Given a graph and collection of subgraphs T (called tiles), we consider covering with copies of tiles in T so that each vertex 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 ).
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 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.
R.K. was supported by NSF Grant DMS-1940932 and the Simons Foundation Grant 327929.
We thank Jim Propp, Robin Pemantle, and Wilhelm Schlag for helpful conversations.
"The multinomial tiling model." Ann. Probab. 50 (5) 1986 - 2012, September 2022. https://doi.org/10.1214/22-AOP1575