Random walks on SL_2(C): spectral gap and limit theorems

Abstract

We obtain various new limit theorems for random walks on SL_2(C) under low moment conditions. For non-elementary measures with a finite second moment, we prove a Local Limit Theorem for the norm cocycle, yielding the optimal version of a theorem of E. Le Page. For measures with a finite third moment, we obtain the Local Limit Theorem for the matrix coefficients, improving a recent result of Grama-Quint-Xiao and the authors, and Berry-Esseen bounds with optimal rate $O(1/\sqrt{n})$ for the norm cocycle and the matrix coefficients. The main tool is a detailed study of the spectral properties of the Markov operator and its purely imaginary perturbations acting on different function spaces. We introduce, in particular, a new function space derived from the Sobolev space $W^{1,2}$ that provides uniform estimates.

64 pages. Revised version. To appear in Probab. Theory Relat. Fields

References in corpus (3)

• A zero-one law for invariant measures and a local limit theorem for coefficients of random walks on the general linear group
• Equilibrium states of endomorphisms of ${\mathbb{P}}^k$ I: existence and properties
• Moser-Trudinger inequalities and complex Monge-Ampere equation