相关论文: Non-Realizability of the Poisson Boundary

In this paper we introduce for a group $G$ the notion of ultralimit of measure class preserving actions of it, and show that its Furstenberg-Poisson boundaries can be obtained as an ultralimit of actions on itself, when equipped with…

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…

Let $G$ be a finite abelian group, let $0 < \alpha < 1$, and let $A \subseteq G$ be a random set of size $|G|^\alpha$. We let $$ \mu(A) = \max_{B,C:|B|=|C|=|A|}|\{(a,b,c) \in A \times B \times C : a = b + c \}|. $$ The issue is to determine…

Let $\Gamma$ be a Gromov hyperbolic group, endowed with an arbitrary left-invariant hyperbolic metric, quasi-isometric to a word metric. The action of $\Gamma$ on its boundary $\partial\Gamma$ endowed with the Patterson-Sullivan measure…

We find upper and lower bounds of the multiplicities of irreducible admissible representations $\pi$ of a semisimple Lie group $G$ occurring in the induced representations $Ind_H^G\tau$ from irreducible representations $\tau$ of a closed…

We identify the Poisson boundary of the dual of the universal compact quantum group A_u(F) with a measurable field of ITPFI factors.

In this paper we extend the construction of random walks with a prescribed Poisson boundary to the case of measures in the class of a generalized Gibbs state. The price for dropping the $\alpha$-quasiconformal assumptions is that we must…

Let $G$ be a compact Lie group. Suppose $g_1, \dots, g_k$ are chosen independently from the Haar measure on $G$. Let $\mathcal{A} = \cup_{i \in [k]} \mathcal{A}_i$, where, $\mathcal{A}_i := \{g_i\} \cup \{g_i^{-1}\}$. Let…

We characterize $C^*$-simplicity for countable groups by means of the following dichotomy. If a group is $C^*$-simple, then the action on the Poisson boundary is essentially free for a generic measure on the group. If a group is not…

The classical Poisson functor associates to every infinite measure preserving dynamical system $(X,\mu,T)$ a probability preserving dynamical system $(X^*,\mu^*,T_*)$ called the Poisson suspension of $T$. In this paper we generalize this…

We prove that finite entropy random walks on the torsion-free Baumslag group in dimension $d=2$ have non-trivial Poisson boundary. This is in contrast with the torsion case where the situation for simple random walks on Baumslag groups is…

Let G be a second countable, locally compact group and let f be a continuous Herz-Schur multiplier on G. Our main result gives the existence of a (not necessarily uniformly bounded) strongly continuous representation on a Hilbert space,…

We study the question, ``For which reals $x$ does there exist a measure $\mu$ such that $x$ is random relative to $\mu$?'' We show that for every nonrecursive $x$, there is a measure which makes $x$ random without concentrating on $x$. We…

We show under weak hypotheses that $\partial X$, the Roller boundary of a finite dimensional CAT(0) cube complex $X$ is the Furstenberg-Poisson boundary of a sufficiently nice random walk on an acting group $\Gamma$. In particular, we show…

Let $(X,\mathcal{A}, \mu)$ be a probability measure space and let $T_i,$ $1\leq i\leq H,$ be invertible bi measurable measure preserving transformations on this measure space. We give a sufficient condition for the product of $H$ bounded…

A sequence $x_1,\dots,x_n,\dots$ of discrete-valued observations is generated according to some unknown probabilistic law (measure) $\mu$. After observing each outcome, one is required to give conditional probabilities of the next…

We prove that the Poisson boundary of a simple random walk on the Schreier graph of action of $F$ on $\mathbb{D}$, where $\mathbb{D}$ is the set of dyadic numbers in $[0, 1]$, is non-trivial. This gives a new proof of the result of…

We study random walks on the lampshuffler group $\mathrm{FSym}(H)\rtimes H$, where $H$ is a finitely generated group and $\mathrm{FSym}(H)$ is the group of finitary permutations of $H$. We show that for any step distribution $\mu$ with a…

Given a probability measure on a finitely generated group, its Martin boundary is a natural way to compactify the group using the Green function of the corresponding random walk. For finitely supported measures in hyperbolic groups, it is…

We establish uniform bounds on the multiplicities of irreducible admissible representations appearing in spaces of functions on symmetric spaces over $p$-adic fields. These multiplicities can exceed one and depend intricately on the group,…