Speed of Random Walks in Dirichlet Environment on a Galton-Watson Tree

TL;DR

This paper analyzes the speed of a transient random walk in Dirichlet environment on a Galton-Watson tree, providing formulas for the stationary distribution and conditions for positive speed.

Contribution

It introduces a formula for the environment's stationary distribution and characterizes when the walk has positive speed based on offspring distribution moments.

Findings

Derived the stationary distribution of the environment from the particle perspective.

Provided a necessary and sufficient condition for positive speed.

Established moment conditions on the offspring distribution for positive speed.

Abstract

This paper deals with a transient random walk in Dirichlet environment, or equivalently a linearly edge reinforced random walk, on a Galton-Watson tree. We compute the stationary distribution of the environment seen from the particle of an edge reinforced random walk. We obtain a formula for the speed and give a necessary and sufficient condition for the walk to have a positive speed under some moment conditions on the offspring distribution of the tree.