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Article Dans Une Revue Communications in Mathematical Physics Année : 2017

Singular Behavior of the Leading Lyapunov Exponent of a Product of Random $${2 \times 2}$$ 2 × 2 Matrices

Résumé

We consider a certain infinite product of random 2×2 matrices appearing in the solution of some 1 and 1 + 1 dimensional disordered models in statistical mechanics, which depends on a parameter ε > 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, 1641-2654 (1983)) that the Lyapunov exponent behaves like Cε 2α in the limit ε 0, where α ∈ (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 ε. 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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Dates et versions

hal-03912018 , version 1 (23-12-2022)

Identifiants

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Giuseppe Genovese, Giambattista Giacomin, Rafael Leon Greenblatt. Singular Behavior of the Leading Lyapunov Exponent of a Product of Random $${2 \times 2}$$ 2 × 2 Matrices. Communications in Mathematical Physics, 2017, 351 (3), pp.923-958. ⟨10.1007/s00220-017-2855-4⟩. ⟨hal-03912018⟩
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