Related papers: Determining subgroups via stationary measures

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…

We consider a random walk on a homogeneous space $G/\Lambda$ where $G$ is a non-compact simple Lie group and $\Lambda$ is a lattice. The walk is driven by a probability measure $\mu$ on $G$ whose support generates a Zariski-dense subgroup.…

Gromov's theorem states that a finitely generated group has polynomial growth if and only if it is virtually nilpotent. A key ingredient in its proof is the small doubling property. In this work, we study entropy analogues of this property…

We study stationary actions of locally compact measured groups through the structure and regularity of their Radon-Nikodym cocycles. We start with two dynamical consequences of stationarity. Extending a theorem of Furstenberg-Glasner from…

We construct Patterson-Sullivan measure and a natural metric on the unit space of a hyperbolic groupoid. In particular, this gives a new approach to defining SRB measures on Smale spaces using Gromov hyperbolic graphs.

We prove many new cases of Zimmer's conjecture for actions by lattices in non-$\mathbb{R}$-split semisimple Lie groups $G$. By prior arguments, Zimmer's conjecture reduces to studying certain probability measures invariant under a minimal…

Cubical complexes are metric spaces constructed by gluing together unit cubes in an analogous way to the construction of simplicial complexes. We construct Brownian motion on such spaces, define random walks, and prove that the transition…

We consider the problems of measurable isomorphisms and joinings, measurable centralizers and quotients for certain classes of ergodic group actions on infinite measure spaces. Our main focus is on systems of algebraic origin: actions of…

We describe a new construction of a family of measures on a group with the same Poisson boundary. Our approach is based on applying Markov stopping times to an extension of the original random walk.

We study Linial-Meshulam random 2-complexes, which are two-dimensional analogues of Erd\H{o}s-R\'enyi random graphs. We find the threshold for simple connectivity to be p = n^{-1/2}. This is in contrast to the threshold for vanishing of the…

We study random walks on groups with the feature that, roughly speaking, successive positions of the walk tend to be "aligned". We formalize and quantify this property by means of the notion of deviation inequalities. We show that deviation…

Random walks on graphs are a fundamental concept in graph theory and play a crucial role in solving a wide range of theoretical and applied problems in discrete math, probability, theoretical computer science, network science, and machine…

The aim of this work is to study the convergence to equilibrium of an $(h,\rho)$-subelliptic random walk on a closed, connected Riemannian manifold $(M,g)$ associated with a subelliptic second-order differential operator $A$ on $M$. In such…

We show that the stationary measure for some random systems of two piecewise affine homeomorphisms of the interval is singular, verifying partially a conjecture by Alsed\`a and Misiurewicz and contributing to a question of Navas on the…

We study time-inhomogeneous random walks on finite groups in the case where each random walk step need not be supported on a generating set of the group. When the supports of the random walk steps satisfy a natural condition involving…

Finding a non-sofic hyperbolic group will resolve two major problems in geometric group theory: Are there non sofic groups? Are there non residually finite hyperbolic groups? In this paper, we propose a new probabilistic approach to this…

A random walk $w_n$ on a separable, geodesic hyperbolic metric space $X$ converges to the boundary $\partial X$ with probability one when the step distribution supports two independent loxodromics. In particular, the random walk makes…

In this paper we study convergence of random walks, on finite quantum groups, arising from linear combination of irreducible characters. We bound the distance to the Haar state and determine the asymptotic behavior, i.e. the limit state if…

We consider stationary measures of the one-dimensional discrete-time quantum walks (QWs) with two chiralities, which is defined by a 2 times 2 unitary matrix U. In our previous paper [15], we proved that any uniform measure becomes the…

In this paper, we extend a result of Kesten and Spitzer (1979). Let us consider a stationary sequence $(\xi\_k:=f(T^k(.)))\_k$ given by an invertible probability dynamical system and some centered function $f$. Let $(S\_n)\_n$ be a simple…