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Uniform estimates for random matrix products and applications

Omar Hurtado, Sidhanth Raman

TL;DR

This work develops a framework of uniform large deviation estimates for random matrix products over local fields, including non-Archimedean settings, by introducing Wasserstein-type topologies that quantify perturbations of the underlying matrix laws. It proves continuity results for the top Lyapunov exponent $\lambda_1(\mu)$ and the variance in the Benoist–Quint central limit theorem, as well as a new lower semi-continuity result for the gap $\lambda_1(\mu)-\lambda_2(\mu)$, using continuity of exterior-power pushforwards. The results apply to both heavy-tailed distributions and compactly supported cases, and they yield uniform large deviation bounds that are robust under perturbations, with concrete applications to one-dimensional random Schrödinger operators (localization for heavy-tailed potentials) and to statistical data for random geodesics on hyperbolic surfaces, including continuity in Teichmüller space. Overall, the paper extends classical Archimedean results to non-Archimedean settings, provides new uniformity in heavy-tailed regimes, and links probabilistic stability to geometric and spectral applications. The methods offer a versatile toolkit for studying how Lyapunov-type quantities and invariant measures respond to perturbations in both algebraic and geometric contexts.

Abstract

For certain natural families of topologies, we study continuity and stability of statistical properties of random walks on linear groups over local fields. We extend large deviation results known in the Archimedean case to non-Archimedean local fields and also demonstrate certain large deviation estimates for heavy tailed distributions unknown even in the Archimedean case. A key technical result, which may be of independent interest, establishes lower semi-continuity for the gap between the first and second Lyapunov exponents. As applications, we are able to obtain a key technical step towards a localization proof for heavy tailed Anderson models (the full proof appearing in a companion article), and show continuity/stability (taking the geometric data as input) of various statistical data associated to hyperbolic surfaces.

Uniform estimates for random matrix products and applications

TL;DR

This work develops a framework of uniform large deviation estimates for random matrix products over local fields, including non-Archimedean settings, by introducing Wasserstein-type topologies that quantify perturbations of the underlying matrix laws. It proves continuity results for the top Lyapunov exponent and the variance in the Benoist–Quint central limit theorem, as well as a new lower semi-continuity result for the gap , using continuity of exterior-power pushforwards. The results apply to both heavy-tailed distributions and compactly supported cases, and they yield uniform large deviation bounds that are robust under perturbations, with concrete applications to one-dimensional random Schrödinger operators (localization for heavy-tailed potentials) and to statistical data for random geodesics on hyperbolic surfaces, including continuity in Teichmüller space. Overall, the paper extends classical Archimedean results to non-Archimedean settings, provides new uniformity in heavy-tailed regimes, and links probabilistic stability to geometric and spectral applications. The methods offer a versatile toolkit for studying how Lyapunov-type quantities and invariant measures respond to perturbations in both algebraic and geometric contexts.

Abstract

For certain natural families of topologies, we study continuity and stability of statistical properties of random walks on linear groups over local fields. We extend large deviation results known in the Archimedean case to non-Archimedean local fields and also demonstrate certain large deviation estimates for heavy tailed distributions unknown even in the Archimedean case. A key technical result, which may be of independent interest, establishes lower semi-continuity for the gap between the first and second Lyapunov exponents. As applications, we are able to obtain a key technical step towards a localization proof for heavy tailed Anderson models (the full proof appearing in a companion article), and show continuity/stability (taking the geometric data as input) of various statistical data associated to hyperbolic surfaces.

Paper Structure

This paper contains 22 sections, 53 theorems, 185 equations.

Table of Contents

  1. Introduction
  2. Main abstract results
  3. Background on random matrix products
  4. Random Schrödinger operators
  5. Statistics of random geodesics on hyperbolic surfaces
  6. Outline of the paper
  7. Acknowledgements
  8. Preliminaries and basic properties of the Wasserstein type topologies
  9. Conventions
  10. Concave Wasserstein distances
  11. Random matrix products
  12. Uniform large deviation estimates for martingales and Markov chains
  13. Uniform large deviations for martingale differences
  14. Uniform estimates for Markovian random walks on $G$-spaces
  15. Large deviations for linear cocycles and estimates on the invariant measure
  16. ...and 7 more sections

Key Result

Theorem 1.1

For any Borel probability measure $\mu$ on $\mathop{\mathrm{\mathrm{GL}}}\nolimits_d(\mathbb{R})$ satisfying (logmoment) and i.i.d. matrices $M_i$ with law $\mu$, almost surely, where

Theorems & Definitions (96)

  • Theorem 1.1: FurstenbergKesten
  • Theorem 1.3: bqclt
  • Definition 1.4
  • Proposition 1.5
  • Theorem 1.6
  • Remark 1.7
  • Theorem 1.8
  • Remark 1.9
  • Theorem 1.10
  • Theorem 1.11
  • ...and 86 more