General random walk in a random environment defined on Galton-Watson trees

TL;DR

This paper studies the behavior of a random walk on Galton-Watson trees with randomly assigned edge weights, extending previous results to Markovian environments and environments that evolve over time.

Contribution

It introduces a new method for analyzing transience and recurrence of random walks on trees with complex, Markovian, and evolving environments.

Findings

Reproved Lyons and Pemantle's result for i.i.d. weights.

Extended analysis to Markovian environments.

Analyzed random walks in evolving environments.

Abstract

We consider the motion of a particle on a Galton Watson tree, when the probabilities of jumping from a vertex to any one of its neighbours is determined by a random process. Given the tree, positive weights are assigned to the edges in such a way that, viewed along any line of descent, they evolve as a random process. In order to introduce our method for proving transience or recurrence, we first suppose that the weights are i.i.d., reproving a result of Lyons and Pemantle. We then extend the argument to allow a Markovian environment, and finally to a random walk on a Markovian environment that changes the environment. Our approach involves studying the typical behaviour of processes on fixed lines of descent, which we then show determines the behaviour of the process on the whole tree.