Analytic expansions of max-plus Lyapunov exponents

We give an explicit analytic series expansion of the (max, plus)Lyapunov ex ponent gamma (p) of a sequence of independent and identically distributed r andom matrices, generated via a Bernoulli scheme depending on a small param eter p. A key assumption is that one of the matrices has a unique normalize d eigenvector. This allows us to obtain a representation of this exponent a s the mean value of a certain random variable. We then use a discrete analo gue of the so-called light-traffic perturbation formulas to derive the expa nsion. We show that it is analytic under a simple condition on p. This also provides a closed form expression for all derivatives of gamma (p) at p = 0 and approximations of gamma (p) of any order, together with an error esti mate for finite order Taylor approximations. Several extensions of this are discussed, including expansions of multinomial schemes depending on small parameters (p(1),...,p(m)) and expansions for exponents associated with ite rates of a class of random operators which includes the class of nonexpansi ve and homogeneous operators. Several examples pertaining to computer and c ommunication sciences are investigated: timed event graphs, resource sharin g models and heap models.