  We study the invariant measure of a discretized stochastic Allen-Cahn
equation in d+1 dimensions in the low noise limit. We consider a cuboidal
domain and impose the two stable phases as boundary conditions at two opposite
faces. We then take a joint limit where the temperature and the mesh of the
discretization go to zero while the size of the domain grows. Our main result
is that the invariant measures concentrate exponentially fast around the
minimizers of the free energy functional if the domain does not grow too fast.
