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September 2022 The multinomial tiling model
Richard Kenyon, Cosmin Pohoata
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Ann. Probab. 50(5): 1986-2012 (September 2022). DOI: 10.1214/22-AOP1575

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 vG 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 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 Zd 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

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Richard Kenyon. Cosmin Pohoata. "The multinomial tiling model." Ann. Probab. 50 (5) 1986 - 2012, September 2022. https://doi.org/10.1214/22-AOP1575

Information

Received: 1 April 2021; Revised: 1 January 2022; Published: September 2022
First available in Project Euclid: 24 August 2022

Digital Object Identifier: 10.1214/22-AOP1575

Subjects:
Primary: 60C05

Keywords: phase transition , quasicrystal , tiling

Rights: Copyright © 2022 Institute of Mathematical Statistics

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Vol.50 • No. 5 • September 2022
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