New examples of ballistic RWRE in the low disorder regime

We give a new criterion for ballistic behavior of random walks in random environments which are low disorder perturbations of the simple symmetric random walk on $\mathbb{Z}^d$, for $d\geq 2$. This extends the results established by Sznitman in 2003 and, in particular, allow us to give new examples of ballistic RWREs in dimension $d=3$ which do not satisfy Kalikow's condition. Essentially, this new criterion states that ballisticity occurs whenever the average local drift of the walk is not too small when compared to the standard deviation of the environment. Its proof relies on applying coarse-graining methods together with a variation of the Azuma-Hoeffding concentration inequality in order to verify the fulfillment of a ballisticity condition by Berger, Drewitz and Ram\'irez.


INTRODUCTION AND MAIN RESULTS
1.1. Introduction. The random walk in a random environment is one of the fundamental models describing the movement of a particle in disordered media (see [13,14] for a comprehensive overview of the model). For walks on Z d with d ≥ 2, few results exist giving explicit formulas for basic associated quantities such as the velocity, asymptotic direction or variance, or conditions characterizing specific long-term behavior such as transience/recurrence, directional transience and ballistic movement. In particular, it is still a widely open problem to explicitly characterize the (law of the) small disorder necessary to produce ballistic behavior whenever added to the jump probabilities of the simple symmetric random walk, see [12,9]. In this article we focus on this particular question and generalize previously known conditions by exploring the use of refined concentration inequalities which are variations of the well-known Azuma-Hoeffding inequality (see [2] for an extensive review of general concentration inequalities and their applications).
In the case of an i.i.d. random environment in dimension d ≥ 3, Sznitman was able to derive in [12] conditions on the small disorder which guarantee that the perturbed walk is ballistic. Essentially, he showed that as long as the average local drift in some direction ℓ of the perturbed random walk is not too small with respect to ǫ, the L ∞ -norm of the perturbation, one has ballisticity in direction ℓ (see Section 1.3 below for a precise statement). In the present article we improve on this by showing that, in fact, one only needs the average drift to be not too small but compared instead to σ, the standard deviation of the environment, which is always a smaller quantity than ǫ (and, furthermore, could potentially be much smaller).
As a consequence of this improvement we are able to obtain new examples of RWREs with ballistic behavior. As a matter of fact, to construct these examples it is necessary to show that they satisfy certain conditions generally known as ballisticity conditions. Two important ballisticity conditions are Kalikow's condition, introduced in [7], which is a requirement on the averaged 1.2. The model. Fix an integer d ≥ 2 and for each x = (x 1 , . . . , x d ) ∈ Z d let |x| := |x 1 | + · · · + |x d | denote its ℓ 1 -norm. Let V := {x ∈ Z d : |x| = 1} be the set of canonical vectors in R d and denote by P the set of all probability vectors p = (p(e)) e∈V on V, i.e. p ∈ [0, 1] V such that ∑ e∈V p(e) = 1. In addition, consider the product space Ω := P Z d with its Borel σ-algebra, denoted by B(Ω). We will call any element ω = (ω(x)) x∈Z d ∈ Ω an environment (on Z d ). For each x ∈ Z d , ω(x) is a probability vector on V, whose components we denote by ω(x, e), i.e. ω(x) = (ω(x, e)) e∈V . The random walk in the environment ω starting from x ∈ Z d is then defined as the Markov chain X = (X n ) n∈N 0 on Z d which starts from x and is given by the transition probabilities P x,ω (X n+1 = y + e|X n = y) = ω(y, e) for each y ∈ Z d , e ∈ V.
We denote its law by P x,ω . We assume throughout that the space of environments Ω is endowed with a probability measure P, called the environmental law. We shall call P x,ω the quenched law of the random walk, and also refer to the semi-direct product P x := P ⊗ P x,ω on Ω × (Z d ) N 0 given by as the averaged or annealed law of the random walk. In general, we will call the sequence (X n ) n∈N 0 under the annealed law a random walk in a random environment (RWRE) with environmental law P.

Assumptions 1.
Throughout the sequel we shall make the following assumptions on P: A1. The family of random probability vectors (ω(x)) x∈Z d is i.i.d. with some common law µ on P.
Equivalently, P is the product measure on Ω with marginal distribution µ. A2. P-almost surely, every weight ω(x) is a small perturbation of the weights of the simple symmetric random walk, i.e.
(The 4d factor in (1) is merely a more convenient normalization for our purposes.) Observe that, by (A1), (1) implies that there exists an event Ω ǫ with P(Ω ǫ ) = 1 such that on Ω ǫ one has that In particular, P is uniformly elliptic with ellipticity constant i.e. P-almost surely one has that ω(x, e) ≥ κ for all x ∈ Z d and e ∈ V. The goal of this article is to study transcience/ballisticity properties of X on fixed directions. Recall that, given ℓ ∈ S d−1 , one says that the random walk X is transient in direction ℓ if lim n→∞ X n · ℓ = +∞ P 0 − a.s., and that it is ballistic in direction ℓ if it satisfies the stronger condition Any RWRE which is ballistic in some direction ℓ satisfies also a law of large numbers (see [4]), i.e. there exists a deterministic vector v ∈ R d with v · ℓ > 0 such that This vector v is known as the velocity of the random walk.
In the sequel we will fix a certain direction, let us say e 1 := (1, 0, . . . , 0) ∈ S d−1 for example, and study transience/ballisticity only in this fixed direction. Thus, whenever we speak of transience or ballisticity of X it will be understood that it is with respect to this given direction e 1 . However, we point out that all of our results can be adapted for any other particular direction. and let λ denote the average local drift in direction e 1 by the formula λ = λ(µ) := E( d(0) · e 1 ) = E(ω(0, e 1 ) − ω(0, −e 1 )). A natural question to ask is whether, under Assumptions 1, X is ballistic as soon as λ > 0 holds. Unfortunately, this is not the case, as one can see from [3] where examples of random walks with diffusive behavior and non-vanishing λ are constructed. However, in [12] it is shown that if d ≥ 3 and λ is not too small with respect to ǫ then the random walk is indeed ballistic and, in fact, satisfies the so-called (T') condition for ballisticity (see Section below 2 for further details). The validity of (T') not only implies ballisticity of X, but also a CLT and large deviation controls for the sequence ( X n n ) n∈N , see [11]. More precisely, in [12] it is shown that, under Assumptions 1, given any η ∈ (0, 1) there exists some ǫ 0 = ǫ 0 (η, d) ∈ (0, 1) such that if ǫ ≤ ǫ 0 and then X satisfies condition (T') in direction e 1 and is, thus, ballistic in this direction. Our goal in this article is to extend the results from [12] and provide new instances of ballistic behavior, by considering the standard deviation of the environment ω, defined as where, for each x ∈ Z d , δ(x) := ( δ(x, e)) e∈V denotes the centered vector given by Our first result is an extension of the main result from [12] for the case d = 3.
Note that, since σ ≤ ǫ automatically holds by properties of the L p norms, (6) is indeed weaker than the λ ≥ ǫ 2.5−η condition stated in [12] for d = 3. Furthermore, as opposed to Sznitman's original result, (6) shows that it is possible to have ballistic behavior for environments with drift λ as small as one wants with respect to ǫ, as long as their standard deviation σ is accordingly small. Indeed, Theorem 2 suggests that, in order to see ballistic behavior, what is important is that λ is not too small but with respect to σ rather than ǫ.
Next, we investigate the validity of Kalikow's condition for ballisticity. We refer to Section 2 for a precise definition of this condition, but for now recall the reader that it is in general strictly stronger than (T'), and that it implies slightly stronger results on the sequence ( X n n ) n∈N 0 , see [10]. We have the following result, which is valid for all dimensions d ≥ 2.
Kalikow's condition is satisfied in direction e 1 . In particular, X is ballistic in direction e 1 .
Note that, whenever σ ≪ ǫ, Theorem 3 refines [8, Theorem 2], where it was shown that Kalikow's condition holds if λ > 1 d ǫ 2 . Combining both theorems together with Sznitman's result, we obtain the following corollary which yields the full map of scenarios of ballisticity known so far in this context for all d ≥ 2.
Theorem 5 is important mainly for two reasons. First, it shows that the condition λ ≥ c 1 (d)σ 2 from Theorem 3 is essentially optimal for the validity of Kalikow's condition, in the sense that there exists c 2 (d) > 0 such that if λ ≤ c 2 (d)σ 2 then one can already find examples of RWREs which do not satisfy Kalikow's condition. But, most importantly, in combination with Theorem 2 it also shows new examples (apart from those already given in [12]) of ballistic walks for d = 3 which satisfy (T') but not Kalikow's condition. Indeed, as a direct consequence of Theorems 2-5, we obtain the following result. Corollary 6. Given any ǫ 0 > 0 one can construct a RWRE in dimension d = 3 verifying Assumptions 1 and such that: C1. ǫ ≤ ǫ 0 and λ ≤ ǫ 2.5 (so that the hypothesis (4) from [12] is not satisfied) C2. (T') is verified in direction e 1 but Kalikow's condition fails in all directions.
One such example can be constructed as in [12] by first fixing ρ ∈ (0, 1] and then setting µ to be the law of the random probability weight ω(0) on P given by for some constant λ > 0 and random probability vector p = (p(e)) e∈V to be specified later. It is simple to check that, if the law µ of p satisfies: then for any η ∈ (0, 1 2 ) one can choose constants k 1 , k 2 , k 3 > 0 (depending only on η, ǫ 0 and ρ) in such a way if µ is taken to also satisfy then there exists a nonempty interval of values of λ for which the associated walk satisfies (C1) and the hypotheses of both Theorems 2 and 5, so that (C2) also holds. We omit the details. The proof of Theorem 2 is an adaptation of the approach developed in [12], which consists of verifying the effective criterion given in [11] for the validity of (T'). Instead of this criterion, we will verify that the polynomial condition from [1] holds, which is equivalent to (T') and more convenient for our purposes. Crucial to this verification are estimates on the average value and size of fluctuations of the Green's operator of the RWRE killed upon exiting a slab of diameter proportional to ǫ −1 , for ǫ as in (1). In [12] it is shown that these quantities can be suitably controlled by ǫ, that is, by the L ∞ -norm of the perturbation. However, to obtain our results we will need to show that these can still be controlled by σ, the L 2 -norm of the perturbed environment, which can be, in principle, a much smaller quantity. This task requires redoing some of the estimates from [12] but now in L 2 (and thus refining them from their original L ∞ -form), and introducing new tools to the analysis, like the generalized version of Azuma-Hoeffding's inequality used in Lemma 9. The proof of Theorem 3 is inspired by that of [8,Theorem 2] and is based on the version of Kalikow's formula proved in [8] together with a careful application of Kalikow's criteria for ballisticity. Finally, Theorem 5 follows from adapting the argument in [12, Theorem 5.1] to the L 2 -setting.
Finally, we mention that the ideas here can also be used to extend the bounds on the velocity developed in [8]. Indeed, the same approach used here goes through in [8] to show that: B1. In dimension d = 3, given two quantities δ < η ∈ (0, 1) there exists ǫ 0 ∈ (0, 1) and c 0 > 0 depending on δ and η such that if ǫ ≤ ǫ 0 and λ ≥ σǫ 1.5−η then X satisfies (T') in direction e 1 and is thus ballistic in direction e 1 with a velocity v satisfying B2. For all dimensions d ≥ 2, given any η ∈ (0, 1) there exists ǫ 0 = ǫ 0 (d, η) ∈ (0, 1) such that if ǫ ≤ ǫ 0 and λ > (4d + η)σ 2 then X satisfies Kalikow's condition in direction e 1 and is this ballistic in direction e 1 with a velocity v satisfying However, since this extension of the results [8] is completely analogous to the one of [12] we shall do here, we will omit the details of how to prove (B1-B2). The reader interested in a proof will know how to proceed from [8] after reading Sections 3 and 4 below.
The remainder of the article is organized as follows. In Section 2 we introduce general notation and establish a few preliminary facts about the RWRE model, including the precise definitions of Kalikow's and (T') conditions for ballisticity. Afterwards, Sections 3, 4 and 5 are each devoted to the proofs of Theorems 2, 3 and 5, respectively.

PRELIMINARIES
In this section we introduce the general notation to be used throughout the article, as well as review some preliminary notions about RWREs we will require for the proofs.
2.1. General notation. Given any subset B ⊂ Z d , we define its (outer) boundary as ∂B := {x ∈ Z d − B : |x − y| = 1 for some y ∈ B} and the first exit time of the random walk from B as Similarly, for x ∈ Z d we define the hitting time H x and first return time H x of x respectively as Furthermore, for L ∈ N we define the L-slab in direction e 1 as where we consciously omit the dependence on L from the notation U for simplicity. Finally, for each M ∈ N we also define the box together with its frontal side ∂ + B M := {y ∈ ∂B M : y · e 1 ≥ M} , and its middle-frontal part

Green's functions and operators.
Let us now introduce some notation we shall use related to the Green's functions of the RWRE and of the simple symmetric random walk (SSRW). Given a subset B ⊆ Z d , the Green's functions of the RWRE and SSRW killed upon exiting B are respectively defined for x, y ∈ B ∪ ∂B as where ω 0 is the corresponding weight of the SSRW, given for all x ∈ Z d and e ∈ V by we can define the corresponding Green's operator on L ∞ (B) by the formula Notice that g B , and therefore also G B , depends on ω only though its restriction ω| B to B. Finally, it is straightforward to check that if U is the slab defined in (9) then both g U and G U are well-defined for all environments ω ∈ Ω ǫ , where Ω ǫ is the full P-probability event under which (2) holds.
2.3. Ballisticity conditions. We now recall the two conditions for ballisticity we shall work with: condition (T'), originally introduced by Sznitman in [11], and Kalikow's condition from [7]. For simplicity, instead of giving the original formulation of (T') by Sznitman, we will state here the polynomial condition (P) for ballisticity, which was shown in [1] to be equivalent to (T') (and, recently in [6], to condition (T), which is the requirement that the exit probability from slabs along the opposite side of the direction of movement of the random walk, decays exponentially fast). The interested reader is invited to consult a more detailed exposition about such conditions in [8].
For simplicity, we consider only ballisticity in direction e 1 .

Condition (P).
We will say that condition (P) is satisfied (in direction e 1 ) if there exists M ≥ M 0 such that for some K > 15d + 5, where M 0 := exp{100 + 4d(log κ) 2 } and κ is the uniform ellipticity constant of the RWRE, which in our case can be taken to be κ = 1 4d . We mention once again that in [1] it was shown that (P) is equivalent to (T').

Introducing Kalikow's walk. Given a nonempty connected strict subset B
Z d , for each x ∈ B we define Kalikow's walk on B (starting from x) as the random walk starting from x which is killed upon exiting B and has transition probabilities determined by the environment ω x B ∈ P B given by It is straightforward to check that by the uniform ellipticity of P we have so that the environment ω x B is well-defined. In accordance with our present notation, we will denote the law of Kalikow's walk on B by P x,ω x B and its Green's function by g B (x, ·, ω x B ). Given a direction ℓ ∈ S d−1 , we will say that Kalikow's condition holds (in direction ℓ) if where d B,0 denotes the drift of Kalikow's walk in B at 0 defined as If Kalikow's condition holds in some direction ℓ ∈ S d−1 then the walk X is ballistic in direction ℓ, see [8] for further details. Furthermore, in [11] it is shown that Kalikow's condition implies (T').

PROOF OF THEOREM 2
We will divide the proof of Theorem 2 into three parts, each carried out in a separate subsection. Throughout this section we shall assume that the environmental law P satisfies Assumptions 1. 0)). The first step is to give a lower bound on E(G U [ d · e 1 ](0)), the expectation of Green's operator on the local drift at 0. The precise bound we need is contained in the following proposition, which is a generalization of [12,

Remark 8. Since σ ≤ ǫ by standard properties of the L p norms, Proposition 7 is indeed a generalization of [12, Proposition 3.1] because the required lower bound on the drift λ in (16) is now smaller.
To prove Proposition 7, we will follow the approach used in the proof of [12, Proposition 3.1], albeit with some modifications. To begin, for x ∈ Z d , e ∈ V and ω ∈ Ω ǫ let us set and Observe that, if ½ denotes the function constantly equal to 1 on Z d , since by definition of d one has

Now, a standard Markov chain calculation yields the formula
where T U , H x and H x are as in (7) and (8), respectively. Using the decomposition withω x ∈ Ω ǫ the environment given by the formulā But, since for all e ∈ V we have the inequality by the uniform ellipticity in (3), we see that for ǫ ≤ 1 In particular, using that |(1 − u) −1 − (1 + u)| ≤ 2u 2 whenever |u| ≤ 1 2 , we may write where δ(x) := ( δ(x, e)) e∈V and · 2 is the standard ℓ 2 -norm in R d , so that where, to obtain the second inequality, we have used the fact that Hence, by taking c 2 > 0 sufficiently large so that 1 5 dλ ≥ 2048 3 ǫd 5 σ 2 when (16) holds, we see that E(|C|) ≤ 1 5 dλL 2 is satisfied as claimed. Thus, in order to conclude the proof it will suffice to show that To this end, we first notice that, since for all x ∈ U we have ∑ e∈V δ(x, e) = 0, B remains unchanged if we replace the term P x+e,ω (H x > T U ) in the innermost sum with Since the latter term is independent of ω(x), we obtain that Now, on the one hand we have On the other hand, by (21) and the fact that for all y, we have that the leftmost expectation in (23) can be bounded from above by If there exist a constant b > 0 and a sequence (v n ) n∈N ⊆ R ≥0 such that for every n ∈ N |F n − F n−1 | ≤ b and E((F n − F n−1 ) 2 |G n−1 ) ≤ v 2 n , then for any u > 0 one has that Proof. Call F ∞ := lim inf n→+∞ F n . Upon noticing that from Theorem 2.1 and Remark 2.1 on [5] one obtains the bound The remaining bound follows by symmetry, working with −F n instead of F n .
The variant of [12, Proposition 3.2] we require for our purposes is the following.
Proof. We follow the proof of [12, Proposition 3.2], applying the martingale method introduced therein. First, let us enumerate the elements of U as U := {x n : n ∈ N}. Now, define the filtration and also the bounded (see [12, Proposition 2.2] for a justification) G n -martingale (F n ) n∈N 0 given for each n ∈ N 0 by for all b and (v n ) n∈N with |F n − F n−1 | ≤ b and E((F n − F n−1 ) 2 |G n−1 ) ≤ v 2 n for every n. Thus, let us find such b and v n . To this end, for each n ∈ N and all environments ω, ω ′ ∈ Ω ǫ with ω ≡ ω ′ off x n , i.e. which coincide at every x i with i = n, let us define If µ denotes the single site distribution of ω U = (ω(x i )) i∈N under P, then it is simple to see that where ω (i>n) := (ω(x i )) i>n , µ ⊗(i>n) denotes its distribution under the law P and ω ′ ≡ ω off x n . Furthermore, it follows from the proof of [12, Proposition 3.2] that for L ≥ 2 as in the statement of the proposition one has for some constant c(d) > 0. In particular, since g 0,U (0, ·) is a bounded function, (26) and (27) together yield that for some constant c(d, α) > 0. Now, on the other hand, by using Fubini's theorem and Minkowski's integral inequality (the latter applied twice), we obtain But by (27) we have that (ω(x n , e))) e∈V and δ ′ (x n ) is defined analogously but with ω ′ instead of ω. Finally, it follows from [12, displays (3. 3.3. Conclusion of the proof. We now finish the proof of Theorem 2 by checking that (T') holds. We will split the proof in two cases, depending on whether σ < ǫ 1.5 or σ ≥ ǫ 1.5 .
Hence, it suffices to prove the result assuming that σ ≥ ǫ 1.5−η . Note that, in this case, we have the inequality λ ≥ max{σǫ 1.5−η , ǫ 3−η } =: λ 0 . We will prove the result by verifying the polynomial condition (P) in (11). To this end, we will use Propositions 7 and 10 above, as well as the estimates derived in [12] to establish the validity of the so-called effective criterion.
First, let us fix a constant θ > 0 with value to be specified later and set  (10)) we consider B(x) := B + x, i.e the translate of B centered at x, then by construction of B we have that for any ω ∈ Ω Thus, it follows from the translation invariance of P that, in order to check that condition (P) holds, it will suffice to show that for some constant c = c(d, η) > 0 if ǫ is taken sufficiently small (but depending only on d and η).
Now, sinceω(y, e) ≥ 1 2d (1 − ǫ 2 ) for all e ∈ V by Assumption (A2), a straightforward computation using (36) then yields the estimate E ( d(y, ω) · e 1 )r(y, ω) Finally, using (2) where the expectation on the right-hand side of (38) can be further bounded from above by Gathering all estimates obtained, we conclude that (1− ǫ 2 ) 3 σ 2 so that ε K > 0 (where ǫ K is the one defined in (14)) provided that Since a simple computation shows that for all ǫ ∈ (0, 1) we have then immediately gives the result.