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.