Large deviations for transient random walks in random environment on a Galton-Watson tree

www.imstat.org/aihp 2010, Vol. 46, No. 1, 159–189

DOI:10.1214/09-AIHP204

© Association des Publications de l’Institut Henri Poincaré, 2010

Large deviations for transient random walks in random environment on a Galton–Watson tree

Elie Aidékon

Laboratoire de Probabilités et Modèles Aléatoires, Université Paris VI, 4 Place Jussieu, F-75252 Paris Cedex 05, France.

E-mail:[email protected]

Received 26 January 2008; revised 4 November 2008; accepted 14 January 2009

Abstract. Consider a random walk in random environment on a supercritical Galton–Watson tree, and letτ_nbe the hitting time of generationn. The paper presents a large deviation principle forτ_n/n, both in quenched and annealed cases. Then we investigate the subexponential situation, revealing a polynomial regime similar to the one encountered in one dimension. The paper heavily relies on estimates on the tail distribution of the first regeneration time.

Résumé. Nous considérons une marche aléatoire en milieu aléatoire sur un arbre de Galton–Watson. Soitτnle temps d’atteinte du niveaun. Le papier présente un principe de grandes déviations pourτn/n, dans les cas quenched et annealed. Nous étudions ensuite le régime sous-exponentiel, qui fait apparaître un régime polynomial rappelant la dimension 1. Le papier repose principalement sur les estimations de la queue de distribution du premier temps de renouvellement.

MSC:60K37; 60J80; 60F15; 60F10

Keywords:Random walk in random environment; Law of large numbers; Large deviations; Galton–Watson tree

1. Introduction

We consider a super-critical Galton–Watson treeTof rooteand offspring distribution(q_k, k≥0)with finite mean

m:=

k≥0kq_k>1. For any vertexxofT, we call|x|the generation ofx,(|e| =0)andν(x)the number of children of x; we denote these children by x_i,1≤i≤ν(x). We let ν_minbe the minimal integer such thatq_νmin>0 and we suppose thatνmin≥1 (thusq0=0). In particular, the tree survives almost surely. Following Pemantle and Peres [14], on each vertexx, we pick independently and with the same distribution a random variableA(x), and we define:

• ω(x, x_i):= ^A(xi)

1+ν(x)

i=1A(xi),∀1≤i≤ν(x),

• ω(x,^←x ):= ¹

1+ν(x) i=1A(xi).

To deal with the casex=e, we add a parent^←e to the root and we setω(^←e , e)=1. Once the environment built, we define the random walk(Xn, n≥0)starting fromy∈Tby

P_ω^y(X₀=y)=1,

P_ω^y(Xn+1=z|Xn=x)=ω(x, z).

The walk(X_n, n≥0)is aT-valued Random Walk in Random Environment (RWRE). To determine the transience or recurrence of the random walk, Lyons and Pemantle [11] provides us with the following criterion. LetAbe a generic random variable having the distribution ofA(e).

Theorem A (Lyons and Pemantle [11]). The walk(X_n)is transient ifinf_[0,1]E[A^t]>_m¹,and is recurrent otherwise.

In the transient case, letvdenote the speed of the walk, which is the deterministic realv≥0 such that

nlim→∞

|X_n|

n =v, a.s.

Define

i:=ess infA, s:=ess supA.

We make the hypothesis that 0< i≤s <∞. Under this assumption, we gave a criterion in [1] for the positivity of the speedv. Let

Λ:=Leb

t∈R: E A^t

≤ 1 q1

(Λ= ∞ifq₁=0). (1.1)

Theorem B ([1]). Assumeinf_[0,1]E[A^t]>_m¹,and letΛbe as in(1.1).

(a) IfΛ <1,the walk has zero speed.

(b) IfΛ >1,the walk has positive speed.

When the speed is positive, we would like to have information on how hard it is for the walk to have atypical be- haviours, which means to go a little faster or slower than its natural pace. Such questions have been discussed in the setting of biased random walks on Galton–Watson trees, by Dembo et al. in [5]. The authors exhibit a large deviation principle both in quenched and annealed cases. Besides, an uncertainty principle allows them to obtain the equality of the two rate functions. For the RWRE in dimensions one or more, we refer to Zeitouni [17] for a review of the subject.

In our case, we consider a random walk which always avoids the parent^←e of the root, and we obtain a large deviation principle, which, following [5], has been divided into two parts.

We suppose in the rest of the paper that

[inf0,1]E A^t

> 1

m, (1.2)

Λ >1, (1.3)

which ensures that the walk is transient with positive speed. Before the statement of the results, let us introduce some notation. Define for anyn≥0 andx∈T,

τn:=inf

k≥0:|Xk| =n ,

D(x):=inf{k≥1:Xk−1=x, Xk=^←x}, inf∅:= ∞.

LetPdenote the distribution of the environmentωconditionally onT, andQ:= P(·)GW (dT). Similarly, we denote byP^xthe distribution defined byP^x(·):= P_ω^x(·)P(dω)and byQ^xthe distribution

Q^x(·):=

P^x(·)GW (dT).

Theorem 1.1 (Speed-up case). There exist two continuous,convex and strictly decreasing functions I_a≤I_q from [1,1/v]toR+,such thatI_a(1/v)=I_q(1/v)=0and fora < b,b∈ [1,1/v],we have almost surely,

nlim→∞

1 nlnQ^e

τ_n n ∈ ]a, b]

= −Ia(b), (1.4)

nlim→∞

1 nlnP_ω^e

τ_n n ∈ ]a, b]

= −I_q(b). (1.5)

Theorem 1.2 (Slowdown case). There exist two continuous,convex functionsI_a≤I_qfrom[1/v,+∞[toR₊,such thatI_a(1/v)=I_q(1/v)=0and for any1/v≤a < b,we have almost surely,

nlim→∞

1 nlnQ^e

τ_n

n ∈ [a, b[

= −I_a(a), (1.6)

nlim→∞

1 nlnP_ω^e

τ_n

n ∈ [a, b[

= −I_q(a). (1.7)

Besides,ifi > ν⁻_min¹,thenIaandIq are strictly increasing on[1/v,+∞[.Wheni≤ν_min⁻¹,we haveIa=Iq=0on the interval.

As pointed by an anonymous referee, it would be interesting to know whenI_aandI_qcoincide. We do not know the answer in general. However, the computation of the value of the rate functions atb=1 reveals situations where the rate functions differ. Let

ψ (θ ):=ln

EQ

_ν(e)

i=1

ω(e, ei)^θ

.

Thenψ (0)=ln(m)andψ (1)=ln(EQ[ν(e)

i=1ω(e, ei)]).

Proposition 1.3. We have

I_a(1)= −ψ (1), (1.8)

I_q(1)= − inf

]0,1]

1

θψ (θ ). (1.9)

In particular,I_a(1)=I_q(1)if and only ifψ(1)≤ψ (1).OtherwiseI_a(1) < Iq(1).

Quite surprisingly, we can exhibit elliptic environments on a regular tree for which the rate functions differ. This could hint that the uncertainty of the location of the first passage in [5] does not hold anymore for a random environ- ment. Here is an explicit example. Consider a binary tree (q2=1). LetAequal 0.01 with probability 0.8 and 500 with probability 0.2. Then we check that the walk is transient, butψ(1) > ψ (1)so thatIa(1)=Iq(1)on such an environment.

Theorem1.2exhibits a subexponential regime in the slowdown case wheni≤ν_min⁻¹. The following theorem details this regime. Let

S^e(·):=Q^e

·|D(e)= ∞ .

Theorem 1.4. We place ourself in the casei < ν_min⁻¹.

(i) Suppose that either “i < ν_min⁻¹ andq₁=0” or “i < ν_min⁻¹ ands <1.” There exist constantsd₁, d₂∈(0,1)such that for anya >1/vandnlarge enough,

e^−nd1 <S^e(τ_n> an) <e^−nd2. (1.10)

(ii) Ifq₁>0ands >1 (i.e.whenΛ <∞),the regime is polynomial and we have for anya >1/v,

nlim→∞

1 ln(n)ln

S^e(τ_n> an)

=1−Λ. (1.11)

We mention that in one dimension, which can be seen as a critical state of our model whereq1=1, such a poly- nomial regime is proved by Dembo et al. [6], our parameterΛtaking the place of the well-knownκ of Kesten et al.

[9]. We did not deal with the critical casei=ν_min⁻¹. Furthermore, we do not have any conjecture on the optimal values ofd₁andd₂and do not know if the two values are equal.

The rest of the paper is organized as follows. Section2describes the tail distribution of the first regeneration time, which is a preparatory step for the proof of the different theorems. Then we prove Theorems1.1and1.2in Section3, which includes also the computation of the rate functions at speed 1 presented in Proposition1.3. Section4is devoted to the subexponential regime with the proof of Theorem1.4.

2. Moments of the first regeneration time

We define the first regeneration time Γ₁:=inf

k >0: ν(X_k)≥2, D(Xk)= ∞, k=τ_|Xk|

as the first time when the walk reaches a generation by a vertex having more than two children and never returns to its parent. We propose in this section to give information on the tail distribution ofΓ₁underS^e. We first introduce some notation used throughout the paper. For anyx∈T, let

N (x):=

k≥0

1_{X_k=x}, (2.1)

T_x:=inf{k≥0: X_k=x}, T_x^∗:=inf{k≥1: X_k=x}. .

This permits to define β(x):=P_ω^x(T←

x = ∞), γ (x):=P_ω^x

T←

x =T_x^∗= ∞

. (2.2)

The following fact can be found in [5] (Lemma 4.2) in the case of biased random walks, and is directly adaptable in our setting.

Fact A. The first regeneration height|X_Γ1|admits exponential moments under the measureS^e(·).

2.1. The casei > ν_min⁻¹

This section is devoted to the casei > ν_min⁻¹, whereΓ₁is proved to have exponential moments.

Proposition 2.1. Suppose thati > ν_min⁻¹.There existsθ >0such thatE_Se[e^{θ Γ1}]<∞.

Proof. The proof follows the strategy of Proposition 1 of Piau [16]. We couple the distance of our RWRE to the root (|Xn|)_n≥0with a biased random walk(Y_n)_n≥0onZas follows. Letp:=₁₊^iν_iν^min_min, and letu_n, n≥0, be a family of i.i.d.

uniformly distributed[0,1]random variables. We setX0=eandY0=0. IfX_kandY_kare known, we construct

X_k+1=x_i if

i−1

=1

ω(x, x)≤u_k<

i

=1

ω(x, x), X_k+1=^←x otherwise,

Yk+1=y+21_{u_k≤p}−1,

where x:=X_k ∈T andy :=Y_k ∈Z. Then (X_n)_n≥0 has the distribution of our T-RWRE indeed, and (Y_n)_n≥0 is a random walk on Z which increases of one unit with probabilityp >1/2 and decreases of the same value with probability 1−p. Notice also that on the event{D(e)= ∞}, we have

|Xk+1| − |Xk| ≥Yk+1−Yk.

It implies that the first regeneration timeR1of(Y_n)_n≥0defined by R1:=inf{k >0:Y< Y_k∀ < k, Y_m≥Y_k∀m > k}

is necessarily a regeneration time for(X_n, n≥0), which proves in turn that S^e(Γ1> n)≤Q^e(R1> n).

To complete the proof, we must ensure thatQ^e(R1> n)is exponentially small, which is done in [6], Lemma 5.1.

2.2. The cases “i < ν_min⁻¹,q₁=0” and “i < ν_min⁻¹,s <1”

Wheni < ν_min⁻¹, if we assume also thatq₁=0 or s <1, we prove thatΓ₁has a subexponential tail. This situation covers, in particular, the case of RWRE on a regular tree.

Proposition 2.2. Suppose thati < ν_min⁻¹ andq1=0,then there exist1> α1> α2>0such that fornlarge enough,

e^−nα1 <S^e(Γ₁> n) <e^−nα2. (2.3)

The same relation holds with some1> α₃> α₄>0in the case “i < ν_min⁻¹ ands <1.”

Proof of Proposition2.2: lower bound. We only suppose thati < ν_min⁻¹, which allows us to deal with both cases of the lemma. Define for somep ∈(0,1/2)andb∈N,

w₊:=Q ν

i=1

A(e_i)≥ 1−p

p , ν(e)≤b

,

w₋:=Q ν

i=1

A(ei)≤ p

1−p , ν(e)≤b

.

By (1.2), E_Q[ν(e)

i=1A(e_i)]>1 and therefore Q(ν(e)

i=1A(e_i) >1) >0. Since ess infA < ν_min⁻¹, it guarantees that Q(ν(e)

i=1A(e_i) <1) >0. Consequently, by choosingp close enough of 1/2 andblarge, we can takew₊ andw₋ positive. Letc:=_{6 ln(b)}¹ , and defineh_n:= cln(n). A treeTis said to ben-good if:

• any vertexx of theh_nfirst generations verifiesν(x)≤band_ν(x)

i=1A(x_i)≥ ¹⁻_p^p ,

• any vertexx of thehnfollowing generations verifiesν(x)≤bandν(x)

i=1A(xi)≤₁₋^p_p.

We observe that Q(Tisn-good)≥w^h₊^nbhnw₋^hnb2hn ≥e^−n1/3+o(1) which is stretched exponential, i.e. behaving like e^−nr+o(1)for somer∈(0,1). Define the events:

E1:= {at timeτ_hnwe cannot find an edge of level smaller thanh_ncrossed only once}

∩ {D(e) > τ_hn},

E₂:= {the walk visits the levelh_nntimes before reaching the root or the level 2hn}, E₃:= {after thenth visit of levelh_n,the walk reaches level 2hnbefore levelh_n}, E4:= {after timeτ2h_nthe walk never comes back to level 2hn−1}.

Suppose that the tree isn-good. Since Ais supposed bounded, there exists a constantc₁>0 such that for any x neighbour ofy, we have

ω(x, y)≥ c₁

ν(x). (2.4)

It yields thatP_ω^e(E₁)⁻¹=O(n^K)for someK >0 (where O(n^K)means that the function is bounded by a factor of n→n^K). Combine (2.4) with the strong Markov property at timeτ_hnto see that

P_ω^e(E₃|E₁∩E₂)⁻¹=O(n^K),

whereKis taken large enough. We emphasize that the functions O(n^K)are deterministic. Still by Markov property, P_ω^e(E₁∩E₂∩E₃∩E₄)=E_ω^e

1E₁∩E₂∩E₃β(X_τ2hn)

. (2.5)

Let(Y_n)_n≥0be the random walk onZstarting from zero with Pω

Y_n+1=k+1|Y_n=k

=1−Pω

Y_n+1=k−1|Y_n=k

=p.

We introduceT_i :=inf{k≥0: Y_k=i}, andp_nthe probability that(Y_n)_n≥0visitsh_nbefore−1:

p_n:=P_ω

T₋₁< T_h

n

.

By a coupling argument similar to that encountered in the proof of Proposition2.1, we show that in ann-good tree, P_ω^e(E₁∩E₂)≥P_ω^e(E₁)(p_n)ⁿ=O

n^K₋1 p_nn

, (2.6)

which gives

P_ω^e(E₁∩E₂∩E₃)≥O n^K₋1

p_nn

. (2.7)

Observing thatQ^e(Γ1> n, D(e)= ∞)≥EQ[1_{Tisn-good}1E₁∩E₂∩E₃∩E₄], we obtain by (2.5) Q^e

Γ₁> n, D(e)= ∞

≥E_Q^e

1_{Tisn-good}1E₁∩E₂∩E₃β(X_τ2hn)

=E_Q^e

1_{Tisn-good}P_ω^e(E₁∩E₂∩E₃) E_Q[β], by independence. By (2.7),

Q^e

Γ₁> n, D(e)= ∞

≥O(n^K)⁻¹Q(Tisn-good)(p_n)ⁿ.

We already know thatQ(Tisn-good)has a stretched exponential lower bound, and it remains to observe that the same holds for(p_n)ⁿ. But the method of gambler’s ruin shows thatp_n≥1−(₁₋^p_p )^hn, which gives the required lower

bound by our choice ofhn.

Let us turn to the upper bound. We divide the proof in two, depending on which case we deal with.

Proof of Proposition 2.2: upper bound in the case q1=0. Assume that q₁=0 (the condition i < ν_min⁻¹ is not required in the proof). The proof of the following lemma is deferred. Recall the notation introduced in (2.2),γ (e):=

P_ω^e(T←

e =T_e^∗= ∞)≤β(e).

Lemma 2.3. Whenq1=0,there exists a constantc2∈(0,1)such that for largen, E_Q

1−γ (e)_n

≤e^−nc2.

Denote byπ_kthekth distinct site visited by the walk(X_n, n≥0). We observe that Q^e(Γ1> n³)≤Q^e(Γ1> τ_n)+Q^e

more thann²distinct sites are visited beforeτ_n

(2.8) +Q^e

∃k≤n²: N (π_k) > n .

SinceQ^e(Γ₁> τ_n)=Q^e(|X_Γ1|> n), it follows from Fact A thatQ^e(Γ₁> τ_n)decays exponentially. For the second term of the right-hand side, beware that

Q^e

more thann²distinct sites are visited beforeτ_n

≤ n

k=1

Q^e(more thanndistinct sites are visited at levelk).

If we denote byt_i^kthe first time when theith distinct site of levelkis visited, we have, by the strong Markov property, P_ω^e(more thannsites are visited at levelk)=P_ω^e

t_n^k<∞

≤P_ω^e

t_n^k₋₁<∞, D(X_tk

n−1) <∞

=E^e_ω 1_{t^k

n−1<∞}

1−β(X_tk n−1)

. The independence of the environments entails that

E_Q^e 1_{tk

n−1<∞}

1−β(X_tk n−1)

=Q^e

t_n^k₋₁<∞

E_Q[1−β]. Consequently,

Q^e

t_n^k<∞

≤Q^e

t_n^k₋₁<∞

E_Q[1−β]

(2.9)

≤

E_Q[1−β]n−1

, which leads to

Q^e

more thann²sites are visited beforeτ_n

≤n

E_Q[1−β]n−1

, (2.10)

which is exponentially small. We remark, for later use, that Eq. (2.9) holds without the assumptionq₁=0. For the last term of Eq. (2.9), we write

Q^e

∃k≤n²: N (π_k) > n

≤

n²

k=1

Q^e

N (π_k) > n . LetU:=

n≥0(N^∗)ⁿbe the set of words, where(N)⁰:= {∅}. Each vertexxofTis naturally associated with a word ofU, andTis then a subset ofU(see [13] for a more complete description). For anyk≥1,

Q^e

N (π_k) > n

=

x∈U

Q^e

x∈T, N (x) > n, x=π_k

≤

x∈U

E_Q

1_{x∈T}P_ω^e(x=π_k)

1−γ (x)n , with the notation of (2.2). By independence,

Q^e

N (π_k) > n

≤

x∈U

E_Q

1_{x∈T}P_ω^e(x=π_k) E_Q

1−γ (e)n

=EQ

1−γ (e)n .

Apply Lemma2.3to complete the proof.

Proof of Lemma2.3. Letμ >0 be such thatq:=Q(β(e) > μ) >0, and write R:=inf

k≥1: ∃|x| =k, β(x)≥μ .

Letx_Rbe such that|x_R| =Randβ(x_R)≥μand we suppose for simplicity thatx_Ris a descendant ofe₁. We see that γ (e)≥ω(e, e₁)β(e₁)≥_ν(e)^c1 β(e₁)by Eq. (2.4). In turn, Eq. (2.1) of [1] implies that for any vertexx, we have

1

β(x)=1+ 1

ν(x)

i=1A(x_i)β(x_i)≤1+ 1 ess infA

1 β(x_i)

for any 1≤i≤ν(x). By recurrence on the path frome₁tox_R, this leads to 1

β(e₁)≤1+ 1

ess infA+ · · · + 1

ess infA R−1

1 μ. We deduce the existence of constantsc₄, c₅>0 such that

γ (e)≥ c4

ν(e)e^−c5R. (2.11)

It yields that E_Q

1−γ (e)n

1_{ν(e)<^√_n}

≤Q

R > 1 4c5

ln(n)

+e^−n1/4+o(1). We observe that

Q

R > 1 4c5

ln(n)

≤Q

∀|x| = 1 4c5

ln(n), β(x) > μ

. By assumption,q₁=0; thus #{x∈T: |x| = _4c¹

5ln(n)} ≥2^1/4c5ln(n)=:n^c6. As a consequence,Q(∀|x| = _4c¹

5ln(n), β(x) > μ)≤q^nc6. Hence, the proof of our lemma is reduced to find a stretched exponential bound forE_Q[(1− γ (e))ⁿ1_{ν(e)≥^√_n}]. For any x ∈T, denote by V_x^μ the number of children x_i of x such that β(x_i) > μ. For ε∈ (0,Q(β(e) > μ)),

EQ

1−γ (e)n

1_{ν(e)≥^√_n}

≤Q^e

ν(e)≥√

n, V_e^μ< εν(e) +E_Q

1−γ (e)n

1_{Ve^μ_≥εν(e)} .

We apply Cramér’s theorem to handle with the first term on the right-hand side. Turning to the second one, the bound is clear once we observe the general inequality,

γ (e)= ν(e)

k=1

ω(e, e_k)β(e_k)≥ c₁ ν(e)

ν(e)

k=1

β(e_k)≥ c₁μ

ν(e)V_e^μ, (2.12)

which is greater thanc1μεon{V_e^μ≥εν(e)}.

Remark 2.3. As a by-product,we obtain thatE_Q[(1−γ (e))ⁿ1_{ν(e)≥^√_n}] ≤e^−nc3 without the assumptionq₁=0.

Proof of Proposition2.2: upper bound in the cases<1. We follow the strategy of the case “q1=0”. The proof boils down to the estimate of

Q^e

N (π_k) > n, D(e)= ∞

=Q^e

N (πk) > n, ν(πk) <√

n, D(e)= ∞ +Q^e

N (πk) > n, ν(πk)≥√

n, D(e)= ∞ .

Letx∈Tand consider the RWRE(X_n, n≥0)when starting from^←x. Inspired by Lyons et al. [12], we propose to couple it with a random walk(Y_n , n≥0)onZ. We first defineX _nas the restriction ofX_non the path[[^←e , x]]. Beware thatX _nexists only up to a timeT, which corresponds to the time when the walk(X_n, n≥0)escapes the path[[^←e , x]], id est leaves the path and never comes back to it. After this time, we setX _n=Δfor someΔin some spaceE. Then (X_n )n≥0is a random walk on[[^←e , x]] ∪ {Δ}, whose transition probabilities are, ify /∈ {^←e , x, Δ},

P_ω^←x

X _n+1=y₊|X _n=y

= ω(y, y₊)

ω(y, y₊)+ω(y,^←y )+

yk=y₊ω(y, y_k)β(y_k) ,

P_ω^←x

X _n+1=^←y|X_n =y

= ω(y,^←y )

ω(y, y₊)+ω(y,^←y )+

yk=y₊ω(y, y_k)β(y_k) ,

P_ω^←x

X _n+1=Δ|X _n=y

=

ν(y)

k=1ω(y, yk)β(yk) ω(y, y₊)+ω(y,^←y )+

yk=y₊ω(y, y_k)β(y_k) ,

wherey₊is the child ofywhich lies on the path[[^←e , x]]. Besides, the walk is absorbed onΔand reflected on^←e andx. We recall thats:=ess supA. We construct the adequate coupling with a biased random walk(Y_n )_n≥0onZ, starting from|x| −1, increasing with probabilitys/(1+s), decreasing otherwise and such thatY_n ≥ |X_n |as long asX _n=Δ (which is always possible sinceP_ω(X_{n +1}=y₊|X _n=y)≤ ₁₊^s_s). After timeT, we letY_n move independently. By coupling and then by gambler’s ruin method, it leads to

P_ω^←x(Tx< T←

e)≤P_ω^|x|−1

∃n≥0: Y_n = |x|

=s.

It follows that 1−P_ω^x

T_x^∗< T←

e

≥ω(x,^←x )

1−P_ω^←x(T_x< T←

e)

≥c₁(1−s) ν(x) , by Eq. (2.4). Hence,

Q^e

N (π_k) > n, ν(π_k)≤√

n, D(e)= ∞

=

x∈U

E_Q

1_{ν(x)≤^√_n}P_ω^e

x=π_k, D(e) > T_x P_ω^x

N (x) > n, D(e)= ∞

≤

x∈U

E_Q

P_ω^e(x=π_k)

1−c1(1−s)

√n n

=

1−c1(1−s)

√n n

, which decays stretched exponentially. On the other hand,

Q^e

N (π_k) > n, ν(π_k)≥√

n, D(e)= ∞

≤Q^e

ν(πk)≥√ n, V_π^μ

k< εν(πk) +Q^e

N (πk) > n, V_π^μ

k≥εν(πk) with the notation introduced in the proof of Lemma2.3. We have

Q^e

ν(π_k)≥√ n, V_π^μ

k< εν(π_k)

=Q

ν(e)≥√

n, V_e^μ< εν(e) , which is stretched exponential by Cramér’s theorem. We also observe that

Q^e

N (πk) > n, V_π^μ

k≥εν(πk)

≤E_Q^e 1_{V^μ

πk≥εν(x)}

1−γ (πk)n

=EQ

1_{V^μ

e≥εν(x)}

1−γ (e)n

≤(1−cμε)ⁿ,

by Eq. (2.12). This completes the proof.

2.3. The caseΛ <∞

In this part, we suppose thatΛ <∞, whereΛis defined by Λ:=Leb

t∈R: E A^t

≤ 1 q₁

.

We prove that the tail distribution ofΓ1is polynomial.

Proposition 2.4. IfΛ <∞,then

nlim→∞

1 ln(n)ln

S^e(Γ1> n)

= −Λ. (2.13)

Proof. Lemma 3.3 of [1] already gives lim inf

n→∞

1 ln(n)ln

S^e(Γ₁> n)

≥ −Λ.

Hence, the lower bound of (2.13) is known. The rest of the section is dedicated to the proof of the upper bound.

We start with three preliminary lemmas. We first prove an estimate for one-dimensional RWRE, that will be useful later on. Denote by(Rn, n≥0)a generic RWRE onZsuch that the random variablesA(i),i≥0 are independent and have the distribution ofA, when we set fori≥0,

A(i):=ωR(i, i+1) ωR(i, i−1)

withω_R(y, z)the quenched probability to jump fromytoz. We denote byP_ω,R^k the quenched distribution associated with(R_n, n≥0)when starting fromk, and byPRthe distribution of the environmentω_R. Letc₇∈(0,1)be a constant whose value will be given later on. For anyk≥≥0 andn≥0, we introduce the notation

p(, k, n):=E_PR

1−c₇P_ω,R

T^∗> T₀∧T_kn

. (2.14)

Lemma 2.5. Let0< r <1,andΛ_r:=Leb{t∈R: E[A^t] ≤¹_r}.Then,for anyε >0,we have fornlarge enough,

k≥≥0

r^kp(, k, n)≤n^−Λr+ε.

Proof. The method used is very similar to that of Lemma 5.1 in [1]. We feel free to present a sketch of the proof. We consider the one-dimensional RWRE(Rn)n≥0. We introduce fork≥≥0, the potentialV (0)=0 and

V ()= −

−1

i=0

ln A(i)

, H₁()= max

0≤i≤V (i)−V (), H₂(, k)= max

≤i≤kV (i)−V ().

We know (e.g. [17]) that e^−H2(+1,k)

k+1 ≤P_ω,R⁺¹(T_k< T)≤e^−H2(+1,k), (2.15)

e^−H1()

k+1 ≤P_ω,R⁻¹(T₋₁< T)≤e^−H1(). (2.16)

It yields that P_ω,R

T^∗> T₀∧T_k

≥e^{−H1()∧H2(,k)+O(lnk)}, where O(lnk)is a deterministic function. Letη∈(0,1).

p(, k, n)≤

1−c₇n^−1+ηn

+PR

H₁()∧H₂(, k)−O(lnk)≥(1−η)ln(n)

≤e^−c8nη+PR

H₁()∧H₂(, k)−O(lnk)≥(1−η)ln(n) .

In Section 8.1 of [1], we proved that for anys∈(0,1),E_PR[e^{Λs(H1()∧H2(,k))}] ≤e^{kln(1/s)+os(k)}, where os(k)is such that os(k)/ktends to 0 at infinity. This implies that, definingos(k):=os(k)−Λ_sO(lnk),

s^kPR

H₁()∧H₂(, k)−O(lnk)≥(1−η)ln(n)

≤s^k

1∧e^{kln(1/s)−Λs(1−η)ln(n)+os(k)}

≤n^{−Λs(1−η)}exp

kln(s)+Λs(1−η)ln(n)

∧os(k) .

Observe that there exists M_s such that for any k and any n, we have (kln(s)+Λ_s(1 −η)ln(n))∧os(k)≤ sup_i≤Msln(n)o(i)+ηlnn, and notice that sup_i≤Msln(n)os(i)is negligible towards ln(n). This leads to, forn large enough,

s^kp(, k, n)≤s^ke^−c8nη+n^{−Λs(1−η)+2η}. Letr∈(0,1)ands > r. We have

r^kp(, k, n)≤r^ke^−c8nη+ r

s k

n^{−Λs(1−η)+2η}.

Lemma2.5follows by choosingηsmall enough andsclose enough tor.

LetZ_nrepresent the size of thenth generation of the treeT. We have the following result.

Lemma 2.6. There exists a constantc₉>0such that for anyH >0, B >0andnlarge enough, E_Q

1−γ (e)n

1_{Z_H>B}

≤n^−c9B. Proof. We have

E_Q

1−γ (e)n

1_{Z_H>B}

≤E_Q

1−γ (e)n

1_{ν(e)≥^√_n} +E_Q

1−γ (e)n

1_{{ZH>B,ν(e)≤}^√_n}

≤e^−nc3 +EQ

1−γ (e)n

1_{{ZH>B,ν(e)≤}^√_n} by Remark2.3. Whenν(e)≤√

n, we have, by (2.11), γ (e)≥ c₄

√ne^−c5R,

withR:=inf{k≥1:∃|x| =k, β(x)≥μ}as before (μ >0 is such thatq:=Q(β(e) > μ) >0). Thus, EQ

1−γ (e)n

1_{{ZH>B,ν(e)≤}^√_n}

≤Q

R > 1 4c5

ln(n)+H, ZH > B

+e^−n1/4+o(1). By considering theZ_Hsubtrees rooted at each of the individuals in generationH, we see that

Q

R > c₁₀ln(n)+H, Z_H> B

=E_GW Q

R > c₁₀ln(n)Z_H

1_{ZH>B}

≤Q

R > c10ln(n)B

.