Singular Behavior of the Leading Lyapunov Exponent of a Product of Random 2 × 2 Matrices

We consider a certain infinite product of random 2 × 2 matrices appearing in the solution of some 1 and 1 + 1 dimensional disordered models in statistical mechanics, which depends on a parameter ε> 0 and on a real random variable with distribution μ. For a large class of μ, we prove the prediction by Derrida and Hilhorst (J Phys A 16:2641, 1983) that the Lyapunov exponent behaves like Cϵ2α in the limit ϵ↘ 0 , where α∈ (0 , 1) and C> 0 are determined by μ. 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 ϵ. 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.