Acceleration of Lamplighter Random Walks

TL;DR

This paper investigates the speed at which lamplighter random walks on wreath products escape to infinity, showing they do so faster than their projection on the base group under certain metrics.

Contribution

It demonstrates that lamplighter random walks on wreath products escape faster than their projections, especially when using the shortest traveling salesman tour metric.

Findings

Lamplighter random walks escape faster than their projections.

The rate of escape exceeds that of the base group's projection.

Results exclude some degenerate cases when the base group is orz.

Abstract

Suppose we are given an infinite, finitely generated group $G$ and a transient random walk on the wreath product $(\mathbb{Z}/ 2\mathbb{Z})\wr G$, such that its projection on $G$ is transient and has finite first moment. This random walk can be interpreted as a lamplighter random walk on $G$. Our aim is to show that the random walk on the wreath product escapes to infinity with respect to a suitable (pseudo-)metric faster than its projection onto $G$. We also address the case where the pseudo-metric is the length of a shortest ``travelling salesman tour''. In this context, and excluding some degenerate cases if $G=\mathbb{Z}$, the linear rate of escape is strictly bigger than the rate of escape of the lamplighter random walk's projection on $G$.