  We prove a stochastic homogenization result for integral functionals defined
on finite partitions assuming the surface tension to be stationary and possibly
ergodic. We also consider the convergence of boundary value problems when we
impose a boundary value just on part of the boundary. As a consequence, we show
that if the homogenized surface tension is isotropic, then one can obtain it by
a multi-cell problem where Dirichlet boundary conditions are imposed only at
the bottom and the top of the cube. We also show that this result fails in the
general anisotropic case.
