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

This paper develops a quantitative regularity theory for the Lyapunov exponents of random products of matrices in $\mathrm{GL}(2,\mathbb{R})$, with extensions to $\mathrm{GL}(d,\mathbb{R})$ for all $d\ge 2$. At every compactly supported measure $\nu$ with simple Lyapunov spectrum, we give an explicit closed-form Hölder exponent ${\beta }_{\ast }(\nu ,\theta )$ and constant in the modulus of continuity of ${\lambda }_{\pm }$ in the Wasserstein-plus-Hausdorff metric, depending only on the eccentricity of $\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}\nu$, the Lyapunov gap, and the Hölder index $\theta$. At every $\nu \in {\mathit{\text{M}}}_c(\mathrm{GL}(2,\mathbb{R}))$ we identify the log-Hölder exponent of Tall and Viana as ${\kappa }_{\ast }(\nu ,\theta )=\theta \mathrm{/}(2+\theta )$ under a natural mixing hypothesis, and $\theta \mathrm{/}(8(1+\theta ))$ 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-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 $\mathrm{GL}(d,\mathbb{R})$ for the top exponent under spectral simplicity, and to the partial sums ${\mathrm{\Lambda }}_k={\lambda }_1+\cdots +{\lambda }_k$ under strong $k$-irreducibility, yielding Hölder continuity of each individual sub-top exponent. A method-optimality proposition shows that ${\beta }_{\ast }$ 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 ${\mathit{\text{M}}}_c(\mathrm{GL}(2,\mathbb{R}))$.