[Paper Review] Random walks on SL_2(C): spectral gap and local limit theorems

Tien‐Cuong Dinh, Lucas Kaufmann+1 more|arXiv (Cornell University)|Jun 8, 2021

Random Matrices and Applications16 references4 citations

TL;DR

This paper establishes new local limit theorems for random walks on SL_2(C) under low moment conditions, using spectral analysis of the Markov operator and its imaginary perturbations. It proves optimal local limit theorems for the norm cocycle under second moment conditions and for matrix coefficients under third moment conditions, introducing a novel function space derived from W^{1,2} for uniform estimates.

ABSTRACT

We obtain 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. We also obtain a Local Limit Theorem for the matrix coefficients under a third moment condition, improving a recent result of Grama-Quint-Xiao. 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.

Motivation & Objective

• To extend local limit theorems for random walks on SL_2(C) to weaker moment conditions than previously known.
• To establish optimal results for the norm cocycle under finite second moment, improving on E. Le Page's theorem.
• To derive a local limit theorem for matrix coefficients under a third moment condition, refining recent work by Grama-Quint-Xiao.
• To develop a new function space based on W^{1,2} that enables uniform estimates in spectral analysis.
• To analyze the spectral properties of the Markov operator and its purely imaginary perturbations across different function spaces.

Proposed method

• Employ spectral theory of the Markov operator and its imaginary perturbations on L^2 and other function spaces.
• Introduce a new function space derived from the Sobolev space W^{1,2} to achieve uniform estimates in spectral analysis.
• Use the spectral gap property to control the decay of correlations and derive limit theorems.
• Apply functional analytic techniques to handle the non-compact nature of SL_2(C) and the non-trivial spectral structure.
• Leverage the structure of the group SL_2(C) and its action on the Riemann sphere to analyze the norm cocycle and matrix coefficients.
• Establish estimates on the resolvent and spectral projections using the new function space to control the operator norm.

Experimental results

Research questions

• RQ1 What is the optimal local limit theorem for the norm cocycle of random walks on SL_2(C) under a finite second moment condition?
• RQ2 Can a local limit theorem for matrix coefficients be established under a third moment condition, and how does it compare to prior results?
• RQ3 How can spectral methods be adapted to non-compact groups like SL_2(C) to derive limit theorems under low moment assumptions?
• RQ4 What function space structure enables uniform estimates in the spectral analysis of Markov operators on SL_2(C)?
• RQ5 What is the role of purely imaginary perturbations of the Markov operator in deriving local limit theorems?

Key findings

• A local limit theorem for the norm cocycle is proven under a finite second moment condition, achieving optimality and improving on E. Le Page's result.
• A local limit theorem for matrix coefficients is established under a third moment condition, refining a recent result by Grama-Quint-Xiao.
• The spectral gap of the Markov operator is used to control the decay of correlations, enabling the derivation of limit theorems.
• A new function space derived from W^{1,2} is introduced, providing uniform estimates essential for the spectral analysis.
• The spectral properties of the Markov operator and its imaginary perturbations are analyzed in detail across different function spaces.
• The method yields quantitative control over the convergence rate in the local limit theorems through spectral estimates.