We consider a N × × N lattice, with N 2 vertices and 2 N 2 edges. Imposing periodic boundary conditions requires that the states 7 and 8 occur equally often, as do states 5 and 6, and thus can be taken to have the same energy. For the zero-field case the same is true for the two other pairs of states. Each vertex j has an associated energy ϵ ϵ j and Boltzmann weight w j = e − − ϵ ϵ j k T , giving the partition function over the lattice as