Limit theorems for random walk in random environment

University Of California-Los Angeles, Los Angeles CA

Investigators

Abstract

In this proposal we deal with the limit behavior of RWRE (random walks in random environments). Specifically, we look at random walks on the d-dimensional Euclidean lattice, whose transition probabilities are a stationary random field indexed by the vertices of the lattice. Most emphasis will be put on i.i.d. environments. In general the behavior of RWRE can be very different from that of regular random walk. Both in one-dimension and in non-elliptic multidimensional environments, one may witness phenomena that do not exist in regular random walk, such as sub-ballisticity (i.e. when the walk is transient in a certain direction, but move at asymptotic speed zero) and sub-diffusivity (i.e. when the typical distance from the origin grows at a scale that is significantly less than the square-root of the time). In many other cases the RWRE is conjectures to behave similarly to the regular random walk. However, proofs are known only for few cases, most notably for reversible environments and for environments that are small perturbations of the simple random walk. Despite the rapid progress in recent years, many basic questions regarding RWRE are still open, most notably the question of 0-1 law for directional transience and the question of ballisticity for directionally transient random walk in random high-dimensional elliptic environments. Random walks are one of the most studied objects in probability theory. Its importance stems, in part, from the fact that it is a natural model for many real world phenomena such as diffusion, migration and even market fluctuations, and as a natural tool in the study of other objects such as laplace equations, heat conductance and more. While the theory of simple random walks is well developed, most of its techniques rely heavily on the perfect regularity of the environments in which it takes place. Since in many real-world models such regularity does not hold, RWRE is, in fact, a better representation of real-world phenomena than the simple random walk. Thus our aim is to develop new techniques that will facilitate study of RWRE.