Random Walks in Random Environments

in the Perturbative Regime

Ofer Zeitouni

Abstract We review some recent results concerning motion in random media satisfying an appropriate isotropy condition, in the perturbative regime in dimension d ≥ 3.

1 Introduction

This talk reports on joint work with E. Bolthausen [1], as well as earlier joint work with A.-S. Sznitman [7]. We consider random walks in random environments on Zd , d ≥ 3, when the environment is a small perturbation of the fixed environment corresponding to simple random walk. More precisely, let P be the set of probability distributions on Zd , charging only neighbors of 0. If ε (0, 1/2d), we set, with {ei }di=1 denoting the standard basis of Rd ,

and transition probabilities

Pω,x (Xn+1 = y|Xn = z) = ωz(y − z).

Ofer Zeitouni

Faculty of Mathematics, Weizmann Institute, Rehovot, Israel and School of Mathematics, University of Minnesota, Minneapolis, USA, e-mail: zeitouni@math.umn.edu

824 Ofer Zeitouni

We are mainly interested in the case of a random ω. Given a probability measure

μ on P , we consider the product measure Pμ on (Ω, F ). We usually drop the index μ in Pμ. In all that follows we make the following basic assumption.

Isotropy Condition μ is invariant under lattice isometries, i.e. μf −1 = μ for any orthogonal mapping f which leaves Zd invariant, and μ(Pε ) = 1 for some ε (0, 1/2d) which will be specified later.

The model of RWRE has been studied extensively. We refer to [6] and [8] for recent surveys. A major open problem is the determination, for d > 1, of laws of large numbers and central limit theorems in full generality (the latter, both under the quenched measure, i.e. for Pμ-almost every ω, and under the annealed measure Pμ Px,ω ). Although much progress has been reported in recent years ([2, 4, 5]), a full understanding of the model has not yet been achieved.

In view of the above state of affairs, attempts have been made to understand the perturbative behavior of the RWRE, that is the behavior of the RWRE when μ is supported on Pε and ε is small. The first to consider such a perturbative regime were [3], who introduced the Isotropy Condition and showed that in dimension d ≥ 3, for small enough ε a quenched CLT holds.1 Unfortunately, the multiscale proof in [3] is rather difficult, and challenging to follow. This in turns prompted the derivation, in [7], of an alternative multiscale approach, in the context of diffusions in random environments. The main result of [7] can be described as follows. Consider a diffusion with random coefficients on Rd ,

dXt = b(Xt , ω)dt + σ (Xt , ω)dWt ,

with W· a d-dimensional Brownian motion and a(x, ω) = σ (x, ω)σ T (x, ω). Assume the local characteristics a, b are uniformly Lipshitz in the space variable, stationary, and of finite range dependence, and further satisfy an isotropy condition of the type described above. One then has the following.

1 As the examples in [2] demonstrate, for every d ≥ 7 and ε > 0 there are measures μ supported

The proof of Theorem 1 is based on a multiscale analysis that includes the appropriate smoothing (with respect to Hölder norms) of the transition density of the diffusion, together with controlling exit measures from boxes, and in particular their large deviations. The latter is a crucial part of the control of exit time from traps. A naturally related question is whether focusing on exit measures from balls, i.e. not considering at all the time to exit, can simplify the analysis on the one hand, and provide sharp (local) estimates on exit on the other. The affirmative answer to these questions in provided in [1], which we summarize in the next section. In contrast with [7], we focus on two ingredients. The first is a propagation of the variational distance between the exit laws of the RWRE from balls and those of simple random walk (which distance remains small but does not decrease as the scale increases). The second is the propagation of the variation distance between the convolution of the exit law of the RWRE with the exit law of a simple random walk from a ball of (random) radius, and the corresponding convolution of the exit law of simple random walk with the same smoothing, which distance decreases to zero as scale increases.

2 Local Limits for Exit Measures

def

Throughout, for x Rd , |x| is the Euclidean norm. If L > 0, we write VL = {x

Zd x L , and for x Zd , V (x) def x V .

: | | ≤ } L = + L

If F , G are functions Zd × Zd → R we write F G for the (matrix) product:

def

F G(x, y) = u F (x, u)G(u, y), provided the right hand side is absolutely summable. We interpret F also as a kernel, operating from the left on functions f :

random walk started from x, and use ΠV (x, ·) for the analogous quantity for the RWRE. Fix once for all a probability density

def

If m > 0, the rescaled density is defined by ϕm(t ) = (1/m)ϕ(t /m). We then let πˆ Ψm (x, ·) denote the exit measure of simple random walk started at x from a ball with random radius R distributed according to ϕm.

For x Zd , t R, and L > 0, we define the random variables

826 Ofer Zeitouni

and with δ > 0, we set

The following theorem is the main result of [1]. It provides a local limit theorem for the exit law.

Theorem 2 ([1]). (d ≥ 3) There exists ε0 > 0, such that if the Isotropy Condition is satisfied with ε ≤ ε0, then for any δ > 0, and for any integer r ≥ 0,

Further, the RWRE Xn is transient, that is

for all x Zd , Px,ω -a.s., limn→∞ |Xn| = ∞.

The Borel-Cantelli lemma implies that under the conditions of Theorem 2,

lim sup DL,Ψt (0) ≤ ct , Pμ-a.s.,

L→∞

where ct is a (random) constant such that ct →t →∞ 0, a.s.

We remark that the rate of decay of probabilities in Theorem 2 is not expected to be optimal, rather it is dictated by the multiscale-scale scheme employed in the proof. We also remark that proving an analogue of the local limit result for the exit measure in Theorem 2 for dimension d = 2 is an open problem; the method of proof employed both in [7] and [1] uses the transience of simple random walk (and the finiteness of associated Green functions) in a crucial way.

References

1.E. Bolthausen and O. Zeitouni, Multiscale analysis for random walks in random environments.

Probab. Theory Relat. Fields 138, 581–645 (2007)

2.E. Bolthausen, A.S. Sznitman, and O. Zeitouni, Cut points and diffusive random walks in random environment. Ann. Inst. Henri Poincaré PR39, 527–555 (2003)

3.J. Bricmont and A. Kupiainen, Random walks in asymmetric random environments. Commun. Math. Phys. 142, 345–420 (1991)

4.A.S. Sznitman, An effective criterion for ballistic behavior of random walks in random environment. Probab. Theory Relat. Fields 122, 509–544 (2002)

5.A.S. Sznitman, On new examples of ballistic random walks in random environment. Ann. Probab. 31, 285–322 (2003)

6.A.S. Sznitman, Topics in random walk in random environment. Notes of Course at School and Conference on Probability Theory, May 2002. ICTP Lecture Series, Trieste, pp. 203–266 (2004)

7.A.S. Sznitman and O. Zeitouni, An invariance principle for isotropic diffusions in random environment. Invent. Math. 164, 455–567 (2006)

8.O. Zeitouni, Random Walks in Random Environment. Lecture Notes in Mathematics, vol. 1837, pp. 190–312. Springer, Berlin (2004)