S (n) j := j i=1 X (n) i for 1 ≤ j ≤ n. Let (a n ) be positive numbers such that a n → ∞ and a 2 n n → 0, n → ∞. We write S (n) j := max 1≤i≤j S (n) i for 1 ≤ j ≤ n. Proposition 3.1 Assume (2.4). Let f : [0, 1] → (0, ∞) be a continuous function. For δ ≥ 0, we consider the event G δ (n) := (1 + δ)S (n) j -S (n) j ≤ a n f ( j n ), ∀1 ≤ j ≤ n . (i) If δ = 0, then lim n→∞ a 2 n n log P G 0 (n) = - π 2 σ 2 8 1 0 ds f 2 (s)
.

Moreover, for any fixed 0 < b < 1,

lim n→∞ a 2 n n log P G 0 (n), S (n) n -S (n) n ≤ b a n f (1) = - π 2 σ 2 8 1 0 ds f 2 (s) . (ii) If δ > 0, then lim n→∞ a 2 n n log P G δ (n) = - π 2 σ 2 2 1 0 ds f 2 (s)
.

We mention that for the centered random walk (S n ) given in (2.3), assumption (2.4) is obviously satisfied. Hence Proposition 2.1 as well as Corollary 3.2 below, hold also for (S n ).

Proof of Proposition 3.1. (i) Let 0 < ε < 1 4 min{b, min 0≤t≤1 f (t)} and let A be a large integer. Consider a sufficiently large n such that sup 0≤s<t≤1, t-s≤2Aa

2 n /n |f (t) -f (s)| ≤ ε. Let m = ⌊ n A 2 a 2 n ⌋. For 0 ≤ k < Am, let r k := k⌊Aa 2 n ⌋ and r Am := n. Note that ⌊Aa 2 n ⌋ ≤ r Am -r Am-1 ≤ 2⌊Aa 2 n ⌋. Let ℓ ∈ [0, A -1] ∩ Z and k ∈ [ℓm, (ℓ + 1)m -1] ∩ Z. For all r k ≤ j < r k+1 , |f ( j n ) -f ( ℓ A )| ≤ ε. Define E (±) n := A-1 ℓ=0 (ℓ+1)m-1 k=ℓm r k+1 -1 j=r k S (n) j -S (n) j ≤ a n (f ( ℓ A ) ± ε) .
Then

P G 0 (n) ≤ P E (+) n , P G 0 (n), S (n) n -S (n) n ≤ b a n f ( j n ) ≥ P E (-) n ∩ 0≤k≤Am {S (n) r k -S (n) r k ≤ εa n } .
Observe that for any r k , conditionally on σ{S

(n) j , 0 ≤ j ≤ r k } and on {S (n) r k -S (n) r k = x}, the reflecting process (S (n) i+r k -S (n) i+r k , 0 ≤ i ≤ r k+1 -r k ) has the same law as (max{x, S (n) i } - S (n) i , 0 ≤ i ≤ r k+1 -r k ).
Using this observation for all k, we see that

P(E (+) n ) ≤ A-1 ℓ=0 (ℓ+1)m-1 k=ℓm P max 0≤i<r k+1 -r k (S (n) i -S (n) i ) ≤ a n (f ( ℓ A ) + ε) , (3.1) P E (-) n ∩ 0≤k≤Am {S (n) r k -S (n) r k ≤ εa n } ≥ A-1 ℓ=0 (ℓ+1)m-1 k=ℓm P Υ k , (3.2) with Υ k := max 0≤i<r k+1 -r k (S (n) i -S (n) i ) ≤ a n (f ( ℓ A )-2ε), S (n)
r k+1 -r k -S (n) r k+1 -r k < εa n , S (n)
r k+1 -r k > εa n .
Now, we prove the upper bound in (i). By (3.1),

a 2 n n log P(E (+) n ) ≤ ma 2 n n A-1 ℓ=0 log P S (n) i -S (n) i ≤ a n (f ( ℓ A ) + ε), ∀ 0 ≤ i < ⌊Aa 2 n ⌋ .
According to Donsker's invariance principle,1 the probability term on the right-hand side converges, when n → ∞, to

P sup 0≤t≤1 (W (t) -W (t)) ≤ 1 σ √ A (f ( ℓ A ) + ε) ,
where W is a standard one-dimensional Brownian motion, and W (t) = sup 0≤s≤t W (s). By Lévy's identity, (W (t) -W (t), t ≥ 0) is distributed as (|W (t)|, t ≥ 0); thus we have

(3.3) P sup 0≤t≤1 (W (t) -W (t)) ≤ u = e -(1+o(1)) π 2 8u 2 , u → 0,
which can be easily deduced from Formula (5.9) of page 342 of Feller [?], taking a = 2u, t = 1 and x = u. As a consequence, for all sufficiently large A, say

A ≥ A 0 = A 0 (ε, σ, f ), log P sup 0≤t≤1 (W (t) -W (t)) ≤ 1 σ √ A (f ( ℓ A ) + ε) ≤ - (1 -ε)π 2 σ 2 A 8(f ( ℓ A ) + ε) 2 . Since m ∼ n a 2 n A 2 , we get, for A ≥ A 0 , lim sup n→∞ a 2 n n log P(E (+) n ) ≤ 1 A 2 A-1 ℓ=0 log P sup 0≤t≤1 (W (t) -W (t)) ≤ 1 σ √ A (f ( ℓ A ) + ε) ≤ - π 2 σ 2 8 1 -ε A A-1 ℓ=0 1 (f ( ℓ A ) + ε) 2 .
Letting A → ∞ and then ε → 0, we get the upper bound in (i):

lim sup n→∞ a 2 n n log P S (n) i -S (n) j ≤ a n f ( j n ), ∀1 ≤ j ≤ n ≤ - π 2 σ 2 8 1 0 ds f 2 (s)
.

To prove the lower bound in (i), we go back to the events Υ k in (3.2). Observe that for each 1 ≤ i ≤ r k+1r k , all the three events in Υ k are non-decreasing with respect to

S (n) i -S (n) i-1
. By the FKG inequality,

P Υ k ≥ P max 0≤i<r k+1 -r k (S (n) i -S (n) i ) ≤ a n (f ( ℓ A ) -2ε) ×P S (n) r k+1 -r k -S (n) r k+1 -r k < εa n P S (n) r k+1 -r k > εa n .
Recall that r k+1r k = ⌊Aa 2 n ⌋ for 0 ≤ k < Am -1, and ⌊Aa 2 n ⌋ ≤ r Amr Am-1 ≤ 2⌊Aa 2 n ⌋. Using Donsker's invariance principle again, we see that there exists a constant c(ε) > 0 such that for all k, P(S

(n) r k+1 -r k -S (n) r k+1 -r k < εa n ) P(S (n) r k+1 -r k > εa n ) ≥ c(ε).
From this, the lower bound in (i) follows in the same way as the upper bound in (i).

(ii) Let us first prove the following fact: for any fixed c > 0, (3.4)

P sup 0≤s≤1 (W (s) -W (s)) ≤ u, W (1) ≤ c u = e -π 2 2u 2 (1+o(1)) , u → 0.
To see why (3.4) holds, we denote by L(t) the local time at 0 of W up to time t, and recall from Borodin and Salminen ( [START_REF] Borodin | Handbook of Brownian motion -Facts and Formulae[END_REF], page 259, Formula 1.16.2) that, for λ > 0, 2λ) .

∞ 0 e -λt P sup s≤t |W (s)| ≤ 1, L(t) ≤ c dt = 1 λ 1 - 1 cosh( √ 2λ) 1 -e -c √ λ 2 coth( √
By analytic continuation, we get that for 0

< λ < π 2 2 , ∞ 0 e λt P sup s≤t |W (s)| ≤ 1, L(t) ≤ c dt = 1 λ 1 cos( √ 2λ) -1 1 -e -c √ λ 2 cotan( √ 2λ) .
This implies, by means of a Tauberian theorem (see, for example, Theorem 3.2 of [START_REF] Hu | Extreme lengths in Brownian and Bessel excursions[END_REF]), that

P sup 0≤s≤t |W (s)| ≤ 1, L(t) ≤ c = e -( π 2 2 +o(1))t , t → ∞,
which, by scaling, is equivalent to P(sup 0≤s≤1 |W (s)| ≤ u, L(1) ≤ u δ ) = e -( π 2 2u +o(1)) , u → 0. By Lévy's identity, the two processes (W -W, W ) and (|W |, L) have the same law; consequently, this implies (3.4). Now let us proceed to prove the upper bound in (ii). Let ε > 0, and let (r k ) be as in the proof of (i), i.e., A is a large integer, m := ⌊ n A 2 a 2 n ⌋, r k := k⌊Aa 2 n ⌋ (for 0 ≤ k < Am) and r Am := n, with n sufficiently large such that |f

( j n ) -f ( ℓ A )| ≤ ε for r k ≤ j ≤ r k+1 and k ∈ [ℓm, (ℓ + 1)m) ∩ Z. Let F (+) n := A-1 ℓ=0 (ℓ+1)m-1 k=ℓm max r k ≤j≤r k+1 (S (n) j -S (n) j ) ≤ a n (f ( ℓ A ) + ε), S (n)
r k+1 ≤ c δ a n , where c = sup 0≤t≤1 f (t). Clearly G δ (n) ⊂ F (+)
n . The Markov property yields that for each k, conditionally on σ{S

(n) j , 0 ≤ j ≤ r k } and on {S (n) r k -S (n) r k = x k , S (n) r k = y k }, the process (S (n) i+r k , S (n) i+r k -S (n) i+r k , 0 ≤ i ≤ r k+1 -r k ) has the same law as (max{x k , S (n) i } + y k , max{x k , S (n) i } -S (n) i , 0 ≤ i ≤ r k+1 -r k ). On F (+) n , we have -y k ≤ x k ≤ (c + ε)a n (recalling that c = sup 0≤t≤1 f (t)), thus y k ≥ -(c + ε)a n .
Therefore, by the Markov property,

P(F (+) n ) ≤ A-1 ℓ=0 (ℓ+1)m-1 k=ℓm P max 0≤i≤r k+1 -r k (S (n) i -S (n) i ) ≤ a n (f ( ℓ A )+ε), S (n) r k+1 -r k ≤ (c+ε+ c δ )a n .
This is the analogue of (3.1) for (ii). From here, the rest of the proof of the upper bound in (ii) is done by using exactly the same arguments as in (i), by applying (3.4) instead of (3.3). We omit the details.

The proof of the lower bound in (ii) is easy. Indeed, let 0 < ε < inf t∈[0, 1] f (t), and let

F (-) n := -(f ( i n ) -ε) ≤ S (n) i a n ≤ ε 1 + δ , ∀0 ≤ i ≤ n . Clearly F (-) n ⊂ G δ (n). By (2.5), we have lim n→∞ a 2 n n log P(F (-) n ) = - π 2 σ 2 2 1 0 dt (f (t) -ε + ε 1+δ ) 2 .
Letting ε → 0 gives the lower bound in (ii).

The following corollary follows from Proposition 3.1 exactly as Corollary 2.2 follows from Proposition 2.1. Corollary 3.2 Assume that (2.4) is satisfied with a n = n 1/3 . Let a > 0 and δ > 0. Then for n → ∞, n j=1 e -a(n-j) 1/3 P S

(n) i -S (n) i ≤ a(n -i) 1/3 , ∀ 1 ≤ i ≤ j = e -min{a, 3π 2 σ 2 8a 2 }(1+o(1))n 1/3 , n j=1 e -a(n-j) 1/3 P (1 + δ)S (n) i -S (n) i ≤ a(n -i) 1/3 , ∀ 1 ≤ i ≤ j = e -min{a, 3π 2 σ 2 2a 2 }(1+o(1))n 1/3 .

Proof of Theorem 1.2

We assume inf t∈[0, 1] ψ(t) = 0 and ψ ′ (1) ≥ 0 in this section. Let θ ∈ (0, 1] be such that ψ ′ (θ) = 0 as in (1.4). By Theorem 1.4 and (1.7), we get that, on the set of non-extinction,

lim inf n→∞ 1 n 1/3 log ̺ n ≥ -α 1/3 θ , P-a.s.,
where

α θ := 3π 2 2θ E[ N i=1 A θ i (log A i ) 2 ] = 3π 2 2θ E[ |x|=1 V (x) 2 e -θV (x)
], and ̺ n := P ω {τ n < τ 0 } is as in (1.7). In view of Fact 1.3, it remains only to check that if ψ ′ (1) > 0 (i.e., if θ < 1), then we have, on the set of non-extinction, We do not assume ψ ′ (1) > 0 for the moment (so θ can be 1, and the inequality (4.2) below can also be used in the proof of Theorem 1.1 in the next section). Let a > 0, n ≥ 1 and δ ≥ 0. For any y with |y| ≤ n, say |y| = j, we introduce the following event:

E δ (y) = (1 + δ)V (y) -V (y) ≥ a θ (n -j) 1/3 ∩ j-1 i=1 (1 + δ)V (y i ) -V (y i ) < a θ (n -i) 1/3 ,
where y i is the unique vertex of [[∅, y]] that is in the i-th generation, whereas V (x) :=

max z∈ ]]∅, x]] V (z) as in (1.8). a(n -i) 1/3 i j V (x i ) -V (x i ) a θ (n -i) 1/3 i j (1 + δ)V (x i ) -V (x i )
= n. Let j = j(x) ∈ [1, n] ∩ Z be the smallest integer such that (1 + δ)V (x j ) -V (x j ) ≥ a θ (n -j) 1/3
. Such a j exists. Moreover, we have T (x) ≥ T (x j ), and E δ (x j ) holds. Consequently,

τ n = inf |x|=n T (x) ≥ min 1≤j≤n inf{T (y) : |y| = j and E δ (y) holds}, so that ̺ n = P ω {τ n < τ 0 } ≤ n j=1 |y|=j 1 E δ (y) P ω {T (y) < τ 0 }. By (1.11), we obtain: (4.2) ̺ n ≤ n j=1 |y|=j 1 E δ (y) ω(∅, y 1 ) e V (y 1 )-V (y) = ω(∅, ← ∅) n j=1 |y|=j 1 E δ (y) e -V (y) ,
which is bounded by n j=1 |y|=j 1 E δ (y) e -V (y) .

We now assume furthermore ψ ′ (1) > 0, so that θ < 1. We choose δ ∈ (0, 1 θ -1). Since (1 + δ)θ < 1, we have

̺ (1+δ)θ n ≤ n j=1 |y|=j 1 E δ (y) e -(1+δ)θV (y) .
Consider the branching random walk V (x) := θV (x) for any x. If we define ψ(t) := log E[ |x|=1 e -t V (x) ], then ψ(1) = ψ ′ (1) = 0. We apply formula (2.3) to ( V (x)), and obtain a centered one-dimensional random walk ( S i , 0

≤ i ≤ n) with σ 2 := E( S 2 1 ) = E[ |x|=1 θ 2 V (x) 2 e -θV (x) ] such that for 1 ≤ j ≤ n (writing S i := max 1≤k≤i S k , ∀i), E |y|=j 1 E δ (y) e -(1+δ)θV (y) = E e S j -(1+δ) S j 1 {(1+δ) S i -S i <a(n-i) 1/3 , ∀i<j, (1+δ) S j -S j ≥a(n-j) 1/3 } ≤ e -a(n-j) 1/3 P (1 + δ) S i -S i < a(n -i) 1/3 , ∀i < j . It follows that E(̺ (1+δ)θ n ) ≤ n j=1 e -a(n-j) 1/3 P (1 + δ) S i -S i < a(n -i) 1/3 , ∀i < j . We choose a := ( 3π 2 σ 2 2 ) 1/3 = θα 1/3 θ . Applying Corollary 3.2 (ii) to ( S i ), we get E(̺ (1+δ)θ n
) ≤ e -(a+o(1))n 1/3 , for n → ∞. By Chebyshev's inequality and the Borel-Cantelli lemma, Palmost surely for n → ∞, ̺ (1+δ)θ n ≤ e -(a+o(1))n 1/3 . Since δ can be arbitrarily small, this implies (4.1), and completes the proof of Theorem 1.2.

Proof of Theorem 1.1: upper bound

We prove that if ψ(1) = ψ ′ (1) = 0, then2

(5.1) lim sup

n→∞ max 0≤k≤n |X k | (log n) 3 ≤ 8 3π 2 σ 2 , P-a.s.,
where

σ 2 := E{ |x|=1 V (x) 2 e -V (x) }.
Let, for any n ≥ 1, (5.2)

β n := P ω {τ n < T← ∅ },
where τ n := inf{i ≥ 1 : |X i | = n} is as before the first time that the walk reaches the n-th generation, whereas T← ∅ := inf{i ≥ 0 :

X i = ←
∅} is the first time that the walk hits ← ∅. There is a simple relation between β n and ̺ n := P ω {τ n < τ 0 }, as stated in the following lemma.

We mention that no condition on ψ is in force for the lemma.

Lemma 5.1 Assume that the walk (X n ) is recurrent. We have, for all n ≥ 1,

(5.3) ̺ n ≤ β n ≤ ̺ n ω(∅, ← ∅)
.

∅ := inf{i > T (k-1) ∅ : X i = ∅} (for k ≥ 1). In words, T (k)
∅ is the k-th return time to the root ∅. [Thus T

(1)

∅ = τ 0 .] Since the walk is recurrent, each T (k)
∅ is well-defined. Recall that β n represents the probability that, starting from the root, the walk visits generation n before hitting ← ∅. By considering the number of returns to ∅ (which can be 0) by the walk before visiting generation n, we have

β n = P ω {τ n < T← ∅ } = ∞ k=0 P ω T (0) ∅ < T (1) ∅ < • • • < T (k) ∅ < τ n < T (k+1) ∅ , τ n < T← ∅ .
Applying the strong Markov property successively at

T (k) ∅ , • • • , T (1)
∅ , we see that the probability on the right-hand side equals [P ω {T

(1) ∅ < (τ n ∧ T← ∅ )}] k P ω {τ n < T (1)
∅ } (notation: u ∧ v := min{u, v}). Therefore

β n = P ω {τ n < T (1) ∅ } 1 -P ω {T (1) ∅ < (τ n ∧ T← ∅ )} = ̺ n 1 -P ω {τ 0 < (τ n ∧ T← ∅ )} . Since 1 -P ω {τ 0 < (τ n ∧ T← ∅ )} ≥ 1 -P ω {τ 0 < T← ∅ } = ω(∅, ← ∅)
, this yields the lemma.

We now turn to the proof of (5.1). Assume ψ(1) = ψ ′ (1) = 0. We claim that it suffices to prove that (5.4) lim sup

n→∞ 1 n 1/3 log E(β n ) ≤ - 3π 2 σ 2 8 1/3
. Indeed, if (5.4) holds, then by Chebyshev's inequality and the Borel-Cantelli lemma, for any ε > 0 and P-almost surely all sufficiently large n,

β n ≤ exp[-(1 -ε)( 3π 2 σ 2 8 ) 1/3 n 1/3 ], which by Lemma 5.1 yields ̺ n ≤ exp[-(1 -ε)( 3π 2 σ 2 8 ) 1/3 n 1/3
]. In view of Fact 1.3, we obtain (5.1). It remains to prove (5.4). Let a := ( 3π 2 σ 2 8 ) 1/3 and n ≥ 1. By (4.2) and Lemma 5.1, y) ,

E(β n ) ≤ n j=1 E |y|=j 1 E 0 (y) e -V (
where

E 0 (y) := V (y) -V (y) ≥ a(n -j) 1/3 ∩ j-1 i=1 V (y i ) -V (y i ) < a(n -i) 1/3 .
Applying (2.3), this leads to (with S j := max 1≤i≤j S i as before):

E(β n ) ≤ n j=1 E e S j 1 {S j -S j ≥a(n-j) 1/3 , S i -S i <a(n-i) 1/3 , ∀i<j} e -S j ≤ n j=1 e -a(n-j) 1/3 P S i -S i < a(n -i) 1/3 , ∀i < j ,
which, according to Corollary 3.2 (i), is bounded by exp[-(1+o( 1))( 3π 2 σ 2 8 ) 1/3 n 1/3 ] for n → ∞. This yields (5.4).

Proof of Theorem 1.1: lower bound

We start by recalling a spinal decomposition for the branching random walk (V (x)). This decomposition has been used in the literature by many authors in various forms, going back at least to Kahane and Peyrière [START_REF] Kahane | Sur certaines martingales de Mandelbrot[END_REF]. The material in this paragraph is borrowed from Lyons, Pemantle and Peres [START_REF] Lyons | Conceptual proofs of L log L criteria for mean behavior of branching processes[END_REF] and Lyons [START_REF] Lyons | A simple path to Biggins' martingale convergence for branching random walk[END_REF]. The starting point is to a change-of-probabilities technique on the space of trees; we refer to the aforementioned references for more precision.

Assume ψ(1) = 0, i.e., E{ |x|=1 e -V (x) } = 1. Let

W n := |x|=n e -V (x) , n ≥ 0.
Clearly, (W n ) is a martingale with respect to the filtration (F n ), where F n is the sigmaalgebra generated by the branching random walk in the first n generations.

By Kolmogorov's extension theorem, there exists a probability Q on F ∞ (the sigmaalgebra generated by the branching random walk) such that for any n, (6.1)

Q | Fn = W n • P | Fn , i.e., Q(A) = E(W n 1 A ), ∀A ∈ F n .
The law of the branching random walk under the new probability Q is called the law of a size-biased branching random walk. It is clear that the size-biased branching random walk survives with probability one.

There is a one-to-one correspondence between a branching random walk and a marked tree. On the enlarged probability space formed by marked trees with distinguished rays, we may construct a probability Q satisfying (6.1), and an infinite ray {w 0 = ∅, w 1 , ..., w n , ..} such that for any n ≥ 1,

← w n = w n-1 (recalling that ← x is the parent of x) and (6.2) Q w n = x F n = e -V (x) W n , ∀ |x| = n.
For any individual x = ∅, let

∆V (x) := V (x) -V ( ← x).
We write, for k ≥ 1, (6.3)

I k := x : |x| = k, ← x = w k-1 , x = w k .
In words, I k is the set of children of w k-1 except w k , or equivalently, the set of the brothers of w k , and is possibly empty. Finally, let us introduce the following sigma-field:

(6.4) G n := σ (∆V (x), x ∈ I k ), V (w k ), w k , I k , 1 ≤ k ≤ n .
The promised spinal decomposition is as follows (xu denoting concatenation of x and u). Although it slightly differs from the spinal decomposition presented in Lyons [START_REF] Lyons | A simple path to Biggins' martingale convergence for branching random walk[END_REF], we feel free to omit the proof. Proposition 6.1 Assume ψ(1) = 0, and fix n ≥ 1. Under probability Q, (i) the random variables (∆V (w k ), ∆V (x), x ∈ I k ), 1 ≤ k ≤ n, are i.i.d.;

(ii) conditionally on G n , the shifted branching random walks ({V (xu) -V (x)} |u|=k , 0 ≤ k ≤ n -|x|), for x ∈ n k=1 I k , are independent, and have the same law as ({V (u)} |u|=k , 0 ≤ k ≤ n -|x|) under P.

We now proceed to (the beginning of) the proof of the lower bound in Theorem 1.1, of which we recall the statement: under the assumption ψ(1) = ψ ′ (1) = 0, we have, on the set of non-extinction, (6.5) lim inf

n→∞ max 0≤k≤n |X k | (log n) 3 ≥ 4 α = 8 3π 2 σ 2 , P-a.s.,
where

σ 2 := E{ |x|=1 V (x) 2 e -V (x) }.
T← ∅ = inf{i ≥ 0 : X i = ← ∅}.
We claim that it suffices to prove that (6.6) lim inf

n→∞ 1 n 1/3 log E(β n ) ≥ - 3π 2 σ 2 8 1/3
.

It is indeed easy to check that (6.6) implies (6.5): Let S := {the system survives},

S n := {the system survives at least until generation n}. Clearly S ⊂ S n for any n. Recall that there exists (see [START_REF] Hu | Minimal position and critical martingale convergence in branching random walks, and directed polymers on disordered trees[END_REF], p. 755) a constant c > 0 such that for all large n,

P(W n < n -c | S n ) ≤ n -2 .
On the other hand, we have (see [START_REF] Hu | A subdiffusive behaviour of recurrent random walk in random environment on a regular tree[END_REF], p. 543, Remark; the result therein states for the regular tree, but the same proof by convexity obviously holds in the general case)

E e -t βn E(βn)
≤ E(e -tWn ), t ≥ 0.

Since

β n = 0 = W n on S c n , it is equivalent to say that E(e -t βn E(βn) | S n ) ≤ E(e -tWn | S n ).
Therefore, for any ε > 0 and all sufficiently large n,

P β n E(β n ) < e -εn 1/3 S n ≤ e 1 E e -e εn 1/3 Wn S n ≤ n -2 e + e -n -c e εn 1/3 .
Since S ⊂ S n , this implies n P( βn E(βn) < e -εn 1/3 | S ) ≤ 1

P(S )

n P( βn E(βn) < e -εn 1/3 | S n ) < ∞. If (6.6) holds, then by the Borel-Cantelli lemma, on the set S , P-almost surely for all sufficiently large n,

β n ≥ e -εn 1/3 E(β n ) ≥ exp{-[2ε + ( 3π 2 σ 2 8 ) 1/3 ]n 1/3 }, and thus ̺ n ≥ ω(∅, ← ∅) exp{-[2ε + ( 3π 2 σ 2 8 ) 1/3
]n 1/3 } (Lemma 5.1). In view of Fact 1.3, we obtain (6.5), the lower bound in Theorem 1.1.

The rest of the section is devoted to the proof of (6.6). Let as before ̺ n := P ω {τ n < τ 0 }. Since β n ≥ ̺ n (Lemma 5.1), we only need to bound E(̺ n ) from below.

For any vertex x, let P x ω be the (quenched) probability such that P x ω {X 0 = x} = 1. We first prove a formula for ̺ n without the assumption ψ(1) = ψ ′ (1) = 0. We mention that if |x| = n, then under P x ω , τ n is the first return time to generation n. Lemma 6.2 Assume that the walk (X n ) is recurrent. For any n ≥ 1, we have

̺ n = ω(∅, ← ∅) |x|=n e -V (x) ω(x, ← x) P x ω {τ n > τ 0 }.
Proof of Lemma 6.2. The beginning of the proof uses a similar idea as in the proof of Lemma 5.1, except that instead of considering the number of returns to ∅ before hitting generation n, we consider the last site at generation n visited by the walk during an excursion. More precisely, for any x with |x| ≥ 1, let T 0)

x := 0 and

T (k) x := inf{i > T (k-1) x : X i = x} (for k ≥ 1). In words, T (k) x
is the time of the k-th visit at x.

Recall that ̺ n is the (quenched) probability that during an excursion away from the root ∅, the walk hits generation n. By considering the last site at generation n visited by the walk during the excursion, we have

̺ n = |x|=n ∞ k=1 P ω T (k) x < τ 0 < T (k+1)
x , max

T (k) x <i≤τ 0 |X i | < n = |x|=n ∞ k=1 P ω T (k) x < τ 0 , max T (k) x <i≤τ 0 |X i | < n .
Applying the strong Markov property at T

x , we see that the probability on the right-hand side equals P ω {T (k)

x < τ 0 } P x ω {τ n > τ 0 }. Therefore,

̺ n = |x|=n P x ω {τ n > τ 0 } ∞ k=1 P ω {T (k) x < τ 0 } = |x|=n P x ω {τ n > τ 0 }E ω τ 0 -1 i=0 1 {X i =x} . E ω ( τ 0 -1 i=0 1 {X i =x} )
, is the expected number of visits at site x in an excursion, and can therefore be explicitly computed. Indeed one can easily check that, as a function of x, it is invariant with respect to the transition matrix ω(x, y). In the particular setting of Markov chains on trees any invariant measure can be computed, using an easy recurrence. One gets that all the invariant measures are proportional to π(x) := 1 ω(x,

←

x ) e -V (x) , for x = ← ∅ (Note that this formula is also valid for x = ∅, because of the consistent definition of ω(∅, ← ∅)). Therefore, there exists 0 < c(ω) < ∞ such that

E ω τ 0 -1 i=0 1 {X i =x} = c(ω) ω(x, ← x) e -V (x) .
To determine the value of c(ω), we take x := ∅, to see that c(ω) = ω(∅, ← ∅). This yields the lemma.

Assume ψ(1) = 0. We make use of the size-biased branching random walk, and work under the new probability Q. Recall the definitions of Q and w n from (6.1) and (6.2), respectively. By Lemma 6.2,

E(̺ n ) = E Q ω(∅, ← ∅) ω(w n , w n-1 ) P wn ω {τ n > τ 0 } .
We observe that

P wn ω {τ n > τ 0 } = n j=1 P w j ω {τ n > T (w j-1 )} =: n j=1 Y j .
Obviously, Y n = ω(w n , w n-1 ), Y n-1 = ω(w n-1 , w n-2 ). Let j ≤ n -2. By the Markov property, Y j = ω(w j , w j-1 ) + x: ← x =w j ω(w j , x)P x ω {τ n > T (w j-1 )}, whereas by the strong Markov property, P x ω {τ n > T (w j-1 )} = P x ω {τ n > T (w j )} Y j for all x such that ← x = w j . Accordingly,

Y j = ω(w j , w j-1 ) 1 -x: ← x =w j ω(w j , x) P x ω {τ n > T (w j )} = 1 1 + x: ← x =w j B(x)P x ω {τ n < T (w j )} , where B(x) := e -[V (x)-V ( ← x )] = ω( ← x , x) ω( ← x, ⇐ x) .
So, if we write

ξ j := x: ← x =w j , x =w j+1 B(x)P x ω {τ n < T (w j )}, 1 ≤ j ≤ n -2, then Y j = 1 1+ξ j +(1-Y j+1 )B(w j+1 ) , 1 ≤ j ≤ n -2, and E(̺ n ) = E Q { ω(∅, ← ∅) ω(wn, w n-1 ) n j=1 Y j } = E Q {ω(∅, ← ∅) n-1 j=1 Y j }.
Let G n be the sigma-algebra generated by the first n generations of the spine (see (6.4)). By Proposition 6.1, under Q, the random variables ξ 1 , • • • , ξ n-1 are conditionally independent given G n . Moreover, for any 1 ≤ j ≤ n -2, (6.7)

E Q (ξ j | G n ) = x: ← x =w j , x =w j+1 B(x) E(β n-1-j ) ≤ E(β n-1-j )
ω(w j , w j-1 ) .

We now provide a lower bound for E(̺ n ), by replacing (Y j ) 1≤j≤n-1 by a new collection of random variables, denoted by (Z j ) 1≤j≤n-1 and defined as follows: Z n-1 := Y n-1 = ω(w n-1 , w n-2 ) and for 1 ≤ j ≤ n -2, (6.8)

Z j := 1 1 + E Q (ξ j | G n ) + (1 -Z j+1 )B(w j+1 )
.

Since Z n-1 , B(w n-1 ), B(w n-2 ), • • • , B(w 2
) are G n -measurable, it follows by backwards induction on j that each Z j , for 1

≤ j ≤ n -1, is G n -measurable.
Lemma 6.3 Assume ψ(1) = 0. For any n ≥ 3, we have

E Q n-1 j=1 Y j G n ≥ n-1 j=1 Z j , Q-a.s.
Proof of Lemma 6.3. For any c ∈ [0, 1] and a := (a 1 , • • • , a n-1 ) ∈ R n-1 + , we define F c,a n-1 (u n-1 ) := c, u n-1 ∈ R + , and for 1 ≤ j ≤ n -2,

F c,a j (u j , • • • , u n-2 ) := 1 1 + u j + a j+1 [1 -F c,a j+1 (u j+1 , • • • , u n-2 )] , (u j , • • • , u n-2 ) ∈ R n-j-1 + .
Then by backwards induction on j, we have, for 1 ≤ j ≤ n -1,

Y j = F Y n-1 ,B(w) j (ξ j , • • • , ξ n-2 ), Z j = F Z n-1 ,B(w) j (E Q (ξ j | G n ), • • • , E Q (ξ n-2 | G n )),
where B(w) := (B(w 1 ), • • • , B(w n-1 )). Note that both Y n-1 and B(w) are G n -measurable.

Recall that (under

Q) ξ 1 , • • • , ξ n-2 are conditionally independent given G n . By Jensen's inequality, if Φ : R n-2 + → R is coordinate-wise convex, then E Q {Φ(ξ 1 , • • • , ξ n-2 ) | G n )} ≥ Φ(E Q (ξ 1 | G n ), • • • , E Q (ξ n-2 | G n )), Q-a.
∈ R n-1 + , (u 1 , • • • , u n-2 ) → n-1 j=1 F c,a j (u j , • • • , u n-2 ) as a function on R n-2 + , is convex in each of u i .
Since the product of non-negative, coordinate-wise non-increasing, coordinate-wise convex functions is still (non-negative, coordinate-wise non-increasing, and) coordinate-wise convex, we only have to check that for any j ≤ n -2, the function (

u j , • • • , u n-2 ) → F c,a j (u j , • • • , u n-2
) is non-negative (which is obvious), coordinate-wise non-increasing, and coordinate-wise convex. We prove it by induction on j.

By definition,

F c,a n-2 (u n-2 ) = [1 + u n-2 + (1 -c)a n-1 ] -1
, which is obviously non-increasing and convex in u n-2 .

Assume that for 1

≤ j ≤ n -3, (u j+1 , • • • , u n-2 ) → F c,a j+1 (u j+1 , • • • , u n-2
) is coordinatewise non-increasing and coordinate-wise convex. Since

F c,a j (u j , • • • , u n-2 ) = 1 1 + u j + a j+1 [1 -F c,a j+1 (u j+1 , • • • , u n-2 )]
, F c,a j is non-increasing and convex in each of u i (for j ≤ i ≤ n -2): the monotonicity is obvious, whereas the convexity follows from the fact that y → 1 1+u j +(1-y)a j+1 is convex and non-decreasing on [0, 1] and that f • g is convex if f is convex and non-decreasing while g is convex.

Recall

that E(̺ n ) = E Q {ω(∅, ← ∅) n-1 j=1 Y j }. Since ω(∅, ← ∅) is G n -measurable, it follows from Lemma 6.3 that (6.9) E(̺ n ) ≥ E Q ω(∅, ← ∅) n-1 j=1 Z j .
We now give a lower bound for n-1 j=1 Z j by means of a deterministic lemma. The proof of the lemma is in the Appendix. Lemma 6.4 Let n > k ≥ 2. Let b j+1 > 0 and r j ≥ 0 for all 0 ≤ j < n. Define (z j ) 1≤j≤n by z n = 0 and

z j := 1 1 + r j + b j+1 (1 -z j+1 ) , 1 ≤ j ≤ n -1.
Let v(0) := 0 and v(j

) := -j i=1 log b i , 1 ≤ j ≤ n. For any m 0 = 0 < m 1 < ... < m k = n-1, we have n-1 j=1 z j ≥ 2 -k k i=1 (m i -m i-1 ) exp - k i=1 λ i + (m i -m i-1 ) 2 r (i) e v * i ,
where for 1 ≤ i ≤ k (with y + := max{y, 0} for y ∈ R),

r (i) := max m i-1 <j≤m i r j , λ i := max m i-1 <j≤m i (v(j) -v(m i )) + (v(m i ) -v(1 + m i )) + , v * i := max m i-1 <j≤ℓ≤m i (v(j) -v(ℓ)).
We continue with the proof of the lower bound in Theorem 1.1. Recall from (6.9) that

E(̺ n ) ≥ E Q {ω(∅, ← ∅) n-1 j=1 Z j }. Let k ≥ 2 and m 0 := 0 < m 1 < m 2 < ... < m k = n -1.
Taking b j+1 = B(w j+1 ) and r j := E Q (ξ j | G n ), we note from (6.8) that we may take the choice of z j = Z j in Lemma 6.4. Applying this lemma, and arguing that k i=1 (m im i-1 ) ≤ k i=1 n = n k , we find that (6.10)

E(̺ n ) ≥ 1 (2n) k E Q ω(∅, ← ∅) e -k i=1 Λ i -k i=1 (m i -m i-1 ) 2 r (i) e S * i ,
where, for any 1

≤ i ≤ k, r (i) := max m i-1 <j≤m i E Q (ξ j | G n ) ≤ max m i-1 <j≤m i E(β n-1-j ) ω(w j , w j-1 ) ,
Λ i := max m i-1 <j≤m i (S j -S m i ) + (S m i -S 1+m i ) + , S * i := max m i-1 <j≤ℓ≤m i (S j -S ℓ ),
with S j := V (w j ), 0 ≤ j ≤ n. [In the inequality for r (i) , we used (6.7).]

We choose: χ := 1 100 , k := ⌊n

1-χ 3 ⌋, m 0 := 0, m i := n -(k -i) 3 ⌊n χ ⌋ for 1 ≤ i ≤ k -1, and m k := n -1.
Let c > 1 be a constant sufficiently large such that

Q{S 2 ≥ S 1 , ω(w 1 , ∅) ≥ 1 c } > 1 c . Let E (1) n := m k j=m k-1 +1
S j+1 ≥ S j , ω(w j , w j-1 ) ≥ 1 c .

On E

n , we have Λ k ≤ 0, r (k) ≤ c, and S * k = 0, whereas by definition, m km k-1 = ⌊n χ ⌋ -1 ≤ n χ . Therefore, by (6.10),

E(̺ n ) ≥ e -c n 2χ (2n) k E Q ω(∅, ← ∅) e -k-1 i=1 Λ i -k-1 i=1 (m i -m i-1 ) 2 r (i) e S * i 1 E (1) n = e -c n 2χ (2n) k E Q ω(∅, ← ∅) e -k-1 i=1 Λ i -k-1 i=1 (m i -m i-1 ) 2 r (i) e S * i Q(E (1) n ),
the last identity being a consequence of the fact (notation: w -1 := ← ∅) that under Q, (S j -S j-1 , ω(w j-1 , w j-2 )), for j ≥ 1, are independent (they are i.i.d. for j ≥ 2). By the definition of c, Q(E

(1) n ) = [Q{S 2 ≥ S 1 , ω(w 1 , ∅) ≥ 1 c }] m k -m k-1 ≥ ( 1 c ) m k -m k-1 = ( 1 c ) ⌊n χ ⌋-1 . Hence, E(̺ n ) ≥ e -c n 2χ (2n) k c ⌊n χ ⌋-1 E Q ω(∅, ← ∅) e -k-1 i=1 Λ i -k-1 i=1 (m i -m i-1 ) 2 r (i) e S * i .
Let ε ∈ (0, χ 3 ). Write a * := ( 3π 2 σ 2 8 ) 1/3 . By (5.4), there exists some constant c 1 > 0 such that E(β i ) ≤ c 1 e -(a * -ε)(i+1) 1/3 for all i ≥ 1. Thus

r (i) ≤ c 1 e -(a * -ε)(n-m i ) 1/3 min m i-1 <j≤m i ω(w j , w j-1 ) , 1 ≤ i ≤ k. Consider E (2) n := ω(w j , w j-1 ) ≥ e -n ε , ∀ 1 ≤ j ≤ m k-1 ∩ {ω(∅, ← ∅) ≥ e -n ε }. On E (2)
n , we have, for any 1

≤ i ≤ k -1, r (i) ≤ c 1 e -(a * -ε)(n-m i ) 1/3 +n ε , whereas m i -m i-1 ≤ n, thus (m i -m i-1 ) 2 r (i) ≤ e -(a * -2ε)(n-m i ) 1/3 (
for all sufficiently large n; we insist on the fact that i < k). Hence (6.11)

E(̺ n ) ≥ e -c n 2χ -n ε (2n) k c ⌊n χ ⌋-1 E Q e -k-1 i=1 [Λ i +e S * i -(a * -2ε)(n-m i ) 1/3 ] 1 E (2) n . Let, for 1 ≤ i ≤ k -1, E (3)
n,i := S * i < (a * -2ε)(n -m i ) 1/3 , max m i-1 <j≤m i (S j -S m i ) ≤ n ε , |S 1+m i -S m i | ≤ n ε .
On the event E

n,i (for 1 ≤ i ≤ k -1), we have Λ i ≤ n ε + n ε = 2n ε , and, of course,

S * i -(a * -2ε)(n -m i ) 1/3 ≤ 0, so that Λ i + e S * i -(a * -2ε)(n-m i ) 1/3 ≤ 2n ε + 1 ≤ 3n ε .
Going back to (6.11), we obtain: (6.12)

E(̺ n ) ≥ e -c n 2χ -n ε -3n ε (k-1) (2n) k c ⌊n χ ⌋-1 Q E (2) n ∩ k-1 i=1 E (3)
n,i .

By independence,

Q E (2) n ∩ k-1 i=1 E (3) n,i = Q{ω(∅, ← ∅) ≥ e -n ε } k-1 i=1 Q E (3) n,i , min m i-1 <ℓ≤m i ω(w ℓ , w ℓ-1 ) ≥ e -n ε ≥ 1 2 k-1 i=1 Q E (3) n,i , min m i-1 <ℓ≤m i ω(w ℓ , w ℓ-1 ) ≥ e -n ε ,
the last inequality holding for all sufficiently large n (in view of the fact that Q{ω(∅,

← ∅) ≥ e -n ε } → 1, n → ∞). By independence again, for any 1 ≤ i ≤ k -1, Q E (3)
n,i , min

m i-1 <ℓ≤m i ω(w ℓ , w ℓ-1 ) ≥ e -n ε = Q |S 1+m i -S m i | ≤ n ε , ω(w m i , w m i -1 ) ≥ e -n ε × ×Q S * i < (a * -2ε)(n -m i ) 1/3 , max m i-1 <j≤m i (S j -S m i ) ≤ n ε , min m i-1 <ℓ<m i ω(w ℓ , w ℓ-1 ) ≥ e -n ε = Q |S 2 -S 1 | ≤ n ε , ω(w 1 , ∅) ≥ e -n ε × Q F i (n), min 1≤ℓ<∆ i ω(w ℓ , w ℓ-1 ) ≥ e -n ε , where, for 1 ≤ i ≤ k -1, ∆ i := m i -m i-1 , F i (n) := max 1≤ℓ≤∆ i (S ℓ -S ℓ ) < (a * -2ε)(n -m i ) 1/3 , S ∆ i -S ∆ i ≤ n ε , with S ℓ := max 1≤j≤ℓ S j as before. Again, Q{|S 2 -S 1 | ≤ n ε , ω(w 1 , ∅) ≥ e -n ε } is greater than 1 2
for large n because it converges to 1. Therefore, for all large n,

Q E (2) n ∩ k-1 i=1 E (3) n,i ≥ 1 2 k k-1 i=1 Q F i (n), min 1≤ℓ<∆ i ω(w ℓ , w ℓ-1 ) ≥ e -n ε .
To bound the probability expression on the right-hand side, we use the following lemma, which is a uniform version of Proposition 3.1. Its proof is in the Appendix. Lemma 6.5 Let S i -S i-1 , i ≥ 1, be i.i.d. mean-zero random variables (S 0 := 0) with σ 2 := E(S 2 1 ) ∈ (0, ∞). For any δ > 0, there exist r 0 > 1 and 0 < η < 1 such that for all r ∈ [r 0 , η √ n ] ∩ Z, for all events A (n)

i , 1 ≤ i ≤ n, satisfying the following two conditions:

• (S i -S i-1 , A (n) i ), for 1 ≤ i ≤ n, are i.i.d., • P(∩ n i=1 A (n) i ) ≥ 1 -η r , we have η r e -(1+δ) π 2 σ 2 8 n r 2 ≤ P max 1≤i≤n (S i -S i ) < r, S n = S n , n i=1 A (n) i ≤ e -(1-δ) π 2 σ 2 8 n r 2 ,
where S i := max 1≤j≤i S j .

Recall that (A 1 , • • • , A N ) is a random vector distributed as any of (A

1 (x), • • • , A N (x) (x)) defined in (1.1). Since E[ 1 ω(∅, ← ∅) ] = E[1 + N i=1 A i ] = 1 + e ψ(1) = 2, we have, Q{ω(w 1 , ∅) < e -n ε } = P{ω(∅, ← ∅) < e -n ε } ≤ 2e -n ε (by Markov's inequality).
Therefore, for all large n and all 1

≤ i ≤ k -1, Q min 1≤ℓ<∆ i ω(w ℓ , w ℓ-1 ) ≥ e -n ε = [Q{ω(w 1 , ∅) ≥ e -n ε }] ∆ i -1 ≥ (1 -2e -n ε ) ∆ i -1 ≥ (1 -2e -n ε ) n ≥ 1 -e -n ε/2 .
We apply Lemma 6.5 to A

(n) i := {ω(w i , w i-1 ) ≥ e -n ε } (with ∆ i -1 and (a * -2ε)(nm i ) 1/3 playing the roles of n and r, respectively; noting that ∆

i -1 ∼ 3(k -i) 2 n χ and (n -m i ) 1/3 ∼ (k -i)n χ/3
, so the last condition in the lemma on A (n) i is satisfied), to see that for all large n and for all 1

≤ i ≤ k -1, Q F i (n), min 1≤ℓ<∆ i ω(w ℓ , w ℓ-1 ) ≥ e -n ε ≥ exp -(1 + ε) 3π 2 σ 2 8(a * -2ε) 2 n χ/3 , which implies that Q E (2) n ∩ k-1 i=1 E (3) n,i ≥ 1 2 k exp -(k -1)(1 + ε) 3π 2 σ 2 8(a * -2ε) 2 n χ/3 . By definition, k = ⌊n (1-χ)/3 ⌋; hence lim inf n→∞ 1 n 1/3 log Q E (2) n ∩ k-1 i=1 E (3) n,i ≥ -(1 + ε) 3π 2 σ 2 8(a * -2ε) 2 .
This, together with (6.12), yields

lim inf n→∞ 1 n 1/3 log E(̺ n ) ≥ - 3π 2 σ 2 8 1/3
.

Since β n ≥ ̺ n (Lemma 5.1), we obtain (6.6), thus the lower bound in Theorem 1.1.

A Appendix. Proofs of Lemmas 6.4 and 6.5

Proof of Lemma 6.4. Although the lemma is deterministic, our proof is probabilistic. We note that the value of b 1 plays no role in the lemma.

Let (η i ) i≥0 be a Markov chain on {0, 1, • • • , n} with transition probabilities

P η i+1 = k η i = j =      b j+1 1+b j+1 , if k = j + 1, 1 1+b j+1 , if k = j -1. 0 < j < n.
where r := max ℓ<j≤m r j . Since P n-1 (τ η (0) < τ η (n)) ≥ k i=1 P m i (τ η (m i-1 ) < τ η (m i )), (A.1) will be a consequence of (A.4) and (A.5).

To prove (A.5), let Ω ℓ,m be the set of all (finite) paths of η starting from m and hitting ℓ before returning to m (and without being absorbed by ∂). For any γ ∈ Ω ℓ,m , let L ± γ (j) := i≥0 1 (γ i =j, γ i+1 =j±1) and L γ (j) := L + γ (j) + L - γ (j). Then This together with (A.6) implies (A.2), completing the proof of Lemma 6.4.

Proof of Lemma 6.5. We start with the proof of the lower bound. Let a ≥ 2 be an integer whose value will be chosen later on. Let η < 1 a . Let K := ⌊ n ar 2 ⌋, and n i = iar 2 for 0 ≤ i ≤ K -1 and n K := n. Write S # n := max 1≤i≤n (S i -S i ). It is clear that .

P S # n <
For any 1 ≤ j ≤ K, conditionally on σ{S i , A

i , 1 ≤ i ≤ n j-1 } and on {S n j-1 -S n j-1 = x}, the reflecting random walk (S i+n j-1 -S i+n j-1 , 0 ≤ i ≤ n jn j-1 ) has the same law as (max{x, S i } -S i , 0 ≤ i ≤ n jn j-1 ). Accordingly,

P S # n < r, S n = S n , n i=1 A (n) i ≥ q K-1 n,r b n,r ,
where q n,r := P S # ar 2 < (1δ)r, S ar 2 -S ar 2 < δr, S ar 2 > δr, .

We observe that q n,r ≥ P S # ar 2 < (1δ)r, S ar 2 -S ar 2 < δr, S ar 2 > δr + P On the other hand, since the three events {S # ar 2 < (1δ)r}, {S ar 2 -S ar 2 < δr} and {S ar 2 > δr} are non-decreasing with respect to each S i -S i-1 (for 1 ≤ i ≤ n), it follows from the FKG inequality that q n,r ≥ P S # ar 2 < (1δ)r P S ar 2 -S ar 2 < δr P S ar Observe that ar 2 ≤ n Kn K-1 ≤ 2ar 2 . Therefore, by Donsker's invariance principle, for all r ≥ r 0 (with an increased value of r 0 if necessary), P(S # n K -n K-1 < (1δ)r) P(S n K -n K-1 > δr) ≥ c(a, δ) for some constant c(a, δ) > 0, whereas P(S n K -n K-1 = S n K -n K-1 ) = P(S 1 ≥ 0, S 2 ≥ 0, ..., S n K -n K-1 -1 ≥ 0) ≥ c ′ r √ a for some constant c ′ > 0. Taking η := min{ c(a,δ)c ′ 2 √ a , 1 a }, we get b n,r ≥ η r . Consequently,

P S # n < r, S n = S n , n i=1 A (n) i ≥ q K-1 n,

Figure 1 :

 1 Figure 1: The different possible shapes for ψ

x

  |x| := x are the vertices on [[∅, x]] with |x i | = i for any 0 ≤ i ≤ |x|. The condition ψ(1) = 0 yields that for any n ≥ 1 and any measurable function

  converges a.s. to a (strictly) positive random variable when k → ∞, which implies R n ∼ 2η n 1/3 log(EN ) P-a.s. (n → ∞), and max k∈[ n 2 , n] min |x|=k+Rn V (x) ≥ min |x|=n V (x) P-almost surely for all large n. As a consequence, upon the system's survival, we have, P-almost surely for all large n,

Figure 3 :

 3 Figure 3: j = inf{i : E δ (x i ) holds.}

Proof of Lemma 5 . 1 .

 51 The first inequality in (5.3) is trivial. Let us prove the second. Let T

Figure 4 :

 4 Figure 4: A Q-tree

  s. So we only need to show that for any c ∈ [0, 1] and a

P

  m τ η (ℓ) < τ η (m) = γ∈Ω ℓ,m ℓ<j≤m (b j+1 q j ) L + γ (j) (q j ) L - Lγ (j) = E m 1 (τη (ℓ)<τη (m)) (1 + r) -τη(ℓ) ,yielding (A.5) and hence (A.1).It remains to show (A.2). We note thatP m {τ η (ℓ) < τ η (m)} = 1 1+b m+1 P m-1 {τ η (ℓ) < τ η (m)}.To compute P m-1 {τ η (ℓ) < τ η (m)} for the birth-and-death chain, we recall v(j) = -j i=1 log b i for j ≥ 1, and use the same argument as in the second identity of (1.10) to see that P m-1 {τ η (ℓ) < τ η (m)} = e v(m) m i=ℓ+1 e v(i) . Therefore,P m τ η (ℓ) < τ η (mℓ) exp -max ℓ<i≤m (v(i)v(m)) -(v(m)v(m + 1)) + . (A.6)Under P m and conditionally on {τ η (ℓ) < τ η (m)}, τ η (ℓ) is stochastically smaller than the hitting time of ℓ by a Markov chain with the same probability transition as η but reflecting on m. The expectation of the latter hitting time was estimated byGolosov ([14], p. 498, (A.1)). HenceE m τ η (ℓ) τ η (ℓ) < τ η (m) ≤ (mℓ) 2 exp max ℓ<i≤j≤m (v(i)v(j)) ,which, by means of the elementary inequality (1 + r) -u ≥ e -ru for u ≥ 0 and Jensen's inequality, implies that E m (1 + r) -τη(ℓ) τ η (ℓ) < τ η (m) ≥ expr(mℓ) 2 e max ℓ<i≤j≤m (v(i)-v(j)) .

≥

  r, S n = S n , P ∀1 ≤ j ≤ K, maxn j-1 <i≤n j (S i -S i ) < r, S n j -S n j < δr, S n = S n ,

  r := P S # n K -n K-1 < (1δ)r, S n K -n K-1 = S n K -n K-1 > δr,

1 ≥ P S # ar 2 <

 12 (1δ)r, S ar 2 -S ar 2 < δr, S ar 2 > δr -η r .

(σ 2 a 8 . 8 .

 88 W (t) -W (t)) < 1δ σ √ a ≥ e -(1+3δ) π 2Since both W (1) -W (1) and W (1) are distributed as the absolute value of a standard Gaussian random variable, we can even increase the value of a 0 (if necessary) such that for all a ≥ a 0 (thusδ σ √ a is very small), P{W (1) > δ σ √ a } ≥ 1 2 and P{W (1)-W (1) < δ σ √ a } ≥ δ 2σ√ a . Now we fix an arbitrary integer a ≥ a 0 . For all large r (say r ≥ r 0 ) such that r ≤ η √ n, we haveq n,r ≥ e -(1+4δ) π 2 σ 2 aThe probability b n,r can be estimated in a similar way: from the assumptions on (A (n) i ) and the FKG inequality, we deduce that b n,r ≥ P(S # n K -n K-1 < (1δ)r) P(S n K -n K-1 = S n K -n K-1 ) P(S n K -n K-1 > δr) -η r .

r b n,r ≥ η r e -(1+4δ) π 2 σ 2 8 n r 2 ,≤ 1 .σ 2 a 8 ; 8 n r 2 - 1 ,

 8218821 proving the lower bound in the lemma.The upper bound is easier: with the same notation and the choice of a, we haveP S # n < r, S n = S n , P ∀1 ≤ j ≤ K -1, max n j-1 <i≤n j (S i -S i ) < r ≤ P(S # ar 2 < r) K-For all large r, P(S # ar 2 < r) ≤ e -(1-δ) π 2 hence P(S # n < r, S n = S n , ∩ n i=1 A (n) i) ≤ e -(1-δ) π 2 σ 2 yielding the upper bound (by eventually modifying the choice of η in terms of a).

  the upper and lower bounds in (2.13) follow from (2.8) and (2.11), respectively. [In (2.11), we use the fact that b

  2 > δr -# ar 2 < (1δ)r P S ar 2 -S ar 2 < δr P S ar 2 > δr where W is a standard Brownian motion, with W (t) := sup s∈[0, t] W (s) for any t. Recall that sup t∈[0, 1] (W (t) -W (t)) has the same distribution as sup t∈[0, 1] |W (t)|; so (see Formula (5.9), page 342 of Feller[START_REF] Feller | An introduction to probability and its applications[END_REF])P{sup t∈[0, 1] (W (t) -W (t)) ≤ x} = exp{-(1 + o(1)) π 28x 2 }, for x → 0. Consequently, for all sufficiently large a, say a ≥ a 0 , we have

	By Donsker's invariance principle,
	lim r→∞ P S = P sup t∈[0, 1] (W (t) -W (t)) <	1 -δ σ √ a	P W (1) -W (1) <	σ	δ √	a	P W (1) >	σ	δ √	a	,
	P sup
	t∈[0, 1]
							η r	.

Finite-dimensional convergence is checked by Lindeberg's condition in the central limit theorem, whereas tightness is proved via a standard argument as in Billingsley[START_REF] Billingsley | Convergence of Probability Measures[END_REF].

On the set of extinction, the upper bound is, in fact, trivially true.

Acknowledgement

We have been kindly informed by Ofer Zeitouni that Theorem 1.4 in the case of regular trees was independently proved by Fang and Zeitouni [10]. We are also grateful to an anonymous referee for her/his very careful reading of the original manuscript.

[The transition probabilities from j = 0 and j = n, having no importance, can be anything. For example, we can take P{η i+1 = 0 | η i = 0} = 1 = P{η i+1 = n | η i = n}.] Define τ η (j) := inf{i ≥ 1 : η i = j} (with inf ∅ := ∞ as usual). Let P j be the probability such that P j {η 0 = j} = 1, and let E j be the expectation with respect to P j . We claim that (∞ being not < ∞)

and for any integers 0 ≤ ℓ < m ≤ n and r ≥ 0,

Plainly Lemma 6.4 will follow from (A.1) and (A.2).

To prove (A.1), we consider a Markov chain (

where ∂ is an absorbing point, such that

with q j := 1 b j+1 +1+r j for all 0 < j < n. [Again, the transition probabilities from j = 0 and j = n can be anything.] Let τ η (j) := inf{i ≥ 1 :

Let us check (A.3): z n-1 = q n-1 , and if z j = P j (τ η (j -1) < τ η (n)) for j ∈ [2, n -1] ∩ Z, then P j-1 (τ η (j -2) < τ η (n)) = q j-1 +b j q j-1 P j (τ η (j -2) < τ η (n)) = q j-1 +b j q j-1 z j P j-1 (τ η (j -2) < τ η (n)) by the Markov property. Hence P j-1 (τ η (j -2) < τ η (n)) = q j-1

1-b j q j-1 z j , which, by the definition of q j-1 , is

We claim that for 0 ≤ ℓ < m < n, (A.5) P m τ η (ℓ) < τ η (m) ≥ E m 1 (τη (ℓ)<τη (m)) (1 + r) -τη(ℓ) ,


M Gamet
email: m.gamet@brgm.fr

GCLs in landfill applications: influence of subgrade, temperature and confining pressure on bentonite hydration

Geosynthetic Clay Liners (GCLs) are often used in landfills applications, for example in landfill covers. In order to act as a barrier, GCLs must hydrate and swell under a confining pressure. According to the French Committee for Geosynthetics, the water content of the bentonite must be at least 100% (wet dry weight). If GCLs are installed at their initial bentonite water content (around 10-15%), the question arises as to the duration of the hydration period (vapour migration from the subgrade). This question was addressed during experiments performed on a natural sodium bentonite GCL and aimed at examining the influence of several parameters: subgrade (sand versus clay), temperature and confining pressure. Results illustrate the strong influence of the subgrade water content and grain size on bentonite water content level and hydration kinetic. As could be expected, sandy soils allow a faster hydration and higher water content than clayey soils, but results provide quantitative information on hydration duration.

INTRODUCTION

Geosynthetic Clay Liners (GCLs) are used in landfill liner systems in bottom, slope or cover lining applications. In order to fulfil their sealing function, GCLs need to be confined under normal stress and hydrated with water in order to achieve very low hydraulic conductivities. Initial hydration, with a nonchemically aggressive hydration fluid, is particularly important in order to confer the hydraulic performances of the GCLs subjected to a prolonged contact with leachate [START_REF] Von Maubeuge | Performances of GCLs. Les Géocomposites Bentonitiques[END_REF][START_REF] Ruhl | Geosynthetic Clay Liner permeated with chemical solutions and leachates[END_REF][START_REF] Didier | Influence of initial hydration conditions on Geosynthetic Clay Liner leachate permeability. Testing and acceptance criteria for GCLs[END_REF][START_REF] Shackelford | Evaluating the hydraulic conductivity of GCLs permeated with non-standard liquids[END_REF][START_REF] Guyonnet | Geosynthetic clay liner interaction with leachate: correlation between permeability, microstructure, and surface chemistry[END_REF]. Indeed, the most detrimental situation for a GCL used in a landfill bottom liner system would be a direct contact with landfill leachate. Several methods for achieving this initial hydration are possible. An active process can be ensured by rainfall or controlled moistening before or after installation of the confining layer. But if GCLs are installed at their initial bentonite moisture (10-15%) and then covered by a geomembrane, as is the most common practice, there is a passive hydration of bentonite by vapour transfer of the water present in the subgrade.

In this paper we present the results of several series of tests performed to determine the stabilized water contents and hydration rates of a GCL in contact with various subgrades. The study addresses the influence of soil water content and grain size, confining pressure and ambient temperature. The GCL tested in this study was a needle-punched GCL containing 5 kg/m² of a granular natural sodium bentonite.

WATER CONTENT IN GCL IN EQUILIBRIUM WITH THE SUBGRADE

Testing procedure

Tests were carried out in the laboratory at T=20°±2°C on 9.5 cm diameter samples. Sample weights and thicknesses were measured before introduction into the test cell. The testing cell is a PVC cylinder with an internal diameter of 9.5 cm and a height of 30 cm.

GB

The colu known w applicat the pist (Figure F

In

The initi sand wa piston w

Once sw Table 1 Figures and the bentonit the subg

BR-C

umn is filled water conte tion of a norm ton is ensure 1). fter 24 days ed after 24 da

Influence of confining pressure on equilibrium water content

The objective was to determine the influence of the normal stress σ on the stabilized water content of a GCL (w b ) in contact with standard sand. We used the same procedure as described previously but we added weights on the pistons in order to increase the applied normal stress σ.

The initial water content of the GCL was w b =9% and the mass per unit area 4.80 kg/m 2 . The soil used was standard sand with an initial water content of 7%. The samples were confined under the following normal stresses σ: 7.0, 9.9, 14.1, 21.2 and 28.2 kPa.

At the end of each test, water contents were measured in the bentonite of the GCL and in the sand at 5 regularly-spaced levels within the column. Results are shown in Table 2 and Figure 4. For all the tests, the water content of the sand at the bottom of the column is higher than the initial water content (w s = 7%) due to the influence of gravity and the low retention capacity of the sand.

A mass balance on water in the sand provides the quantity of water absorbed by the bentonite. The equilibrium bentonite water content w b and the swelling S decreases when the normal stress σ increases (Figure 5). The water content balance decreases by about 12.5% when the normal stress increases from 7 to 28.2 kPa.

HYDRATION KINETICS OF GCL ON SUBGRADE

Testing procedure

This test was carried out in a laboratory at T=5°C, 20°C and 45°C on 10 cm diameter samples. Sample and subgrade weights were measured regularly. The test cell is a sealed HDPE cylindrical container with an internal diameter of 11 cm and a height of 12 cm. -fine grain sand at 5% and 10% water contents (wet dry weight); -plastic clay at 15% and 25% water contents.

Cells were filled by compacting subgrades at known water contents. The GCL sample was then installed on top of the subgrade (see Figure 6). An optional steel plate allowed the application of a confining pressure on the GCL (σ=4 kPa).

3.2 Influence of water content and subgrade grain size Figure 7 shows, for a given temperature, the evolution of GCL water content w b for each subgrade. For a given temperature, the sandy subgrade results in a higher final GCL water content and a faster hydration than the clayey subgrade, although the initial water content in the sand is lower than in the clay. At 45°C, the GCL's water content is 170% on wetter (w = 10%) sandy soil and 94% on wetter (w = 25%) clayey soil after 124 days. Open porosity of coarse grain soil is more important than clayey soil, allowing an easier vapour migration from subgrade to GCL. This shows the major influence of the retention capacity of the soil for hydration kinetics of bentonites.

Figure 7 also shows that hydration of GCLs depends on subgrade water content. As previously shown, GCL equilibrium water content is closely related to subgrade water content, while hydration rate also seems to be influenced. For a given temperature different subgrades, local slopes of hydration curves rise with subgrade water content: at 20°C, the GCL average hydration rate is 0.27% per day (%/d) for the 15% moisture clayey subgrade and over 0.50%/d for the 25% moisture clayey subgrade.

Influence of temperature

Figure 8 shows the influence of temperature T on GCL hydration. At 5°C, final water content does not exceed 40% in GCLs in contact with sandy subgrade, and 20% on clayey subgrade. At 45°C, it reaches 170% for the 10% wetted sand, whereas water content of the three other samples is about 90%. At 5°C, the equilibrium water content is reached within 30 days. At 20°C and 45°C, equilibrium is not achieved, except for the dryer sandy subgrade (w=5%).

Figure 8. Influence of temperature on GCL hydration Hydration rates are influenced by temperature, since water transfer is primarily achieved by vapour transfer. Figure 9 shows, for each subgrade, the evolution of GCL average hydration rate versus temperature. This average rate rises with temperature T, and also with subgrade grain size and water content. The measured rates are rather low, confirming that, even in good hydration conditions (wet sandy soil, high temperature), GCLs hydration is a slow process. Consistent with results in Table 2, Figure 10 shows that confining pressure has an influence on GCL hydration. For a given temperature T, the application of a σ=4 kPa vertical stress changes slightly final water content of GCLs samples. On the other hand, confining pressure seems to be influent on hydration rate: stabilized water content is reached 30 days earlier under a vertical stress σ=4 kPa at 45°C.

CONCLUSIONS

The results of this study show that hydration of GCL by subgrade water requires periods of time on the order of several weeks. Results also suggest that both final water content and hydration rate of GCLs increase with subgrade grain size, subgrade initial water content and temperature. Final water content decreases with confining pressure, whereas hydration rate rises. The minimum water content of 100%, sometimes recommended in geotechnical guidelines, is reached for coarse grain subgrades (sand) or high ambient temperatures (45°C). Such conditions are not typical of European landfills, where subgrades are mostly fine grained soils (typically clays and loam) and temperatures are colder.

This study is by no means exhaustive, as it concerns only one type of GCL and tests were performed for a limited number of hydration conditions. Further studies should examine a larger range of GCLs (with different types of bentonite mineralogy or grain size, different geotextiles, etc.), and of subgrades (silts, marls …).

However, some recommendations can be proposed based on the results obtained. Firstly, particularly under desiccating climatic conditions (due to heat or wind), subgrade water content should be controlled and possibly maintained by sprinkling, until the GCL is installed. Then, GCLs could be hydrated before geomembrane implementation. [START_REF] Didier | Influence d'une préhydratation sans confinement sur les performances hydrauliques des GSB[END_REF], show that the hydraulic conductivity of a needle punched GCL is not reduced when the GCL is hydrated with sprinkling hydration without confinement. Finally, confining pressure (drainage layer) should be implemented as fast as possible, in order to speed up GCL hydration.

Figure

  Figure 1. App nfluence of in ial water con as used at v was 7 kPa for welling had s gives the re

Figure 2 .

 2 Figure 2. Final water content of the bentonite (Wb) versus initial water content of the sand (Ws).

Figure 3 .

 3 Figure 3. Swelling (S) versus water content of the bentonite (Wb).

Figure 5 .Figure 6 .

 56 Figure 5. Bentonite water content versus normal stress (left). Swelling versus time (right)

Figure 7 .

 7 Figure 7. Influence of subgrade grain size and water content for three constant temperatures

Figure 9 .

 9 Figure 9. Average hydration rate of GCLs versus temperature. 3.4 Influence of confining pressure

Table 1

 1

	. Fin
	content Be
	afte
	(%
	39
	66
	90
	100
	118
	148
	160
	near relation
	content (w b ;
	gure 3) The
	ial and the a

rd Internati Geos

	zburg, German
	and (Standa
	op of the sa
	between the
	allows a mon
	n equilibrium
	content
	er unit area 5
	%. The norm
	easured in th
	GCL
	Swelling
	after 24 day
	S (mm)
	0.36
	0.87
	2.34
	2.58
	2.62
	2.90
	3.46
	n the sand i
	ween the GC
	ssion lines d

Table 2 .

 2 Final water content and swelling of GCL

			180				4
	Bentonite water content	(%)	30 60 90 120 150	Wb = 8,3Ws + 17,6 Normal stress=7 kPa	Swelling (mm)	1 2 3	S = 0,0229Wb -0,18
			0				0
			0	4 Initial water content of the sand (%) 8 1 2 1 6	2 0	0	50 Bentonite water content (%) 100 150	200
				Normal stress	Bentonite water content	Swelling
				σ (kPa)	w b (%)			S (mm)
				7.0	106.9			1.28
				9.88	105.8			1.25
				14.1	105.22			1.32
				21.16	99.0			0.74
				28.2	95.0			0.94

GBR-C 2k10