# Regular expansion for the characteristic exponent of a product of 2 × 2 random matrices

Abstract : We consider a product of $2 \times 2$ random matrices which appears in the physics literature in the analysis of some 1D disordered models. These matrices depend on a parameter $\epsilon >0$ and on a positive random variable $Z$. Derrida and Hilhorst (J Phys A 16:2641, 1983, §3) predict that the corresponding characteristic exponent has a regular expansion with respect to $\epsilon$ up to — and not further — an order determined by the distribution of $Z$. We give a rigorous proof of that statement. We also study the singular term which breaks that expansion.
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https://hal.archives-ouvertes.fr/hal-01849549
Contributor : Benjamin Havret <>
Submitted on : Thursday, July 26, 2018 - 11:52:41 AM
Last modification on : Friday, April 10, 2020 - 5:26:51 PM

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• HAL Id : hal-01849549, version 1

### Citation

Benjamin Havret. Regular expansion for the characteristic exponent of a product of 2 × 2 random matrices. 2018. ⟨hal-01849549⟩

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