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

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 $\epsilon >0$ 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 $C{\epsilon }^{2\alpha }$ in the limit $\epsilon \searrow 0$, 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 $\epsilon$. 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.

35 pages, 1 figure, LaTeX. Various revisions including many changes to the introduction, a simpler statement of the results of Section 4, and some simplifications of the calculations in Section 5

References in corpus (2)

• Lyapunov exponents, one-dimensional Anderson localisation and products of random matrices
• Products of random matrices and generalised quantum point scatterers

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