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arXiv · 1602.03633

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

Giuseppe Genovese·Giambattista Giacomin·Rafael Leon Greenblatt
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Abstract

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 $μ$. For a large class of $μ$, we prove the prediction by B. Derrida and H. J. Hilhorst (J. Phys. A 16:2641, 1983) that the Lyapunov exponent behaves like $C \varepsilon^{2 α}$ in the limit $\varepsilon \searrow 0$, where $α\in (0,1)$ 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 $\varepsilon$. 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.

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Giuseppe Genovese, Giambattista Giacomin, Rafael Leon Greenblatt. 2016-12-07. Singular behavior of the leading Lyapunov exponent of a product of random $2 \times 2$ matrices. https://doi.org/10.1007/s00220-017-2855-4

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