Random walks on SL2(C): spectral gap and limit theorems

Abstract

We obtain various new limit theorems for random walks on SL2(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 & Eacute;. 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/root 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,W-2 that provides uniform estimates.