Related papers: A spectral gap property for random walks under uni…

The cutoff phenomenon was recently confirmed for random walks on Ramanujan graphs by the first author and Peres. In this work, we obtain analogs in higher dimensions, for random walk operators on any Ramanujan complex associated with a…

Three separation properties for a closed subgroup $H$ of a locally compact group $G$ are studied: (1) the existence of a bounded approximate indicator for $H$, (2) the existence of a completely bounded invariant projection of…

Let $G$ be a simply connected exponential solvable Lie group, $H$ a closed connected subgroup, and let $\tau$ be a representation of $G$ induced from a unitary character $\chi_f$ of $H$. The spectrum of $\tau$ corresponds via the orbit…

Given a simple Lie group $G$, we show that the lattices in $G$ are weakly uniformly discrete. This is a strengthening of the Kazhdan-Margulis theorem. Our proof however is straightforward --- considering general IRS rather than lattices…

We consider (random) walks in a multidimensional orthant. Using the idea of universality in probability theory, one can associate a unique polyhedral domain to any given walk model. We use this connection to prove two sets of new results.…

Let $G$ be a countable group and $\mu$ a probability measure on $G$. We build a new framework to compute asymptotic quantities associated with the $\mu$-random walk on $G$, using methods from harmonic analysis on groups and Banach space…

We give a new and elementary computation of the spectral gap of the Kac walk on the N-sphere. The result is obtained as a by-product of a more general observation which allows to reduce the analysis of the spectral gap of an N-component…

We consider the following situation: G is a finite directed graph, where to each vertex of G is assigned an element of a finite group Gamma. We consider all walks of length N on G, starting from v_i and ending at v_j To each such walk $w$…

We investigate the spectral fluctuation properties of constrained ensembles of random matrices (defined by the condition that a number N(Q) of matrix elements vanish identically; that condition is imposed in unitarily invariant form) in the…

Let $F$ be a non-archimedean local field of odd residue characteristic $p$. Let $G$ be the unramified unitary group $U(2, 1)(E/F)$, and $K$ be a maximal compact open subgroup of $G$. For an $\overline{\mathbf{F}}_p$-smooth representation…

We give a partial solution to a long-standing open problem in the theory of quantum groups, namely we prove that all finite-dimensional representations of a wide class of locally compact quantum groups factor through matrix quantum groups…

The Kac model describes the local evolution of a gas of $N$ particles with three dimensional velocities by a random walk in which the steps correspond to binary collisions that conserve momentum as well as energy. The state space of this…

We extend Friedman's theorem to show that, for any fixed $r>1$, a random $2r$--regular Schreier graph associated with the action of $r$ uniformly random permutations of $[n]$ on $k_{n}$--tuples of distinct elements in $[n]$ has a…

Notions of higher Kazhdan property can be defined in terms of vanishing of unitary group cohomology in higher degrees. Garland's theorem for simple groups over non-archimedean fields provides the first examples of a higher Kazhdan property.…

We show that the spectral gap of a random walk on the domain of normal attraction of an $\alpha$-stable law is of order $\mathcal O(n^{\alpha})$ when restricted to boxes of size $n$. The proof is based on a comparison principle that may be…

The spectral properties of a set of local gauge-invariant composite operators are investigated in the $U(1)$ Higgs model quantized in the 't Hooft $R_{\xi}$ gauge. These operators enable us to give a gauge-invariant description of the…

We provide a simplified proof of the existence, under some assumptions, of a spectral gap for the Perron-Frobenius operator of piecewise uniformly expanding maps on Riemannian manifolds when acting on some Sobolev spaces. Its consequences…

Let $G$ and $T$ be topological groups, $\alpha : T \to \Aut(G)$ a homomorphism defining a continuous action of $T$ on $G$ and $G^\sharp := G \rtimes_\alpha T$ the corresponding semidirect product group. In this paper we address several…

For any locally compact group $G$, we show the existence and uniqueness up to quasi-equivalence of a unitary $C_0$-representation $\pi_0$ of $G$ such that all coefficient functions of $C_0$-representations of $G$ are coefficient functions…

Let $G$ be a simply connected, connected completely solvable Lie group with Lie algebra $\mathfrak{g}=\mathfrak{p}+\mathfrak{m}.$ Next, let $\pi$ be an infinite-dimensional unitary irreducible representation of $G$ obtained by inducing a…