Gaussian fluctuations of products of random matrices distributed close to the identity

TL;DR

This paper studies how products of nearly identity 2x2 random matrices fluctuate around their Lyapunov exponents, providing perturbative formulas for these fluctuations relevant to quantum physics models.

Contribution

It introduces a perturbative analysis of Gaussian fluctuations for products of random matrices close to the identity, with explicit calculations of Lyapunov exponents and variances.

Findings

Gaussian fluctuations are characterized for matrices near identity

Explicit formulas for Lyapunov exponents and variances in small neighborhoods

Applications to one-dimensional random Schrödinger operators

Abstract

Products of random $2\times 2$ matrices exhibit Gaussian fluctuations around almost surely convergent Lyapunov exponents. In this paper, the distribution of the random matrices is supported by a small neighborhood of order $\lambda>0$ of the identity matrix. The Lyapunov exponent and the variance of the Gaussian fluctuations are calculated perturbatively in $\lambda$ and this requires a detailed analysis of the associated random dynamical system on the unit circle and its invariant measure. The result applies to anomalies and band edges of one-dimensional random Schr\"odinger operators.