Analytic expansions of max-plus Lyapunov exponents

Abstract

We give an explicit analytic series expansion of the (max, plus)-Lyapunov exponent γ(p) of a sequence of independent and identically distributed random matrices, 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 form expression for all derivatives of γ(p) at p = 0 and approximations of γ(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 (p1, . . . , pm) 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.