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

• Main Authors: Genovese, Giuseppe, Giacomin, Giambattista, Greenblatt, Rafael Leon
• Format: Journal Article
• Language: English
• Published: 11.02.2016
• Subjects: Mathematics - Mathematical Physics; Mathematics - Probability; Physics - Mathematical Physics
• DOI: 10.48550/arxiv.1602.03633

Communications in Mathematical Physics vol. 351, pp. 923-958 (2017) 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.