  We study the limit behaviour of a sequence of non-convex, vectorial, random
integral functionals, defined on $W^{1,1}$, whose integrands satisfy degenerate
linear growth conditions. These involve suitable random, scale-dependent
weight-functions. Under minimal assumptions on the integrand and on the
weight-functions, we show that the sequence of functionals homogenizes to a
non-degenerate functional defined on $BV$.
