The Erdős-Rényi-Shepp law of large numbers for ballistic random walk in random environment

We consider a one dimensional ballistic nearest-neighbor random walk in a random environment. We prove an Erdős-Rényi– Shepp strong law for the increments. 1. Definitions and main results The classical Erdős-Rényi–Shepp strong law of large numbers [4], [5], asserts as follows. Theorem 1.1 (Erdős-Rényi 1970, Shepp 1964). Consider a random walk Sn = ∑n i=1Xi with Xi i.i.d., satisfying EX1 = 0. Set φ(t) = E[e tX ] and let D φ = {t > 0 : φ(t) < ∞}. Let α > 0 be such that φ(t)e −αt achieves its minimum value for some t in the interior of D φ . Set 1/Aα := − logmin t>0 φ(t)e −αt. Then, Aα > 0 and (1.1) max 0≤j≤n−⌊Aα logn⌋ Sj+⌊Aα logn⌋ − Sj ⌊Aα log n⌋ a.s. → α, a.s. In the particular case of Xi ∈ {−1, 1}, the assumptions of the theorem are satisfied for any α ∈ (0, 1). The theorem also trivially generalizes to EX1 ̸= 0, by considering Yi = Xi − EXi. Theorem 1.1 is closely related to the large deviation principle for Sn/n given by Cramér’s theorem, see e.g. [3] for background. Indeed, with I(x) = supt(tx−log φ(t)) denoting the rate function, one observes that I(α) = 1/Aα and that (1.2) α = inf{x > 0 : I(x) > 1/Aα}. In this paper, we prove an analogous statement for standard one dimensional random walk in random environment (RWRE), in the case of positive velocity. We begin by introducing the model. Fix a realization ω = {ωi}i∈Z with ωi ∈ (0, 1) of a collection of i.i.d. random variables, which we call the environment. With p denoting the law of ω0 and σ(p) its support, denote by P = pZ the law of the environment on Σp := σ(p) Z. We make throughout the following assumption. Condition 1.2 (Uniform Ellipticity). There exists a κ ∈ (0, 1) such that σ(p) ⊂ [κ, 1− κ] almost surely. Date: May 4, 2020. Revised May 19, 2021 and July 8, 2021. This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 692452). 1 2 DARCY CAMARGO, YURI KIFER, AND OFER ZEITOUNI Letting ρi := (1 − ωi)/ωi, we note that the ellipticity assumption gives a deterministic uniform upper and lower bounds on ρi. It will be useful for us to consider also different laws of the environment Σ = [κ, 1 − κ]Z, not necessarily product laws. Such laws will be denoted η. Equipping Σ with the standard shift operator θ, so that (θω)j := ωi+j , the spaces of probability measures (stationary/ ergodic wrt θ) on Σ are denoted M1(Σ) (M s 1 (Σ)/M e 1 (Σ)), respectively; similar definitions hold when Σ is replaced by Σp. On top of ω we consider the RWRE, which is a nearest neighbor random walk {Xt}t∈Z. Conditioned on the environment ω, {Xt} is a Markov chain with transition probabilities π(i, i+ 1) = 1− π(i, i− 1) = ωi. We denote the law of the random walk, started at i ∈ Z and conditioned on a fixed realization of the environment ω, by Pi (the so-called quenched law). For any measure η ∈ M1(Σ), the measure η(dω) ⊗ Pi is referred to as the annealed law, and denoted by P i ; with some abuse of notation, we sometimes say annealed law for the restriction of P i to path space. If η = P then we write Pi for P a,P i . We use similar conventions for expectations, e.g. Ei for expectation with respect to P a,P i , etc. 1.1. The potential V and functional S. Introduce the potential function, which is defined as (1.3) Vω(j) =  j ∑ i=1 log ρi(ω), if j > 0; 0, if j = 0; − 0 ∑ i=j+1 log ρi(ω), if j < 0. and the Lyapunov function, see [2], (1.4) S(n, ω) =  n−1 ∑ i=0 eVω(i), if n > 0, 0, if n = 0, −1 ∑ i=n eVω(i), if n < 0. By definition, for n > m ≥ 0 we can decompose S(n, ω) as (1.5) S(n, ω) = S(m,ω) + eωS(n−m, θω). Another important property of S(n, ω) is its relation to hitting times. Let τA = inf{t > 0 : Xt ∈ A} and abbreviate τ{i} = τi for i ∈ Z. Then, see e.g. [8, (2.1.4)], Px [τ0 > τy] = S(x, ω) S(y, ω) , for y > x > 0. (1.6) Also, for n > 0, (1.7) e max 0≤j≤n−1 Vω(j) ≤ S(n, ω) ≤ ne max 0≤j≤n−1 Vω(j) . ERDŐS-RÉNYI-SHEPP LAW 3 1.2. Rate functions and modified environments. We follow [1] in introducing the function φ(ω, λ) = E0 [e11[τ1 < ∞]], and the hitting time quenched rate function, defined for η ∈ M1(Σ), (1.8) I η (u) = sup λ∈R { λu− ∫ log φ(ω, λ)η(dω) } . We denote the empirical field Rn ∈ M1(Σ) by (1.9) Rn = 1 n n−1 ∑ j=0 δθjω. It is well known, see e.g. [3], that under P , the sequence Rn satisfies a large deviation principle in M1(Σ), equipped with the topology of weak convergence, with rate function h(·|η), the so-called specific relative entropy. We need to consider the RWRE conditioned on not hitting the origin, i.e. conditioned on τ0 = ∞. Using Doob’s h-transform, it is straightforward to check that such conditional law is equivalent to using a transformed environment, namely for all measurable A and i ≥ 1, Pi [A | τ0 = ∞] = Pi [A], where ω̂0 = 1 and, for i ≥ 1, (1.10) ω̂i = ωiS(i+ 1, ω) S(i, ω) . Note that due to (1.4), we have that ω̂i ∈ [0, 1] and ω̂i > ωi. For L a positive integer, consider the following ergodic (with respect to shifts, if the law of ω is ergodic) environment obtained as a transformation of ω, (1.11) ω̂ i := ωiS(L+ 1, θ i−Lω) S(L, θi−Lω) . Here again, ω̂L i ∈ [0, 1] and using (1.5), we conclude that (1.12) ω̂ i > ω̂i, for i > L. The environment ω̂L i at site i approximates the environment at i one would get by conditioning on not hitting i− L. Introduce the function (1.13) I (x, η) = lim L→∞ sup λ≤0 { λ− x ∫ log φ(ω̂, λ)η(dω) } , η ∈ M s 1 (Σ). The existence of the limit in (1.13) is due to the following lemma, whose proof appears in Section 5. Lemma 1.3. For any fixed i, the sequence {ω̂L i } is decreasing in L ∈ Z+. Moreover the limit in (1.13) exists for any η ∈ M s 1 (Σ). For η ∈ M e 1 (Σ), IF (x, η) has a natural interpretation as a rate function for the quenched LDP of the hitting times of the random walk in random environment, conditioned on never hitting the origin, see Appendix A. The superscript F thus stands for the word forward. 4 DARCY CAMARGO, YURI KIFER, AND OFER ZEITOUNI 1.3. Statement of main result. With all needed information gathered, we state the main result of the paper. Theorem 1.4. Let P = pZ satisfy Condition 1.2. Set (1.14) s = sup{θ > 0 : Epρ0 ≤ 1}. Assume that s ∈ (1,∞]. Fix A > 0. Then, for k = k(n) positive integer such that k(n)/ log n → A, (1.15) max 1≤t≤n−k Xt+k −Xt k →n→∞ x∗(A), P0 − a.s., where (1.16) x∗(A) = inf{x > 0 : I∗(x) > 1/A}, and (1.17) I∗(x) = inf η∈Ms 1 (Σp) { I ( x, η ) + xh(η|P ) } . (Compare (1.15) and (1.16) with (1.1) and (1.2).) Let (1.18) vp := 1− Ep(ρ0) 1 + Ep(ρ0) . We remark, see [8], that the condition s ∈ (1,∞] is equivalent to Ep(ρ0) < 1 and is also equivalent to the convergence (1.19) Xn n →n→∞ vp > 0, P − a.s.. That is, we are dealing here with the transient ballistic case. It also implies that Ep log ρ0 < 0. We now claim that it follows from the definitions that x 7→ I∗(x) is an increasing (not necessarily strictly) function on R+, with I∗(x) = 0 for x ∈ [0, vP ] and I ∗(x) →x→∞ ∞, with I∗ continuous on its domain and strictly increasing in the set {x : ∞ > I∗(x) > 0}. Indeed, I∗(x) ≤ IF (x, P ) = 0 for x ≤ vP , see Proposition A.1 and note that under P1 , for such x, the event τxk ≤ k in the proposition has probability which is bounded below uniformly in k. Further, I∗(·), being the infimum of increasing functions, is clearly increasing. Let G(x, η) = IF (x, η) + xh(η|P ). The continuity of I∗ follows immediately by noting that for given ε > 0 there exists an η ∈ M s 1 (Σp) so that I∗(x) ≥ G(x, η)− ε, and therefore I∗(y) ≤ G(y, η) ≤ I∗(x)+IF (y, η)−I (x, η)+(y−x)h(η|P )+ε →y↘x I∗(x)+ε, which gives right continuity of I∗ and therefore continuity since I∗ is increasing. Finally, the strict monotonicity in the indicated domain follows from the fact that if I∗(x) > 0 then for all η ∈ M s 1 (Σp), G(x, η) > 0 and therefore, for all y > x, using the convexity of IF (·, η), G(y, η) ≥ G(x, η) + (y − x) x ( I (x|η) + xh(η|P ) ) ≥ I∗(x) ( 1 + (y − x) x ) , which implies that I∗(y) = infη∈Ms 1 (Σp)G(y, η) > I ∗(x). The last claim in turns implies that x∗(A) is well defined and also that AI∗(x∗(A)) = 1. It is also obvious from Theorem 1.4 that x∗(A) ≤ 1. ERDŐS-RÉNYI-SHEPP LAW 5 1.4. Proof strategy. The standard proof of Theorem 1.1 and of its extensions to sums of weakly dependent random variables usually consists of an upper and of a lower bounds for increments within time intervals (which we refer to as temporal blocks) of length Aα log n. The former relies only on the upper large deviations bound for such sums while the latter in addition to the lower large deviations bound requires also sufficiently weak dependence which enables to split the sum into weakly dependent disjoint blocks (this step is, of course, trivial in the independent case). In this way the corresponding random walk is split into weakly dependent temporal blocks. Such a temporal splitting is not possible in our case of random environment, since (under the annealed measure) increments of the random walk in disjoint time intervals are strongly correlated. So instead, in the proof of Theorem 1.4 we use a spatial decoupling of the walk in order to obtain both upper and lower bounds on maximal increments. This leads to several complications. First, the increments of the walk in different spatial blocks are not independent. Secondly, and more important, the walk may visit a block many times, and the probability to do so depends not only on the environment in the block but also on adjacent blocks. The first difficulty is relatively easily dispensed with by appealing to a standard non-backtracking estimate (Lemma 2.2). This allows us to consider only blocks of size c log n for some large c. To address the second issue, we use the environment ω̂, see (1.10), representing the environment under the condition of not backtracking at all, and use it to introduce the crucial quantity χ(k, x, c, η) which serves as a proxy for the probability of having a fast segment of the walk in a block of length xk with k = k(n) such that