Rate of escape on the lamplighter tree

Abstract

Abstract. 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 = * q 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.

The Poisson boundary of lamplighter random walks on trees

“…It is clear that ℓ(P ) ≥ m(P ). Recent [15] and ongoing work by Gilch suggests that ℓ(P ) > m(P ) strictly.…”

“…Wreath products exhibit interesting types of asymptotic behaviour of n-step return probabilities, see Saloff-Coste and Pittet [26], [27], Revelle [29]. The rate of escape has been studied by Lyons, Pemantle and Peres [24], Erschler [11], [12], Revelle [28] and, for simple lamplighter walks on trees, by Gilch [15]. For the spectrum of transition operators, see Grigorchuk and Żuk [16], Dicks and Schick [10] and Bartholdi and Woess [1].…”

Acceleration of Lamplighter Random Walks

“…However, this example is more or less the only counterexample. The author of this article has investigated the rate of escape of lamplighter random walks arising from a simple random walk on homogeneous trees providing tight lower and upper bounds for the rate of escape; see Gilch [9]. In particular, the lamplighter random walk is significantly faster than its projection onto the tree.…”