The infinite valley for a recurrent random walk in random environment

www.imstat.org/aihp 2010, Vol. 46, No. 2, 525–536

DOI:10.1214/09-AIHP205

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

The infinite valley for a recurrent random walk in random environment

Nina Gantert

, Yuval Peres

and Zhan Shi

aCeNos Center for Nonlinear Science, and Institut für Mathematische Statistik, Fachbereich Mathematik und Informatik, Universität Münster, Einsteinstr. 62, D-48149 Münster, Germany. E-mail:gantert@math.uni-muenster.de

bDepartment of Statistics, University of California at Berkeley, Berkeley, CA 94720. E-mail:peres@stat.berkeley.edu cLaboratoire de Probabilités et Modèles Aléatoires, Université Paris VI, 4 place Jussieu, F-75252 Paris Cedex 05, France.

E-mail:zhan.shi@upmc.fr

Received 6 August 2007; revised 27 February 2009; accepted 7 February 2009

Abstract. We consider a one-dimensional recurrent random walk in random environment (RWRE). We show that the – suitably centered – empirical distributions of the RWRE converge weakly to a certain limit law which describes the stationary distribution of a random walk in an infinite valley. The construction of the infinite valley goes back to Golosov, seeComm. Math. Phys.92 (1984) 491–506. As a consequence, we show weak convergence for both the maximal local time and the self-intersection local time of the RWRE and also determine the exact constant in the almost sure upper limit of the maximal local time.

Résumé. Nous prouvons que les mesures empiriques d’une marche aléatoire unidimensionnelle en environnement aléatoire convergent étroitement vers la loi stationnaire d’une marche aléatoire dans une vallée infinie. La construction de cette vallée infinie revient à Golosov, voirComm. Math. Phys.92(1984) 491–506. En applications, nous obtenons la convergence étroite du maximum des temps locaux et du temps local d’intersections de la marche aléatoire en environnement aléatoire; de plus, nous identifions la constante représentant la “limsup” presque sûre du maximum des temps locaux.

MSC:60K37; 60J50; 60J55; 60F10

Keywords:Random walk in random environment; Empirical distribution; Local time; Self-intersection local time

1. Introduction and statement of the results

Let ω=(ω_x)_x∈Z be a collection of i.i.d. random variables taking values in (0,1) and letP be the distribution of ω. For eachω∈Ω =(0,1)^Z, we define the random walk in random environment (abbreviated RWRE) as the time-homogeneous Markov chain(X_n)taking values inZ₊, with transition probabilitiesP_ω(X_n+1=1|X_n=0)=1, P_ω[X_n+1=x+1|X_n=x] =ω_x=1−P_ω[X_n+1=x−1|X_n=x]forx >0, andX₀=0. We equipΩ with its Borel σ-fieldF andZ^N with its Borelσ-fieldG. The distribution of(ω, (X_n))is the probability measurePonΩ ×Z^N defined byP[F×G] =

FP_ω[G]P (dω),F ∈F,G∈G.

Letρ_i=ρ_i(ω):=(1−ω_i)/ω_i. We will always assume that

logρ0(ω)P (dω)=0, (1.1)

P[δ≤ω₀≤1−δ] =1 for someδ∈(0,1), (1.2)

Var(logρ₀) >0. (1.3)

1Research supported in part by the European program RDSES.

The first assumption, as shown in [11], implies that forP-almost allω, the Markov chain(X_n)is recurrent, the second is a technical assumption which could probably be relaxed but is used in several places, and the third assumption ex- cludes the deterministic case. Usually, one defines in a similar way the RWRE on the integer axis, but for simplicity, we stick to the RWRE on the positive integers; see Section4for the results for the usual RWRE model. A key prop- erty of recurrent RWRE is its strong localization: under the assumptions above, Sinai [10] showed thatXn/(logn)² converges in distribution. A lot more is known about this model; we refer to the survey by Zeitouni [12] for limit theorems, large deviations results, and for further references.

Letξ(n, x):= |{0≤j≤n:X_j=x}|denote the local time of the RWRE inxat timenandξ^∗(n):=sup_x∈Zξ(n, x) the maximal local time at timen. It was shown in [7] and [8] that

lim sup

n→∞

ξ^∗(n)

n >0, P-a.s. (1.4)

(Clearly this lim sup is at most 1/2.) In addition, a 0–1 law (see [5]) says that lim sup_n→∞^ξ∗_n⁽ⁿ⁾ isP-almost surely a constant. The constant however was not known. We give its value in Theorem1.1.

Our main result (Theorem1.2below) shows weak convergence for the process(ξ(n, x), x∈Z)– after a suitable normalization – in a function space. In particular, it will imply the following theorem.

Theorem 1.1. LetM:=sup{s: s∈supp(ω0)} ∈(¹₂,1]andw:=inf{s: s∈supp(ω0)} ∈ [0,¹₂).Then lim sup

n→∞

ξ^∗(n)

n = (2M−1)(1−2w)

2(M−w)min{M,1−w}, P-a.s. (1.5) In particular,ifM=1−w,we have

lim sup

n→∞

ξ^∗(n)

n =2M−1

2M , P-a.s. (1.6)

Define the potentialV =(V (x), x∈Z)by

V (x):=

⎧⎨

⎩ _x

i=1logρ_i, x >0,

0, x=0,

−₀

i=x+1logρ_i, x <0

andC_(x,x+1):=exp(−V (x)). For eachω, the Markov chain is an electrical network in the sense of [4], whereC_(x,x+1) is the conductance of the bond(x, x+1). In particular,μ(x):=exp(−V (x−1))+exp(−V (x)), x >0,μ(0)=1 is a (reversible) invariant measure for the Markov chain.

LetV=(V (x), x∈Z)be a collection of random variables distributed asV conditioned to stay non-negative for x >0 and strictly positive forx <0. Due to (1.3), such a distribution is well-defined, see, for example, Bertoin [1] or Golosov [6]. Moreover, it has been shown that

x∈Z

exp

−V (x)

<∞, (1.7)

see [6], p. 494. For each realization of (V (x), x∈Z)consider the corresponding Markov chain onZ, which is an electrical network with conductancesC_(x,x+1):=exp(−V (x)). Intuitively, this Markov chain is a random walk in the

“infinite valley” given byV. As usual,μ(x):=exp(−V (x −1))+exp(−V (x)), x ∈Zis a reversible measure for this Markov chain. But, due to (1.7), we can normalizeμto get a reversible probability measureν, defined by

ν(x):=exp(−V (x −1))+exp(−V (x))

2

x∈Zexp(−V (x)) , x∈Z. (1.8)

Note that in contrast to the original RWRE which is null-recurrent, the random walk in the “infinite valley” given byVis positive recurrent.

Fig. 1. Definition ofb_n.

Let ¹be the space of real-valued sequences := { (x), x∈Z}satisfying :=

x∈Z| (x)|<∞. Let c_n:=min

x≥0: V (x)− min

0≤y≤xV (y)≥logn+(logn)^1/2

and

b_n:=min

x≥0:V (x)= min

0≤y≤cn

V (y)

,

see Figure1. We will consider{ξ(n, x), x∈Z}as a random element of ¹(of course,ξ(n, y)=0 fory <0). Here is our main result.

Theorem 1.2. Consider{ξ(n, x), x∈Z}as a random element of ¹under the probabilityP.Then, ξ(n, b_n+x)

n , x∈Z law

−→ ν, n→ ∞, (1.9)

where−→^law denotes convergence in distribution.In other words,the distributions of{^ξ(n,b_n^n+x), x∈Z}converge weakly to the distribution ofν(as probability measures on ¹).

The following corollary is immediate.

Corollary 1.1. For each continuous functionalf: ¹→Rwhich is shift-invariant,we have f

ξ(n, x) n , x∈Z

law

−→f

ν(x), x∈Z

. (1.10)

Example 1 (Self-intersection local time). Let f

(x), x∈Z

=

x∈Z

(x)².

Corollary1.1yields that 1

n²

x∈Z

ξ(n, x)²−→^law

x∈Z

ν(x)². (1.11)

This confirms Conjecture7.4in[9],at least for the RWRE on the positive integers;for the RWRE on the integer axis, see Section4.

Example 2 (Maximal local time). Let f

(x), x∈Z

=sup

x∈Z

(x).

Corollary1.1yields that ξ^∗(n)

n

−→law sup

x∈Zν(x). (1.12)

2. Proof of Theorem1.2

The key lemma is

Lemma 2.1. For anyK∈N, ξ(n, b_n+x)

n ,−K≤x≤K

−→law

ν(x),−K≤x≤K

. (2.1)

Given Lemma 2.1, the proof of Theorem 1.2 is straightforward. It suffices to show that for every function f: ¹→R,f bounded and uniformly continuous, we have

E

f

ξ(n, b_n+x) n , x∈Z

→E f

ν(x), x∈Z

, n→ ∞. (2.2)

For ∈ ¹, define K by K(x)= (x)I_{−K≤x≤K} (x∈Z). Forf: ¹→R, definefK byfK( )=f ( K)( ∈ ¹).

Due to Lemma2.1, E

f_K

ξ(n, b_n+x) n , x∈Z

→E f_K

ν(x), x∈Z

, n→ ∞. (2.3)

Fixε >0. Sincef is uniformly continuous, there isδsuch that we have for ∈ ¹

K− 1≤δ ⇒ fK( )−f ( )≤ε. (2.4)

Hence E

f_K

ξ(n, b_n+x) n , x∈Z

−f

ξ(n, b_n+x)

n , x∈Z

≤ε (2.5)

provided that E

x:|x|>K

ξ(n, bn+x) n

≤δ. (2.6)

Due to Lemma 2.1, E[

x:|x|>K

ξ(n,b_n+x)

n ] converges to E[

x:|x|>Kν(x)]. But, for K large enough, E[

x:|x|>Kν(x)] ≤δsinceE[ν(·)]is a probability measure onZ. Together with (2.6) and (2.3), this implies (2.2).

Proof of Lemma2.1. We proceed in several steps:

(i) Define

T (y):=inf{n≥1: Xn=y}, (2.7)

the first hitting time ofy. Forε >0, letA_ndenote the event{T (b_n)≤εn}. Then,P[A_n] →1 forn→ ∞.

Proof: see Golosov [6], Lemma 1.

(ii) LetBndenote the event{the RWRE exits[0, cn)before timen} = {T (cn) < n}. Then, forP-a.a.ω,Pω[Bn|X0= b_n] →0 forn→ ∞.

Proof: Due to [6], Lemma 7, we have for allm P_ω

T (c_n) < m|X₀=b_n

≤const·me^{−logn−(logn)1/2}. Takingm=n, we conclude thatP_ω[B_n|X0=b_n] →0.

(iii) Due to (i) and (ii), we can consider, instead ofP_ω, a finite Markov chainP_ω=P_ω⁽ⁿ⁾started fromb_n, in the valley [0, cn]. More precisely, the Markov chainP_ω is the original Markov chainP_ω with reflection atc_n, i.e. for 0< x < c_n,P_ω[X_n+1=x+1|X_n=x] =ω_x=1−P_ω[X_n+1=x−1|X_n=x],P_ω[X_n+1=1|X_n=0] =P_ω[X_n+1= c_n−1|X_n=c_n] =1 andX₀=b_n. The invariant probability measureμ_ω=μ⁽ⁿ⁾_ω ofP_ωis given by

μ_ω(x):=

⎧⎪

⎪⎨

⎪⎪

⎩

1 Zω

e^{−V (x)}+e^{−V (x−1)}

, 0< x < c_n,

1

Z_ωe^{−V (0)}, x=0,

1

Zωe^{−V (cn−1)}, x=cn, whereZω=2cn−1

x=0 e^{−V (x)}.

(iv) Recall (2.7). We note for further reference that for 0≤b < y < i,

P_ω

T (b) < T (i)|X₀=y

=

i−1

j=y

e^{V (j )} _i−1

j=b

e^{V (j )} ₋1

. (2.8)

This follows from direct computation, usingC_(x,x+1)=e^{−V (x)}, see also [12], formula (2.1.4). We now decompose the paths of the Markov chainPω into excursions frombntobn. Letx∈ [−bn, cn−bn],x=0 and denote byYb_n,x

the number of visits tob_n+xbefore returning tob_n. The distribution ofY_bn,x is “almost geometric”: we have P_ω[Y_bn,x=m] =

α(1−β)^m−1β, m=1,2,3, . . . ,

1−α, m=0,

whereα=α_bn,x=P_ω[T (b_n+x) < T (b_n)|X₀=b_n],β=β_bn,x=P_ω[T (b_n) < T (b_n+x)|X₀=b_n+x]. In particular, Eω[Yb_n,x] =α

β =μ_ω(b_n+x)

μ_ω(b_n) , (2.9)

whereμ_ω is the reversible measure for the Markov chain, see above. Note that (2.9) also applies forx =0, with Y_bn,0=1. Now, with Varω(Y_bn,x)denoting the variance ofY_bn,x w.r.t.P_ω,

Varω(Yb_n,x)=α(2−β−α)

β² ≤ 2

β

μ_ω(b_n+x) μ_ω(b_n) ≤ 4

β. (2.10)

Forx >1,

β=(1−ωb_n+x)Pω

T (bn) < T (bn+x)|X0=bn+x−1 .

Taking into account (2.8) yields β=(1−ω_bn+x)

_b

n+x−1

j=b_n

e^{V (j )−V (bn+x−1)} ₋1

, (2.11)

and (2.11) applies also to x =1. In particular, recalling (1.2), there is a constant a =a(K, δ) >0 such that for 1≤x≤K, uniformly inn,

Varω(Y_bn,x)≤a(K, δ). (2.12)

In the same way, one obtains (2.12) for−K≤x≤0. We note for further reference that due to (2.10) and (2.11), there is a constanta=a(δ) >0 such that forx∈ [−b_n, c_n−b_n],

Varω(Y_bn,x)≤a(δ)c_n

logn+(logn)^1/2

. (2.13)

(v) Denote byk_n the number of excursions fromb_n tob_n before timen. It follows from (2.9) that the average lengthγnof an excursion frombntobnunderPω is given by

γn=

cn

y=0

μ_ω(y) μ_ω(b_n)≥2.

Fixε∈(0,1). We show that forP-a.a.ω, P_ω

k_n n − 1

γ_n ≥ε

→0, n→ ∞. (2.14)

Proof: LetY_b⁽¹⁾

n,x, Y_b⁽²⁾

n,x, Y_b⁽³⁾

n,x, . . .be i.i.d. copies ofY_bn,x, and denote byE_n⁽ⁱ⁾=cn−bn

x=−b_nY_b⁽ⁱ⁾

n,x the length of theith excursion fromb_ntob_n. Then,{k_n≥m} ⊆ {_m

i=1E_n⁽ⁱ⁾≤n}and{k_n≤m} ⊆ {_m

i=1E_n⁽ⁱ⁾≥n}. Hence, Pω

k_n n − 1

γ_n ≥ε

=Pω

_n(1/γ

n+ε)

i=1

E_n⁽ⁱ⁾≤n

+Pω

_n(1/γ

n−ε)

i=1

E_n⁽ⁱ⁾≥n

. (2.15)

To handle the first term in (2.15), recallE_n⁽¹⁾, E⁽²⁾_n , . . .are i.i.d. with expectationγ_nunderP_ω and apply Chebyshev’s inequality:

Pω

1 n(1/γn+ε)

n(1/γ_n+ε)

i=1

E_n⁽ⁱ⁾−γn

≤ − εγ_n² 1+γ_nε

≤ Varω(E_n⁽¹⁾) n(1/γn+ε)

(1+γ_nε)²

ε²γ_n⁴ . (2.16)

Now, use the inequality Var(_N

j=1X_j)≤N_N

j=1Var(Xj)and (2.13) to get from (2.16) that Pω

_n(1/γ

n+ε)

i=1

E_n⁽ⁱ⁾≤n

≤c⁴_na(δ)²(logn)³ n

(1+γ_nε)

ε²γ_n³ ≤c⁴_na(δ)²(logn)³ n

2 ε².

Due to Chung’s law of the iterated logarithm, lim infn→∞(^{log logn}_n )^1/2(max0≤x≤nV (x)−min0≤x≤nV (x))is a.s. a strictly positive constant, and this impliesc⁴_n/n→0 forP-a.a.ω. We conclude that the first term in (2.15) goes to 0 for a.a.ω. The second term in (2.15) is treated in the same way.

Further, due to Chebyshev’s inequality, (2.9) and (2.12), we have for−K≤x≤K Pω

1 k_n

kn

i=1

Y_b⁽ⁱ⁾

n,x−μ_ω(b_n+x) μ_ω(b_n)

≥ε

→0, n→ ∞. (2.17)

(vi) Define, for−K≤x≤K,ρ_n,ωby ρn,ω(x)=ξ(n+T (b_n), b_n+x)

n −ξ(T (b_n), b_n+x)

n , −K≤x≤K. (2.18)

Forx=0, we havenρ_n,ω(x)=k_n.

We now estimateρn,ω(x)for−K≤x≤K,x=0:

k_n n

1 k_n

kn

i=1

Y_b⁽ⁱ⁾

n,x≤ρ_n,ω(x)≤k_n+1 n

1 k_n+1

kn+1

i=1

Y_b⁽ⁱ⁾

n,x. (2.19)

We conclude from (2.14) and (2.17) that for−K≤x≤K, P_ω

ρ_n,ω(x)− 1 γ_n

μ_ω(b_n+x) μ_ω(b_n)

≥ε

→0, n→ ∞. (2.20)

(vii) We show that{_γ¹_n^μω_μ^(b_ω(bⁿ⁺_n)^x),−K≤x≤K}converges in distribution to{ν(x),−K≤x≤K}.

Proof: Due to [6], Lemma 4, the finite-dimensional distributions of{V (b_n+x)−V (b_n)}x∈Zconverge to those of {V (x) }x∈Z. Therefore, settingγ_n^{(N )}:=b_n+N

x=bn−N μω(x)

μω(bn), we have for eachN that 1

γ_n^{(N )}

μ_ω(b_n+x)

μ_ω(b_n) ,−K≤x≤K

converges in distribution to e^{−V (x)} +e^{−V (x −1)}

_N

x=−Ne^{−V (x)} +e^{−V (x −1)},−K≤x≤K

.

It remains to show that for N large enough, γ_n^{(N )} and γn are close for n→ ∞ and that, for N large enough, N

x=−N(e^{−V (x)}+e^{−V (x −1)})is close to_∞

x=−∞(e^{−V (x)} +e^{−V (x −1)}). Note thatγ_n^{(N )}≤γnand P

γ_n^{(N )}

γ_n ≥1−ε

→P

N

x=−N(e^{−V (x)}+e^{−V (x −1)}) _∞

x=−∞(e^{−V (x)}+e^{−V (x −1)})≥1−ε

(2.21) forn→ ∞. But, due to (1.7),

_N

x=−N

e^{−V (x)} +e^{−V (x −1)}

≥(1−ε)

∞ x=−∞

e^{−V (x)} +e^{−V (x −1)}

increases to the whole space and we conclude that the last term in (2.21) goes to 1 forN→ ∞.

Putting together (i)–(vii), we arrive at (2.1).

3. Proof of Theorem1.1

We know now that the laws of ^ξ∗_n⁽ⁿ⁾converge to the law ofν^∗(see (1.12)), where ν^∗:=sup

x∈Zν(x)=sup

x∈Z

exp(−V (x −1))+exp(−V (x))

2

x∈Zexp(−V (x)) . (3.1)

We will show that lim sup

n→∞

ξ^∗(n)

n =c, P-a.s., (3.2)

where

c=sup

z: z∈suppν^∗

. (3.3)

In fact, we get lim sup

n→∞

ξ^∗(n)

n ≥c, P-a.s. (3.4)

from the following general lemma, whose proof is easy (and therefore omitted).

Lemma 3.1. Let(Y_n), n=1,2, . . .be a sequence of random variables such that(Y_n)converges in distribution to a random variableY andlim sup_n→∞Y_nis constant almost surely.Then,

lim sup

n→∞ Y_n≥sup{z: z∈suppY}. (3.5)

To show that lim sup

n→∞

ξ^∗(n)

n ≤c, P-a.s., (3.6)

we will use a coupling argument with an environmentωwhose potentialV will be shown to achieve the supremum in (3.3). Define the environmentωas follows.

ω_x:=

w, x >0, M, x≤0.

For the corresponding potentialV, exp(−V (x))is given as follows:

exp

−V (x)

=

⎧⎨

⎩ _w

1−w

x

, x >0,

1, x=0,

_M

1−M

x

, x <0.

For any fixed environmentω∈ [w, M]^Z,Q_ω defines a Markov chain on the integers with transition probabilities Q_ω[X_n+1=x+1|X_n=x] =ω_x=1−Q_ω[X_n+1=x−1|X_n=x],x∈Z, andX₀=0. Then we have the following lemma.

Lemma 3.2. Let the environmentωbe defined as above.AssumeM≤1−w.For allω∈ [w, M]^Z,allx∈Zand all n∈N,the distribution ofξ(n, x)with respect toQ_ω is stochastically dominated by the distribution ofξ(n,0)with respect toQ_ω.In particular,

sup

z: z∈supp sup

x∈Zν(x)

=ν(0):=exp(−V (−1))+exp(−V (0))

2

x∈Zexp(−V (x)) . (3.7)

Proof of Lemma3.2. First step. We show that with T (0):=min{n≥1: Xn=0}, the distribution ofT (0) with respect toQω is stochastically dominated by the distribution of T (0)with respect toQω. We do this by coupling two Markov chains: the Markov chain(Xn)moves according toQω, with X0=0, the Markov chain(Xn)moves according toQω, with X0=0, in such a way that (Xn) returns to 0 before (or at the same time as) (Xn). Let (Xn)move according to Qω. IfXn≤0 and Xn+1< Xn andXn≤0, then also Xn+1< Xn: this is possible since Qω[Xn+1< Xn] =1−M≤Qω[Xn+1< Xn]. IfXn>0 andXn+1> XnandXn>0, then alsoXn+1> Xn: this is possible sinceQω[Xn+1> Xn] =w≤Qω[Xn+1> Xn]. If Xn>0 andXn+1> Xn andXn<0, thenXn+1< Xn: this is possible sinceQω[Xn+1> Xn] =w≤1−M≤Qω[Xn+1< Xn]. Now,(Xn)visits 0 before (or at the same time as) as(Xn)does and this proves the statement.

Second step. We show that for eachn, the distribution ofξ(n,0)with respect toQ_ω is stochastically dominated by the distribution of ξ(n,0)with respect toQ_ω. LetT₁, T₂, . . . be i.i.d. copies ofT (0). Then, Q_ω[ξ(n,0)≥k] = P_ω[_k

j=1T_j≤n] ≤Q_ω[_k

j=1T_j≤n] =Q_ω[ξ(n,0)≥k], where the inequality follows from the first step.

Third step. Forx∈Z, letθ^x be the shift on Ω, i.e.(θ_xω)(y)=ω(x+y),y∈Z. Forx∈Z, the distribution of ξ(n, x) with respect toQ_ω is dominated by the distribution ofξ(n,0)with respect toQ_θxω (more precisely, with T (x):=inf{n: Xn=x} denoting the first hitting time ofx, the distribution of ξ(n, x) with respect toQω is the distribution ofξ((n−T (x)),0)IT (x)≤nwith respect toQθ^xω). Now, apply the second step withθ^xωinstead ofω.

To show (3.7), define, forω∈ [w, M]^Z, the corresponding potentialV as in Section 1. Note that ifV is in the support ofV, the Markov chain defined byQ_ωis positive recurrent with invariant probability measure

ν(x)=exp(−V (x−1))+exp(−V (x))

2

x∈Zexp(−V (x)) , (3.8)

and we haveν(x)=limn→∞1

nξ(n, x), hence (3.7) follows from the stochastic domination.

Intuitively,V is the steepest possible (infinite) valley which maximises the occupation time of 0 (ifM≤1−w).

Letε >0. Since(X_n)is a positive recurrent Markov chain with respect toQ_ω, we have

nlim→∞

ξ(n,0)

n =ν(0), Q_ω-a.s., and

Q_ω

ξ(n,0)

n ≥ν(0)+ε

≤e^−C(ε)n, (3.9)

whereC(ε)is a strictly positive constant depending only onε. We have Q_ω

ξ^∗(n)

n ≥ν(0)+ε

≤

n

x=−n

Q_θ^x_ω

ξ(n,0)

n ≥ν(0)+ε

≤(2n+1)Qω

ξ(n,0)

n ≥ν(0)+ε

≤(2n+1)e^−C(ε)n, (3.10)

where we used Lemma3.2for the second inequality and (3.9) for the last inequality. By the Borel–Cantelli lemma, we conclude that

lim sup

n→∞

ξ^∗(n)

n ≤ν(0), Qω-a.s., for allω∈ [w, M]^Z. (3.11)

We have to show that (3.11) holds trueP-a.s., i.e. for our RWRE onZ₊, with reflection at 0. In order to do so, we need the following modification of Lemma3.2for environments onZ₊. LetK≥1 and let the environmentω^(K) be defined as follows.

ω^(K)_x :=

M, 0< x≤K, w, x > K, 1, x=0.

For the corresponding potentialV^(K), exp(−V^(K)(x))is given as follows:

exp

−V^(K)(x)

=

⎧⎨

⎩ _M

1−M

x

, 0< x≤K, _M

1−M

K _w

1−w

x−K

, x > K,

1, x=0.

The Markov chain onZ₊defined byP_ω(K)is positive recurrent with invariant probability measureν^(K)given by

ν^(K)(x):=

⎧⎪

⎨

⎪⎩

exp(−V^(K)(x−1))+exp(−V^(K)(x)) 1+2

x≥1exp(−V^(K)(x)) , x≥1,

1 1+2

x≥1exp(−V^(K)(x)), x=0.

(3.12)

Lemma 3.3. AssumeM≤1−w.For allK≥1,ifω∈ {1} × [w, M]^Z+ satisfiesω₀=1andb_n(ω)→ ∞,then for alln∈Nsuch thatb_n(ω)≥2K,and allx∈Z,x≥ −b_n+K,the distribution ofξ(n, b_n+x)with respect toP_ω is stochastically dominated by the distribution ofξ(n, K)with respect toP_ω(K)[·|X₀=K].Further,ν^(K)(K)→ν(0)for K→ ∞.

The proof of Lemma3.3is similar to the proof of Lemma3.2. The last statement follows from (3.12).

Letε >0. ChooseKsuch thatν^(K)(K)≤ν(0)+^ε₂. Since(X_n)is a positive recurrent Markov chain with respect toP_ω(K)[·|X₀=K], we have

nlim→∞

ξ(n, K)

n =ν^(K)(K), P_ω(K)[·|X₀=K]-a.s., and

P_ω(K)

ξ(n, K)

n ≥ν^(K)(K)+ε|X₀=K

≤e^−C(ε)n, (3.13)

whereC(ε)is a strictly positive constant depending only onε, and onK. ForP-a.a.ω, choosen₀(ω)such that for n≥n₀(ω),b_n(ω)≥2K and ¹_nK

x=0ξ(n, x)≤ ₄^ε,P_ω-a.s. (this is possible since b_n(ω) increases to∞,P-a.s. for n→ ∞, and_n¹K

x=0ξ(n, x)→0,P-a.s. forn→ ∞sinceP_ωis null-recurrent forP-a.a.ω). We then haveP-a.s. for n≥n₀(ω)

P_ω ξ^∗(n)

n ≥ν(0)+ε

≤P_ω ξ^∗(n)

n ≥ν^(K)(K)+ε 2

≤

n

x=−b_n+K

P_ω

ξ(n, b_n+x)

n ≥ν^(K)(K)+ε 4

≤(2n+1)P_ω(K)

ξ(n, K)

n ≥ν^(K)(K)+ε 4

X0=K

≤(2n+1)e^−C(ε/4)n, (3.14)

where we used Lemma3.2for the third inequality and (3.13) for the last inequality. By the Borel–Cantelli lemma, we conclude that

lim sup

n→∞

ξ^∗(n)

n ≤ν(0), P_ω-a.s., forP-a.a.ω. (3.15)

Together with (3.7), this finishes the proof in the caseM≤1−w: the value ofcis computed easily from (3.7). In the caseM >1−w, Lemma3.2is replaced with the following

Lemma 3.4. Let the environment ωbe defined as above.AssumeM >1−w.For allω∈ [w, M]^Z,allx∈Zand alln∈N, the distribution ofξ(T (1)+n, x)−ξ(T (1), x)with respect to Q_ω is stochastically dominated by the distribution ofξ(T (1)+n,1)with respect toQ_ω.In particular,

sup

z: z∈supp sup

x∈Zν(x)

=ν(1)=exp(−V (0))+exp(−V (1))

2

x∈Zexp(−V (x)) . (3.16)

The rest of the proof is analogous to the caseM≤1−w.

4. Further remarks

1. Consider RWRE on the integers axis, i.e. with transition probabilitiesQ_ω[X_n+1=x+1|X_n=x] =ω_x=1− Q_ω[X_n+1=x−1|X_n=x]forx∈Z. Then, Theorem1.2can be modified in the following way (we refer to [2] for details). First, we have to replacebnin (1.9) withbn defined as follows. We call a triple(a, b, c)witha < b < ca valleyofV if

V (b)= min

a≤x≤cV (x), V (a)= max

a≤x≤bV (x), V (c)= max

b≤x≤cV (x).

The depth of the valley is defined asd_(a,b,c)=(V (a)−V (b))∧(V (c)−V (b)). Call a valley(a, b, c)minimalif for all valleys(a,b,c)witha <a,c < c, we haved_(a,b,c)< d_(a,b,c). Consider the smallest minimal valley(a_n,b_n,c_n)with a_n<0<c_nandd_(an,bn,cn)≥logn+(logn)^1/2, i.e.(a_n, b_n,c_n)is a minimal valley withd_(a

n, bn,cn)≥logn+(logn)^1/2 and for every other valley(an,bn,cn)with these properties, we havean<an or cn<cn. If there are several such valleys, take (a_n, b_n,c_n)such that |b_n| is minimal (and b_n >0 if there are two possibilities). Further, one has to replace the limit measureνin (1.9) withνdefined in as follows. LetV_left=(V_left(x), x∈Z)be a collection of random variables distributed as V conditioned to stay strictly positive forx >0 and non-negative for x <0. (Recall that V=(V (x), x∈Z)is a collection of random variables distributed asV conditioned to stay non-negative forx >0 and strictly positive forx <0.) Set

ν_left(x):=exp(−Vleft(x−1))+exp(−Vleft(x))

2

x∈Zexp(−Vleft(x)) , x∈Z. (4.1)

In other words,ν_leftis defined in the same way asν, replacingVin (1.8) withV_left. Now, letQbe the distribution of νandQleftbe the distribution ofνleftand letνbe a random probability measure onZwith distribution ¹₂(Q+Qleft).

Then, (1.9)–(1.12) hold true for RWRE on the integer axis, if we replacebnwithbnandνwithν.

Since the right-hand side of (1.5) is a function ofM andw, sayϕ(M, w), for which we haveϕ(M, w)=ϕ(1− w,1−M), we see that replacingV withV_leftin (3.1) yields the same constantcas in (3.3) and therefore Theorem1.1 carries over verbatim.

2. We showed that lim sup_n→∞^ξ∗_n⁽ⁿ⁾ is P-a.s. a strictly positive constant and gave its value. We have lim infn→∞ξ^∗(n)

n =0 P-a.s. which is a consequence of (1.12). It was shown in [3] that there is a strictly positive constantasuch that lim infn→∞ξ^∗(n)log log logn

n =a,P-a.s. The value ofais not known.

Acknowledgments

We thank Gabor Pete, Achim Klenke and Pierre Andreoletti for pointing out mistakes in earlier versions of this paper and Amir Dembo for discussions. We also thank the referee for several valuable comments.

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