Quantitative H\"older Regularity, Concentration, and Spectral Applications for Lyapunov Exponents of Random $\operatorname{GL}(2,\mathbb{R})$ Cocycles, with Extensions to $\operatorname{GL}(d,\mathbb{R})$

TL;DR

This paper establishes a comprehensive quantitative regularity framework for Lyapunov exponents of random matrix products in GL(2,R), extending to higher dimensions, with explicit exponents, concentration inequalities, and spectral applications.

Contribution

It introduces explicit H"older exponents and constants for Lyapunov exponents, extending regularity results to higher dimensions and various spectral quantities, with optimality and lower bounds.

Findings

Explicit H"older exponents depend only on support eccentricity, spectral gap, and H"older index.

Derived large deviation principles and concentration inequalities for Lyapunov exponents.

Extended regularity results to higher-dimensional GL(d,R) and partial sums under irreducibility.

Abstract

This paper develops a quantitative regularity theory for the Lyapunov exponents of random products of matrices in $\operatorname{GL}(2,\mathbb{R})$, with extensions to $\operatorname{GL}(d,\mathbb{R})$ for all $d \geq 2$. At every compactly supported measure $\nu$ with simple Lyapunov spectrum, we give an explicit closed-form H\"older exponent $\beta_*(\nu, \theta)$ and constant in the modulus of continuity of $\lambda_\pm$ in the Wasserstein-plus-Hausdorff metric, depending only on the eccentricity of $\mathrm{supp}\,\nu$, the Lyapunov gap, and the H\"older index $\theta$. At every $\nu \in \textit{M}_c(\operatorname{GL}(2,\mathbb{R}))$ we identify the log-H\"older exponent of Tall and Viana as $\kappa_*(\nu, \theta) = \theta/(2+\theta)$ under a natural mixing hypothesis, and $\theta/(8(1+\theta))$ in the perpetuity regime. The same spectral-gap method yields a large deviation principle with explicit rate function, Hoeffding-Azuma concentration inequalities, an extension to Markov-chain driven cocycles with closed-form exponent, and a quantitative log-H\"older modulus of continuity for the integrated density of states of one-dimensional random Schr\"odinger operators with absolutely continuous disorder. The H\"older theory extends to $\operatorname{GL}(d,\mathbb{R})$ for the top exponent under spectral simplicity, and to the partial sums $\Lambda_k = \lambda_1 + \cdots + \lambda_k$ under strong $k$-irreducibility, yielding H\"older continuity of each individual sub-top exponent. A method-optimality proposition shows that $\beta_*$ is the best exponent obtainable from the linear balance of axioms (A1)-(A3) of the spectral-gap method; strict improvement requires either modifying these axioms or adopting a different proof strategy. A lower-bound proposition adapted from Duarte, Klein, and Santos rules out uniform H\"older continuity across $\textit{M}_c(\operatorname{GL}(2,\mathbb{R}))$.