Large deviations for products of random two dimensional matrices

TL;DR

This paper derives large deviation estimates for products of i.i.d. 2D random matrices, providing stability under perturbations and implications for Lyapunov exponents, density of states, and localization in random operators.

Contribution

It introduces large deviation estimates for 2D matrix products without irreducibility assumptions, enhancing understanding of stability and continuity properties.

Findings

Established stable large deviation estimates for 2D matrix products.

Proved uniform local modulus of continuity for Lyapunov exponents.

Implications for the regularity of density of states and localization phenomena.

Abstract

We establish large deviation type estimates for i.i.d. products of two dimensional random matrices with finitely supported probability distribution. The estimates are stable under perturbations and require no irreducibility assumptions. In consequence, we obtain a uniform local modulus of continuity for the corresponding Lyapunov exponent regarded as a function of the support of the distribution. This in turn has consequences on the modulus of continuity of the integrated density of states and on the localization properties of random Jacobi operators.