Singular behavior of the leading Lyapunov exponent of a product of random $2 \times 2$ matrices
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...
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Zusammenfassung: | 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. |
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DOI: | 10.48550/arxiv.1602.03633 |