Functional central limit theorem for random walks in random environment defined on regular trees

Abstract

We study Random Walks in an i.i.d. Random Environment (RWRE) defined on b-regular trees. We prove a functional central limit theorem (FCLT) for transient processes, under a moment condition on the environment. We emphasize that we make no uniform ellipticity assumptions. Our approach relies on regenerative levels, i.e. levels that are visited exactly once. On the way, we prove that the distance between consecutive regenerative levels have a geometrically decaying tail. In the second part of this paper, we apply our results to Linearly Edge-Reinforced Random Walk (LERRW) to prove FCLT when the process is defined on b-regular trees, with b ≥ 4, substantially improving the results of the first author (see Theorem 3 of [4]). introductionRandom Walk in Random Environment (RWRE) is a class of self-interacting processes that attracted much attention from probabilists since the seminal work of Kesten, Kozlov, and Spitzer [10] and Solomon [15], in the 70's. It seems that the initial motivation behind this class of process was related to problems in biology, crystallography and metal physics. The interest in this field grew substantially, and we refer to [3] and [16] for an overview of this beautiful subject.We study random walks in an i.i.d. random environment defined on b-regular trees. We provide a functional central limit theorem (FCLT) for processes that are transient, assuming a moment condition on the environment. Our approach relies on regenerative levels, i.e. levels that are visited exactly once. On the way, we prove that the space between these regenerative levels have a geometrically decaying tail. We emphasize that we make no uniform ellipticity assumptions.