The multinomial tiling model
Abstract
Given a graph $G$ and collection of subgraphs $T$ (called tiles), we consider covering $G$ with copies of tiles in $T$ so that each vertex $v\in G$ 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 "blowup" of $G$). In the limit of large multiplicities we compute asymptotic growth rate of the number of multinomial tilings. We show that the individual tile densities tend to a Gaussian field with respect to an associated discrete Laplacian. We also find an exact discrete Coulomb gas limit when we vary the multiplicities. For tilings of ${\mathbb Z}^d$ with translates of a single tile and a small density of defects, we study a crystallization phenomena when the defect density tends to zero, and give examples of naturally occurring quasicrystals in this framework.
 Publication:

arXiv eprints
 Pub Date:
 April 2021
 DOI:
 10.48550/arXiv.2104.03205
 arXiv:
 arXiv:2104.03205
 Bibcode:
 2021arXiv210403205K
 Keywords:

 Mathematics  Probability;
 Mathematical Physics;
 Mathematics  Combinatorics;
 60C05;
 82B20