  We study the homogenization of a stationary conductivity problem in a random
heterogeneous medium with highly oscillating conductivity coefficients and an
ensemble of simply closed conductivity resistant membranes. This medium is
randomly deformed and then rescaled from a periodic one with periodic
membranes, in a manner similar to the random medium proposed by Blanc, Le Bris
and Lions \cite{BLBL06}. Across the membranes, the flux is continuous but the
potential field itself undergoes a jump of Robin type. We prove that, for
almost all realizations of the random deformation, as the small scale of
variations of the medium goes to zero, the random conductivity problem is well
approximated by that on an effective medium which has deterministic and
constant coefficients and contains no membrane. The effective coefficients are
explicitly represented. One of our main contributions is to provide a solution
to the associated auxiliary problem that is posed on the whole domain with
infinitely many interfaces, in a setting that is neither periodic nor
stationary ergodic in the usual sense.
