Rate of Escape on the Lamplighter Tree

Abstract

Suppose we are given a homogeneous tree ${\mathcal{T}}_q$ of degree $q\ge 3$, 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.

16 pages; accepted for publication in Journal of Mathematical Sciences (N.Y.)