Regular expansion for the characteristic exponent of a product of $2 \times 2$ random matrices

TL;DR

This paper rigorously proves the regular expansion of the characteristic exponent for a product of 2x2 random matrices depending on a parameter, confirming predictions from physics literature and analyzing the singular term that disrupts this expansion.

Contribution

The paper provides a rigorous proof of the regular expansion of the characteristic exponent and investigates the singular term that affects this expansion.

Findings

Confirmed the regular expansion of the characteristic exponent up to a certain order.

Identified and analyzed the singular term disrupting the expansion.

Validated predictions from physics literature with rigorous mathematical proof.

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, \S 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.