Singular behavior of the leading Lyapunov exponent of a product of random 2 x 2 matrices

We consider a certain infinite product of random 2 x 2 matrices appearing in the solution of some 1 and 1 + 1 dimensional disordered models in statistical mechanics, which depends on a parameter epsilon > 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 SE arrow 0, where alpha is an element of ( 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 e. 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.