Multi-Dimensional Random Walks in Random Environment

Abstract

We now return to model a) of the Introduction. Until recently relatively few works have discussed the behavior of multi-dimensional random walks in random environment: Kalikow [29] establishes certain 0 —1 laws and a sufficient criterion for transience, Lawler [40] shows a central limit theorem in the special situation where d(x, ω) ≡ 0, (see Lecture 2), and Bricmont-Kupiainen [9] derives a central limit theorem in the isotropic case, when d≥3, for small perturbations of the simple random walk on ${{\mathbb{Z}}^{d}}$. More recently, some new results have been obtained, cf. Zerner [73], Sznitman-Zerner [68], Sznitman [65], [66], [67]. We shall describe some of them in this and the next lecture. As we shall see, especially in the next lecture, another example of “preponderant role of atypical pockets of low local eigenvalues” emerges in this context.