Analytic expansions of (max,+) Lyapunov exponents

Dohy Hong , INRIA Sophia Antipolis

Abstract

We give an explicit analytic series expansion of the Lyapunov exponent of a class of random i.i.d. matrices in the algebra, based on a Bernoulli scheme (resp. multinomial) depending on a small parameter p (resp. small parameters ). For this purpose we use a representation of this exponent as the mean value of a certain random variable, and then use a discrete analogue of the so-called light-traffic perturbation formulas. We show that this leads to an analytic expansion under a simple condition on p. This also provides a closed form expression for all derivatives of at p=0 and approximations of of any order. Besides, this enables us to give an expression of the expectation of the time between two renovating events of the system.