  We elucidate connections between four models in statistical physics and
probability theory: (1) the toric code model of Kitaev, (2) the uniform
eight-vertex model, (3) random walk on a hypercube, and (4) a classical Ising
model with four-body interaction. As a consequence of our analysis (and of the
GKS-inequalities for the Ising model) we obtain correlation inequalities for
the toric code model and the uniform eight-vertex model.
