Affiliation:
1. Department of Mathematics University of Chicago Chicago Illinois USA
2. Institut für Mathematik Universität Zürich, Zurich Winterthurerstrasse Switzerland
Abstract
AbstractGiven a Gromov‐hyperbolic group endowed with a finite symmetric generating set, we study the statistics of counting measures on the spheres of the associated Cayley graph under linear representations of . More generally, we obtain a weak law of large numbers for subadditive functions, echoing the classical Fekete lemma. For strongly irreducible and proximal representations, we prove a counting central limit theorem with a Berry–Esseen type error rate and exponential large deviation estimates. Moreover, in the same setting, we show convergence of interpolated normalized matrix norms along geodesic rays to Brownian motion and a functional law of iterated logarithm, paralleling the analogous results in the theory of random matrix products. Our counting large deviation estimates address a question of Kaimanovich–Kapovich–Schupp. In most cases, our counting limit theorems will be obtained from stronger almost sure limit laws for Patterson–Sullivan measures on the boundary of the group.
Funder
Statens Naturvidenskabelige Forskningsrad
Reference68 articles.
1. R.Aoun P.Mathieu andC.Sert Random walks on hyperbolic spaces: second order expansion of the rate function at the drift to appear in JEP 2021.
2. Law of large numbers for the spectral radius of random matrix products
3. R.AounandC.Sert Random walks on hyperbolic spaces: concentration inequalities and probabilistic Tits alternative arXiv:2101.08222 v3 2021.
4. H.Baik I.Choi andD. M.Kim Linear growth of translation lengths of random isometries on Gromov hyperbolic spaces and Teichmüller spaces arXiv:2103.13616 2021.
5. Stationary measures and invariant subsets of homogeneous spaces (III)