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

TL;DR

This paper establishes a functional central limit theorem for transient random walks in i.i.d. environments on regular trees, using regenerative levels and applying results to reinforced random walks, without assuming uniform ellipticity.

Contribution

It proves a new FCLT for RWRE on regular trees under minimal conditions and extends results to linearly edge-reinforced random walks, improving prior work.

Findings

FCLT holds for transient RWRE on regular trees under moment conditions.

Distance between regenerative levels has a geometrically decaying tail.

Results apply to linearly edge-reinforced random walks on trees with b ≥ 4.

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 \ge 4$, substantially improving the results of the first author (see Theorem 3 of Collevecchio (2006)).