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 ...

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...

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...

Let Tq be the homogeneous tree with degree q + 1 ≥ 3 and G a finitely generated group whose Cayley graph is Tq. The associated lamplighter group is the wreath product Zr ≀ G, where Zr is the cyclic group of order r. For a large class of random walks on this group, we prove almost sure convergence to a natural geometric boundary. If the probability law governing the random walk has finite first ...

We introduce the discrete affine group of a regular tree as finitely generated subgroup group. describe Poisson boundary random walks on it space configurations. compute isoperimetric profile and Hilbert compression exponent also discuss metric relationship with some lamplighter groups graphs.

Abstract We apply a construction developed in previous paper by the authors order to obtain formula which enables us compute $$\ell ^2$$ ℓ 2 -Betti numbers coming from family of group algebras representable as crossed product algebras. As an application, we whole irrati...

We show that the cyclic lamplighter group C2 oCn embeds into Hilbert space with distortion O ( √ log n ) . This matches the lower bound proved by Lee, Naor and Peres in [14], answering a question posed in that paper. Thus the Euclidean distortion of C2 o Cn is Θ ( √ log n ) . Our embedding is constructed explicitly in terms of the irreducible representations of the group. Since the optimal Eucl...