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Preprints, Working Papers, ... Year : 2023

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.

Dates and versions

hal-03991897 , version 1 (16-02-2023)

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Cite

Tien-Cuong Dinh, Lucas Kaufmann, Hao Wu. Random walks on SL_2(C): spectral gap and limit theorems. 2023. ⟨hal-03991897⟩
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