Quantitative Analyticity for Lyapunov Exponents of Random Products of Matrices with Explicit Polydiscs and Cauchy Coefficient Bounds

Abstract

The top Lyapunov exponent $\lambda_+(A, p)$ of a random product of matrices in $\mathrm{GL}(d, \mathbb{R})$, $d \geq 2$, with simple top spectrum, depends real-analytically on the probability weights $p$ and the matrix coefficients $A$. We establish a quantitative form of this analyticity through a single Kato perturbation argument on the complexified Markov operator on H\"older functions on projective space, yielding seven main theorems with explicit closed-form constants: (i) an explicit polydisc of holomorphy for $p \mapsto \lambda_+(A, p)$ in $\mathbb{C}^N$, giving the quantitative form of the Peres and Bezerra-S\'anchez-Tall analyticity theorem; (ii) closed-form Cauchy bounds on its Taylor coefficients; (iii) joint analyticity in the weights $p$ and the matrix entries $A$, with explicit radii in both; (iv) an extension to Markov-chain driven cocycles, with polydisc radius explicit in the chain spectral gap; (v) explicit polynomial boundary-decay rates as $p$ approaches $\partial \Delta_N$, conditional on a spectral-gap-decay hypothesis; (vi) extension to $\mathrm{GL}(d, \mathbb{R})$ for all $d \geq 2$ via the Fubini-Study metric; and (vii) a Grassmannian variant giving quantitative analyticity of the partial sums $\Lambda_k = \lambda_1 + \cdots + \lambda_k$ under strong $k$-irreducibility, hence of each individual sub-top Lyapunov exponent. The polydisc radius is method-optimal within the Kato class, and a Bernstein-type result shows the Cauchy growth $\alpha! \cdot M_*/r_*^{|\alpha|}$ is sharp up to constants. A two-matrix example with numerical values connects the bounds to the H\"older estimates of the companion paper Thiam (Nov. 2025).