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


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.


We thank Jim Propp, Robin Pemantle, and Wilhelm Schlag for helpful conversations.


Download Citation

Richard Kenyon. Cosmin Pohoata. "The multinomial tiling model." Ann. Probab. 50 (5) 1986 - 2012, September 2022.


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

MathSciNet: MR4474506
zbMATH: 1515.60042
Digital Object Identifier: 10.1214/22-AOP1575

Primary: 60C05

Keywords: phase transition , quasicrystal , tiling

Rights: Copyright © 2022 Institute of Mathematical Statistics


This article is only available to subscribers.
It is not available for individual sale.

Vol.50 • No. 5 • September 2022
Back to Top