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

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.