Asymptotic Lyapunov exponents for large random matrices

TL;DR

This paper extends the understanding of the asymptotic behavior of Lyapunov exponents for large products of independent random matrices with general distributions, providing explicit convergence rates.

Contribution

It generalizes previous Gaussian-specific results to broader classes of random matrices and introduces a new method linking structures and dynamics for analysis.

Findings

Convergence of N^{-1} log ||A_N ... A_1|| to a non-random limit for general distributions

Explicit rate of convergence established for the Lyapunov exponents

Method based on a simple connection between structures and dynamics

Abstract

Suppose that A_1,\dots, A_N are independent random matrices whose atoms are iid copies of a random variable \xi of mean zero and variance one. It is known from the works of Newman et. al. in the late 80s that when \xi is gaussian then N^{-1} \log ||A_N \dots A_1|| converges to a non-random limit. We extend this result to more general matrices with explicit rate of convergence. Our method relies on a simple connection between structures and dynamics.