Related papers: Random walks on products of trees and certain t.d.…

For any countable group with infinite conjugacy classes we construct a family of forests on the group. For each of them there is a random walk on the group with the property that its sample paths almost surely converge to the geometric…

Let $\mu$ be a probability measure on $\text{Out}(F_N)$ with finite first logarithmic moment with respect to the word metric, finite entropy, and whose support generates a nonelementary subgroup of $\text{Out}(F_N)$. We show that almost…

We introduce the discrete affine group of a regular tree as a finitely generated subgroup of the affine group. We describe the Poisson boundary of random walks on it as a space of configurations. We compute isoperimetric profile and Hilbert…

The group of affine transformations with rational coefficients acts naturally on the real line, but also on the $p$-adic fields. The aim of this note is to show that, for random walks whose laws have a finite first moment, all these actions…

Let T be the homogeneous tree with degree and G a finitely generated group whose Cayley graph is T. The associated lamplighter group is the wreath product of the cyclic group of order r with G. For a large class of random walks on this…

We give sufficient conditions for the non-triviality of the Poisson boundary of random walks on $H(\mathbb{Z})$ and its subgroups. The group $H(\mathbb{Z})$ is the group of piecewise projective homeomorphisms over the integers defined by…

We consider a finite-dimensional, locally finite CAT(0) cube complex X admitting a co-compact properly discontinuous countable group of automorphisms G. We construct a natural compact metric space B(X) on which G acts by homeomorphisms, the…

We give a geometric description of the Poisson boundaries of certain extensions of free and hyperbolic groups. In particular, we get a full description of the Poisson boundaries of free-by-cyclic groups. We rely upon the description of…

The Poisson boundary of a group G with a probability measure \mu is the space of ergodic components of the time shift in the path space of the associated random walk. Via a generalization of the classical Poisson formula it gives an…

We study the Poisson-Furstenberg boundary of random walks on permutational wreath products. We give a sufficient condition for a group to admit a symmetric measure of finite first moment with non-trivial boundary, and show that this…

The usual random walk on a group (homogeneous both in time and in space) is determined by a probability measure on the group. In a random walk with random transition probabilities this single measure is replaced with a stationary sequence…

We prove that random walks on Thompson's group $F$ driven by strictly non-degenerate finitely supported probability measures $\mu$ have a non-trivial Poisson boundary. The proof consists in an explicit construction of two different…

The main goal of this paper is to determine the Poisson boundary of lamplighter random walks over a general class of discrete groups $\Gamma$ endowed with a rich boundary. The starting point is the Strip Criterion of identification of the…

We consider a transitive action of a finitely generated group $G$ and the Schreier graph $\Gamma$ defined by this action for some fixed generating set. For a probability measure $\mu$ on $G$ with a finite first moment we show that if the…

We show that the Poisson boundary of random walks of finite entropy on Zariski-dense discrete subgroups of semisimple Lie groups equals the Furstenberg boundary of the corresponding symmetric spaces equipped with the hitting measure,…

The main aim of the present set of notes is to give new, short and essentially self-contained proofs of some classical, as well as more recent, results about random walks on groups. For instance, we shall see that the drift characterization…

By developing the entropy theory of random walks on equivalence relations and analyzing the asymptotic geometry of horospheric products we describe the Poisson boundary for random walks on random horospheric products of trees.

The group of affine transformations with rational coefficients, $aff(Q)$, acts naturally on the real line, but also on the $p$-adic fields. The aim of this note is to show that all these actions are necessary and sufficient to represent…

We extend some properties of random walks on hyperbolic groups to random walks on convergence groups. In particular we prove that if a convergence group $G$ acts on a compact metrizable space $M$ with the convergence property then we can…

We consider random walks on finitely or countably generated free semigroups, and identify their Poisson boundaries for classes of measures which fail to meet the classical entropy criteria. In particular, we introduce the notion of…