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

TL;DR

This paper establishes new limit theorems for random walks on SL_2(C) with low moment conditions, including local limit theorems and Berry-Esseen bounds, by analyzing spectral properties of associated operators.

Contribution

It introduces novel limit theorems for random walks on SL_2(C) under minimal moment assumptions, improving existing results and employing new spectral analysis techniques.

Findings

Proves a Local Limit Theorem for the norm cocycle with finite second moment.

Establishes a Local Limit Theorem for matrix coefficients with finite third moment.

Derives optimal Berry-Esseen bounds with rate O(1/√n).

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.