We derive Donsker-Vardhan type results for functionals of the occupation times when the underlying random walk on Z is in the domain of attraction of an operatorstable law on R. Applications to random walks on wreath products (also known as lamplighter groups) are given.

We answer positively a question of Kaimanovich and Vershik from 1979, showing that the final configuration of lamps for simple random walk on the lamplighter group over Z (d ≥ 3) is the Poisson boundary. For d ≥ 5, this had been shown earlier by Erschler (2011). We extend this to walks of more general types on more general groups.

We determine the precise asymptotic behaviour (in space) of the Green kernel of simple random walk with drift on the Diestel–Leader graph DL(q, r), where q, r 2. The latter is the horocyclic product of two homogeneous trees with respective degrees q + 1 and r + 1. When q = r , it is the Cayley graph of the wreath product (lamplighter group) Zq Z with respect to a natural set of generators. We d...

Suppose that G is a finite, connected graph and X is a lazy random walk on G. The lamplighter chain X associated with X is the random walk on the wreath product G = Z2 oG, the graph whose vertices consist of pairs (f, x) where f is a labeling of the vertices of G by elements of Z2 and x is a vertex in G. There is an edge between (f, x) and (g, y) in G if and only if x is adjacent to y in G and ...

Given a finite graph G, a vertex of the lamplighter graph G♦ = Z2 o G consists of a zero-one labeling of the vertices of G, and a marked vertex of G. For transitive G we show that, up to constants, the relaxation time for simple random walk in G♦ is the maximal hitting time for simple random walk in G, while the mixing time in total variation on G♦ is the expected cover time on G. The mixing ti...

We show that the measure on markings of Zn, d ≥ 3, with elements of {0,1} given by i.i.d. fair coin flips on the range R of a random walk X run until time T and 0 otherwise becomes indistinguishable from the uniform measure on such markings at the threshold T = 1 2 Tcov(Z d n). As a consequence of our methods, we show that the total variation mixing time of the random walk on the lamplighter gr...

Recently, several papers have been devoted to the analysis of lamplighter random walks, in particular when the underlying graph is the infinite path Z. In the present paper, we develop a spectral analysis for lamplighter random walks on finite graphs. In the general case, we use the C2-symmetry to reduce the spectral computations to a series of eigenvalue problems on the underlying graph. In th...

Let L X be a lamplighter graph, i.e., the graph-analogue of a wreath product of groups, and let P be the transition operator (matrix) of a random walk on that structure. We explain how methods developed by Saloff-Coste and the author can be applied for determining the p-norms and spectral radii of P , if one has an amenable (not necessarily discrete or unimodular) locally compact group of isome...

The main goal of this paper is to determine the Poisson boundary of lamplighter random walks over a general class of discrete groups Γ endowed with a “rich” boundary. The starting point is the Strip Criterion of identification of the Poisson boundary for random walks on discrete groups due to Kaimanovich [16]. A geometrical method for constructing the strip as a subset of the lamplighter group ...