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.