  We present a thorough study of the static properties of 2D models of spin-ice
type on the square lattice or, in other words, the sixteen-vertex model. We use
extensive Monte Carlo simulations to determine the phase diagram and critical
properties of the finite dimensional system. We put forward a suitable
mean-field approximation, by defining the model on carefully chosen trees. We
employ the cavity (Bethe-Peierls) method to derive self-consistent equations,
the fixed points of which yield the equilibrium properties of the model on the
tree-like graph. We compare mean-field and finite dimensional results. We
discuss our findings in the context of experiments in artificial two
dimensional spin ice.
