The multinomial tiling model

Abstract

Given a graph $\mathcal{G}$ and collection of subgraphs T (called tiles), we consider covering $\mathcal{G}$ with copies of tiles in T so that each vertex $\mathit{v}\in \mathcal{G}$ is covered with a predetermined multiplicity. The multinomial tiling model is a natural probability measure on such configurations (it is the uniform measure on standard tilings of the corresponding “blow-up” of $\mathcal{G}$).

In the limit of large multiplicities, we compute the asymptotic growth rate of the number of multinomial tilings. We show that the individual tile densities tend to a Gaussian field defined by an associated discrete Laplacian. We also find an exact discrete Coulomb gas limit when we vary the multiplicities.

For tilings of ${\mathbb{Z}}^{\mathit{d}}$ with translates of a single tile and a small density of defects, we study a crystallization phenomenon when the defect density tends to zero, and give examples of naturally occurring quasicrystals in this framework.