Analytic expansions of max-plus Lyapunov exponents

Abstract

We give an explicit analytic series expansion of the (max, plus)-Lyapunov exponent $\gamma(p)$ of a sequence of independent and identically distributed randommatrices, generated via a Bernoulli scheme depending on a small parameter $p$. A key assumption is that one of the matrices has a unique normalized eigenvector. This allows us to obtain a representation of this exponent as the mean value of a certain random variable.We then use a discrete analogue of the so-called light-traffic perturbation formulas to derive the expansion.We show that it is analytic under a simple condition on $p$. This also provides a closed formexpression for all derivatives of $\gamma(p)$ at $p = 0$ and approximations of $\gamma(p)$ of any order, together with an error estimate for finite order Taylor approximations. Several extensions of this are discussed, including expansions of multinomial schemes depending on small parameters $(p_1,\dots, p_m)$ and expansions for exponents associated with iterates of a class of random operators which includes the class of nonexpansive and homogeneous operators. Several examples pertaining to computer and communication sciences are investigated: timed event graphs, resource sharing models and heap models.