Transient Random Walks in Random Environment on a Galton-Watson Tree

TL;DR

This paper analyzes the behavior of transient random walks in random environments on Galton-Watson trees, providing criteria for positive speed and exploring cases with zero speed, along with results on reinforced random walks.

Contribution

It offers a sharp criterion for positive asymptotic speed and characterizes zero-speed scenarios, also proving positive speed for reinforced walks on regular trees.

Findings

Criteria for positive asymptotic speed

Existence of zero-speed cases with sublinear growth

Positive speed for reinforced random walks on regular trees

Abstract

We consider a transient random walk $(X_n)$ in random environment on a Galton--Watson tree. Under fairly general assumptions, we give a sharp and explicit criterion for the asymptotic speed to be positive. As a consequence, situations with zero speed are revealed to occur. In such cases, we prove that $X_n$ is of order of magnitude $n^{\Lambda}$, with $\Lambda \in (0,1)$. We also show that the linearly edge reinforced random walk on a regular tree always has a positive asymptotic speed, which improves a recent result of Collevecchio \cite{Col06}.