Singular behavior of the leading Lyapunov exponent of a product of random $2 \times 2$ matrices

TL;DR

This paper analyzes the singular behavior of the leading Lyapunov exponent for a product of random 2x2 matrices, confirming a predicted power-law dependence on a small parameter in certain disordered models.

Contribution

It rigorously proves the asymptotic behavior of the Lyapunov exponent as the parameter approaches zero, validating previous predictions by Derrida and Hilhorst.

Findings

Lyapunov exponent behaves like C ε^{2α} as ε → 0

Established closeness of invariant measure to the predicted distribution

Provided bounds and contractivity properties for the Markov chain

Abstract

We consider a certain infinite product of random $2 \times 2$ matrices appearing in the solution of some $1$ and $1+1$ dimensional disordered models in statistical mechanics, which depends on a parameter $\varepsilon>0$ and on a real random variable with distribution $\mu$. For a large class of $\mu$, we prove the prediction by B. Derrida and H. J. Hilhorst (J. Phys. A 16:2641, 1983) that the Lyapunov exponent behaves like $C \varepsilon^{2 \alpha}$ in the limit $\varepsilon \searrow 0$, where $\alpha \in (0,1)$ and $C>0$ are determined by $\mu$. Derrida and Hilhorst performed a two-scale analysis of the integral equation for the invariant distribution of the Markov chain associated to the matrix product and obtained a probability measure that is expected to be close to the invariant one for small $\varepsilon$. We introduce suitable norms and exploit contractivity properties to show that such a probability measure is indeed close to the invariant one in a sense which implies a suitable control of the Lyapunov exponent.