Acceleration of Lamplighter Random Walks

Suppose we are given an infinite, finitely generated group G and a
transient random walk on the wreath product (Z/2Z) o G, such that its projec-
tion 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 suit-
able (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.