Acceleration of Lamplighter Random Walks

#### L.A. Gilch

2008, v.14, Issue 4, 465-486

ABSTRACT

Suppose we are given an infinite, finitely generated group $G$ and a transient random walk on the wreath product $(Z/2Z)\wr G$, 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 $G=Z$, the linear rate of escape is strictly bigger than the rate of escape of the lamplighter random walk's projection on $G$.

Keywords: random walks,lamplighter groups,rate of escape