We consider a certain infinite product of random 2 x 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 Derrida and Hilhorst (J Phys A 16:2641, 1983) that the Lyapunov exponent behaves like C-epsilon(2 alpha) in the limit epsilon SE arrow 0, where alpha is an element of ( 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 e. 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 that implies a suitable control of the Lyapunov exponent.

Genovese, G., Giacomin, G., Greenblatt, R.l. (2017). Singular behavior of the leading Lyapunov exponent of a product of random 2 x 2 matrices. COMMUNICATIONS IN MATHEMATICAL PHYSICS, 351(3), 923-958 [10.1007/s00220-017-2855-4].

Singular behavior of the leading Lyapunov exponent of a product of random 2 x 2 matrices

Rafael Leon Greenblatt
2017-01-01

Abstract

We consider a certain infinite product of random 2 x 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 Derrida and Hilhorst (J Phys A 16:2641, 1983) that the Lyapunov exponent behaves like C-epsilon(2 alpha) in the limit epsilon SE arrow 0, where alpha is an element of ( 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 e. 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 that implies a suitable control of the Lyapunov exponent.
2017
Pubblicato
Rilevanza internazionale
Articolo
Esperti anonimi
Settore MAT/07
Settore MAT/06
English
https://link.springer.com/article/10.1007/s00220-017-2855-4
Genovese, G., Giacomin, G., Greenblatt, R.l. (2017). Singular behavior of the leading Lyapunov exponent of a product of random 2 x 2 matrices. COMMUNICATIONS IN MATHEMATICAL PHYSICS, 351(3), 923-958 [10.1007/s00220-017-2855-4].
Genovese, G; Giacomin, G; Greenblatt, Rl
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2108/343723
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