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

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 $ν$ with simple Lyapunov spectrum, we give an explicit closed-form Hölder exponent $β_*(ν, θ)$ and constant in the modulus of continuity of $λ_\pm$ in the Wasserstein-plus-Hausdorff metric, depending only on the eccentricity of $\mathrm{supp}\,ν$, the Lyapunov gap, and the Hölder index $θ$. At every $ν\in \textit{M}_c(\operatorname{GL}(2,\mathbb{R}))$ we identify the log-Hölder exponent of Tall and Viana as $κ_*(ν, θ) = θ/(2+θ)$ under a natural mixing hypothesis, and $θ/(8(1+θ))$ 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-Hölder modulus of continuity for the integrated density of states of one-dimensional random Schrödinger operators with absolutely continuous disorder. The Hölder theory extends to $\operatorname{GL}(d,\mathbb{R})$ for the top exponent under spectral simplicity, and to the partial sums $Λ_k = λ_1 + \cdots + λ_k$ under strong $k$-irreducibility, yielding Hölder continuity of each individual sub-top exponent. A method-optimality proposition shows that $β_*$ 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 Hölder continuity across $\textit{M}_c(\operatorname{GL}(2,\mathbb{R}))$.