Rate of Escape on the Lamplighter Tree

TL;DR

This paper investigates the rate at which a random walk on a lamplighter group over a homogeneous tree escapes to infinity, providing bounds on its speed using group-theoretic and probabilistic methods.

Contribution

It offers new bounds for the escape rate of random walks on lamplighter groups over trees, advancing understanding of their asymptotic behavior.

Findings

Derived lower and upper bounds for the escape rate

Analyzed the behavior of random walks on lamplighter groups over trees

Enhanced understanding of transience and speed in these groups

Abstract

Suppose we are given a homogeneous tree $\mathcal{T}_q$ of degree $q\geq 3$, where at each vertex sits a lamp, which can be switched on or off. This structure can be described by the wreath product $(\mathbb{Z}/2)\wr \Gamma$, where $\Gamma=\ast_{i=1}^q \mathbb{Z}/2$ is the free product group of $q$ factors $\mathbb{Z}/2$. We consider a transient random walk on a Cayley graph of $(\mathbb{Z}/2)\wr \Gamma$, for which we want to compute lower and upper bounds for the rate of escape, that is, the speed at which the random walk flees to infinity.