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

• 成果类型: Article
• 署名作者: Dinh, Tien-Cuong; Kaufmann, Lucas; Wu, Hao
• 署名单位: National University of Singapore; Institute for Basic Science - Korea (IBS); Centre National de la Recherche Scientifique (CNRS); Universite de Orleans
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-023-01191-y
• 发表日期: 2023
• 页码: 877-955
• 关键词: coefficients; PRODUCTS

摘要

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 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/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.