Quadri-tilings of the Plane

We introduce quadri-tilings and show that they are in bijection with dimer models on a family of graphs R ^* arising from rhombus tilings. Using two height functions, we interpret a sub-family of all quadri-tilings, called triangular quadri-tilings, as an interface model in dimension 2+2. Assigning “critical" weights to edges of R ^*, we prove an explicit expression, only depending on the local geometry of the graph R ^*, for the minimal free energy per fundamental domain Gibbs measure; this solves a conjecture of Kenyon (Invent Math 150:409–439, 2002). We also show that when edges of R ^* are asymptotically far apart, the probability of their occurrence only depends on this set of edges. Finally, we give an expression for a Gibbs measure on the set of all triangular quadri-tilings whose marginals are the above Gibbs measures, and conjecture it to be that of minimal free energy per fundamental domain.