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

TL;DR

This paper provides explicit quantitative bounds and analytic properties of the top Lyapunov exponent for random matrix products, extending to Markov chains and higher dimensions with optimal constants.

Contribution

It establishes explicit polydisc bounds, Cauchy coefficient estimates, and joint analyticity results for Lyapunov exponents, including extensions to Markov-driven cocycles and higher dimensions.

Findings

Explicit polydisc of holomorphy for Lyapunov exponents.

Closed-form Cauchy bounds on Taylor coefficients.

Joint analyticity in weights and matrix entries.

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).