Rate of escape of random walks on free products

Abstract

Suppose we are given the free product V of a finite family of finite or countable sets (V/), e jj and probability measures on each V h which govern random walks on it. We consider a transient random walk on the free product arising naturally from the random walks on the Vj. We prove the existence of the rate of escape with respect to the block length, that is, the speed at which the random walk escapes to infinity, and furthermore we compute formulae for it. For this purpose, we present three different techniques providing three different, equivalent formulae.2000 Mathematics subject classification: primary 60G50; secondary 20E06, 60B15.

Asymptotic Entropy of Random Walks on Free Products

“…We write also ξ i := ξ i (1), ξ min := min i∈I ξ i and ξ max := max i∈I ξ i . Observe that ξ i < 1; see [11,Lemma 2.3]. We have F (x i , y i |z) = F i x i , y i |ξ i (z) for all x i , y i ∈ V i ; see Woess [29,Prop.…”

Renewal theory for random walks on surface groups

“…Moreover, analyticity of the rate of escape also follows in certain cases where explicit formulae for the rate of escape are known, see Mairesse and Mathéus [27] and Gilch, [14] and [15]. Mairesse and Mathéus [28] show that the rate of escape for some random walks on the Braid group B 3 = a, b|aba = bab is continuous but not(!)…”

Rate of escape on the lamplighter tree

“…For this purpose, we apply a technique going back to Furstenberg [6], which was used by Ledrappier [11, Section 4 b] for free groups, and also by the author [7] for free products of groups. By Lebesgue's Dominated Convergence Theorem we have…”