Abstract
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.
Funding Statement
R.K. was supported by NSF Grant DMS-1940932 and the Simons Foundation Grant 327929.
Acknowledgments
We thank Jim Propp, Robin Pemantle, and Wilhelm Schlag for helpful conversations.
Citation
Richard Kenyon. Cosmin Pohoata. "The multinomial tiling model." Ann. Probab. 50 (5) 1986 - 2012, September 2022. https://doi.org/10.1214/22-AOP1575
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