We present in this paper an approach for computing the homogenized behavior
of a medium that is a small random perturbation of a periodic reference
material. The random perturbation we consider is, in a sense made precise in
our work, a rare event at the microscopic level. It however affects the
macroscopic properties of the material, and we indeed provide a method to
compute the first and second-order corrections. To this end, we formally
establish an asymptotic expansion of the macroscopic properties. Our
perturbative approach shares common features with a defect-type theory of solid
state physics. The computational efficiency of the approach is demonstrated