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

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 $\textit{M}_c(\operatorname{GL}(2,\mathbb{R}))$.