Related papers: Random walks on dense subgroups of locally compact…

Given a finitely generated group, the well-known Stability Problem asks whether the non-triviality of the Poisson-Furstenberg boundary (which is equivalent to the existence of non-constant bounded harmonic functions) depends on the choice…

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…

We present a generalization of Hirschman's entropic uncertainty principle for locally compact abelian groups to unimodular locally compact quantum groups. As a corollary, we strengthen a well-known uncertainty principle for compact groups,…

We begin by giving a new proof of the equivalence between the Liouville property and vanishing of the drift for symmetric random walks with finite first moments on finitely generated groups; a result which was first established by…

We extend results of Y. Benoist and J.-F. Quint concerning random walks on homogeneous spaces of simple Lie groups to the case where the measure defining the random walk generates a semigroup which is not necessarily Zariski dense, but…

We prove that the Poisson boundary of a random walk with finite entropy on a non-elementary hyperbolic group can be identified with its hyperbolic boundary, without assuming any moment condition on the measure. We also extend our method to…

We clarify the relation between noncommutative Poisson boundaries and Furstenberg-Hamana boundaries of quantum groups. Specifically, given a compact quantum group $G$, we show that in many cases where the Poisson boundary of the dual…

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…

Let $(G,\mu)$ be a discrete group with a generating probability measure. Nevo shows that if $G$ has property (T) then there exists an $\epsilon>0$ such that the Furstenberg entropy of any $(G,\mu)$-stationary ergodic space is either zero or…

We obtain a description of Poisson--Furstenberg boundaries for (random walks on) fundamental groups of compact graph-manifolds. Together with previously known results due to V.A. Kaimanovich and others, this allows one to obtain…

We construct random walks on simple Lie groups that quickly converge to the Haar measure for all moments up to order $t$. Specifically, a step of the walk on the unitary or orthognoal group of dimension $2^{\mathsf n}$ is a random Pauli…

We introduce and study a generalization of the notion of the Furstenberg boundary of a discrete group $\Gamma$ to the setting of a general unitary representation $\pi: \Gamma \to B(\mathcal H_\pi)$. This space, which we call the…

We study the decay of convolution powers of probability measures without second moment but satisfying some weaker finite moment condition. For any locally compact unimodular group G and any positive function $\rho:G \rightarrow…

In this paper we introduce Patterson--Sullivan systems, which consist of a group action on a compact metrizable space and a quasi-invariant measure which behaves like a classical Patterson--Sullivan measure. For such systems we prove a…

We investigate analogues of some of the classical results in homogeneous dynamics in non-linear setting. Let $G$ be a closed subgroup of the group of automorphisms of a biregular tree and $\Gamma<G$ a discrete subgroup. For a large class of…

Let $G$ be a connected simple real Lie group, $\Lambda_{0}\subseteq G$ a lattice and $\Lambda \unlhd \Lambda_{0}$ a normal subgroup such that $\Lambda_{0}/\Lambda\simeq \mathbb{Z}^d$. We study the drift of a random walk on the…

Let $G$ be a real Lie group, $\Lambda<G$ a lattice and $H<G$ a connected semisimple subgroup without compact factors and with finite center. We define the notion of $H$-expanding measures $\mu$ on $H$ and, applying recent work of…

We study fine structure related to finitely supported random walks on infinite finitely generated discrete groups, largely motivated by dimension group techniques. The unfaithful extreme harmonic functions (defined only on proper space-time…

Any Zariski dense countable subgroup of $SL(d,R)$ is shown to carry a non-degenerate finitely supported symmetric random walk such that its harmonic measure on the flag space is singular. The main ingredients of the proof are: (1) a new…

An invariant random subgroup of the countable group {\Gamma} is a random subgroup of {\Gamma} whose distribution is invariant under conjugation by all elements of {\Gamma}. We prove that for a nonamenable invariant random subgroup H, the…