  This paper is concerned with the behavior of the ergodic constant associated
with convex and superlinear Hamilton-Jacobi equation in a periodic environment
which is perturbed either by medium with increasing period or by a random
Bernoulli perturbation with small parameter. We find a first order Taylor's
expansion for the ergodic constant which depends on the dimension d. When d = 1
the first order term is non trivial, while for all d $\ge$ 2 it is always 0.
Although such questions have been looked at in the context of linear uniformly
elliptic homogenization, our results are the first of this kind in nonlinear
settings. Our arguments, which rely on viscosity solutions and the weak KAM
theory, also raise several new and challenging questions.
