THE MULTINOMIAL TILING MODEL
成果类型：
Article
署名作者：
Kenyon, Richard; Pohoata, Cosmin
署名单位：
Yale University
刊物名称：
ANNALS OF PROBABILITY
ISSN/ISSBN：
0091-1798
DOI：
10.1214/22-AOP1575
发表日期：
2022
页码：
1986-2012
关键词：
摘要：
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 is an element of( )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 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 Z(d) 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.