Rate of Escape on the Lamplighter Tree

Suppose we are given a homogeneous tree Tq of degree q ≥ 3, where at each vertex sits a lamp, which can be switched on or off. This structure can be described by the wreath product (Z/2) ≀ Γ, where Γ = ∗ i=1 Z/2 is the free product group of q factors Z/2. We consider a transient random walk on a Cayley graph of (Z/2) ≀ Γ, 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.