  We consider the scaling of entanglement entropy in random Projected Entangled
Pairs States (PEPS) with an internal symmetry given by a finite group G. We
systematically demonstrate a correspondence between this entanglement entropy
and the difference of free energies of a classical Ising model with an addition
non-local term. This non-local term counts the number of domain walls in a
particular configuration of the classical spin model. We argue that for that
overwhelming majority of such states, this gives rise to an area law scaling
with well-defined topological entanglement entropy. The topological
entanglement entropy is shown to be log|G| for a simply connected region A and
which manifests as a difference in the number of domain walls of ground state
energies for the two spin models.
