The multinomial tiling model

TL;DR

This paper introduces the multinomial tiling model for covering graphs with subgraphs, analyzes its asymptotic behavior, and explores phenomena like Gaussian tile densities, Coulomb gas limits, and crystallization in lattice tilings.

Contribution

It develops the multinomial tiling model, computes asymptotic growth, and reveals Gaussian and Coulomb gas limits, along with crystallization phenomena in large-scale tilings.

Findings

Asymptotic growth rate of multinomial tilings computed.

Tile densities tend to a Gaussian field.

Crystallization phenomena observed in lattice tilings.

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 $v\in G$ 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 asymptotic growth rate of the number of multinomial tilings. We show that the individual tile densities tend to a Gaussian field with respect to an associated discrete Laplacian. We also find an exact discrete Coulomb gas limit when we vary the multiplicities. For tilings of ${\mathbb Z}^d$ with translates of a single tile and a small density of defects, we study a crystallization phenomena when the defect density tends to zero, and give examples of naturally occurring quasicrystals in this framework.