Quadri-tilings of the plane

Abstract

We introduce {\em quadri-tilings} and show that they are in bijection with dimer models on a {\em family} of graphs $\{R^{\ast }\}$ arising from rhombus tilings. Using two height functions, we interpret a sub-family of all quadri-tilings, called {\em triangular quadri-tilings}, as an interface model in dimension 2+2. Assigning "critical" weights to edges of $R^{\ast }$, we prove an explicit expression, only depending on the local geometry of the graph $R^{\ast }$, for the minimal free energy per fundamental domain Gibbs measure; this solves a conjecture of \cite{Kenyon1}. We also show that when edges of $R^{\ast }$ 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 {\em all} triangular quadri-tilings whose marginals are the above Gibbs measures, and conjecture it to be that of minimal free energy per fundamental domain.

Revised version, minor changes. 30 pages, 13 figures

References in corpus (2)

• Local statistics of lattice dimers
• The Laplacian and $\bar{\partial }$ operators on critical planar graphs

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