Lyapunov exponent for products of random Ising transfer matrices: the balanced disorder case

TL;DR

This paper investigates the behavior of the top Lyapunov exponent for products of 2x2 random matrices relevant to disordered Ising models, especially in the balanced disorder case, revealing critical behavior and continuity properties.

Contribution

It provides a precise analysis of the Lyapunov exponent's behavior near a diagonal matrix in the balanced disorder case, highlighting its log-Hölder continuity.

Findings

Sharp behavior of Lyapunov exponent in large interaction limit

Identification of critical behavior in balanced disorder case

Lyapunov exponent is log-Hölder continuous

Abstract

We analyze the top Lyapunov exponent of the product of sequences of two by two matrices that appears in the analysis of several statistical mechanics models with disorder: for example these matrices are the transfer matrices for the nearest neighbor Ising chain with random external field, and the free energy density of this Ising chain is the Lyapunov exponent we consider. We obtain the sharp behavior of this exponent in the large interaction limit when the external field is centered: this balanced case turns out to be critical in many respects. From a mathematical standpoint we precisely identify the behavior of the top Lyapunov exponent of a product of two dimensional random matrices close to a diagonal random matrix for which top and bottom Lyapunov exponents coincide. In particular, the Lyapunov exponent is only $\log$-H\"older continuous.