Singular Behavior of the Leading Lyapunov Exponent of a Product of Random ${2 \times 2}$ Matrices

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 Derrida and Hilhorst (J Phys A 16:2641, 1983) that the Lyapunov exponent behaves like ${C \epsilon^{2 \alpha}}$ in the limit ${\epsilon \searrow 0}$, 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 ${\epsilon}$. 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 that implies a suitable control of the Lyapunov exponent.