Acceleration of lamplighter random walks

Suppose we are given an infinite, finitely generated group G and a transient random walk with bounded range on the wreath product (Z/2Z) ≀ G, such that its projection on G is transient. This random walk can be interpreted as a lamplighter random walk, where there is a lamp at each element of G, which can be switched on and off, and a lamplighter walks along G and switches lamps randomly on and off. Our aim is to show that the lamplighter random walk escapes with respect to a suitable (pseudo-)metric on the wreath product faster to infinity than its projection onto G. For this purpose, we show that the asymptotic linear rate of burning lamps is non-zero, providing an acceleration of the lamplighter. If lamp switches are not charged by the pseudo-metric and if G 6= Z, we prove that the rate of escape with respect to the pseudo-metric, which becomes the length of a shortest “travelling salesman tour”, is strictly bigger than the rate of escape of the lamplighter random walk’s projection on G. We prove the same for nondegenerate cases if G = Z. Furthermore, we prove for G having infinitely many ends the acceleration with respect to a Markovian distance, which arises from probabilities on (Z/2Z) ≀ G and the metric on G.