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 B. Derrida and H. J. Hilhorst (J. Phys. A 16, 1641-2654 (1983)) that the Lyapunov exponent behaves like Cε 2α 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 which implies a suitable control of the Lyapunov exponent.