Domino Tilings of the Torus

TL;DR

This paper investigates the enumeration and classification of domino tilings on a torus, utilizing height functions, flux analysis, and Kasteleyn matrices to understand tiling behavior and distribution patterns.

Contribution

It introduces a characterization of flux values for domino tilings on a torus and applies Kasteleyn matrices to count tilings with specific fluxes, extending to infinite lattices.

Findings

Characterization of flux values for torus tilings

Application of Kasteleyn matrices to count tilings with prescribed flux

Analysis of the limit distribution of tilings as lattice scales

Abstract

We consider the problem of counting and classifying domino tilings of a quadriculated torus. The counting problem for rectangles was studied by Kasteleyn and we use many of his ideas. Domino tilings of planar regions can be represented by height functions; for a torus given by a lattice L, these functions exhibit arithmetic L-quasiperiodicity. The additive constants determine the flux of the tiling, which can be interpreted as a vector in the dual lattice (2L)*. We give a characterization of the actual flux values, and of how corresponding tilings behave. We also consider domino tilings of the infinite square lattice; tilings of tori can be seen as a particular case of those. We describe the construction and usage of Kasteleyn matrices in the counting problem, and how they can be applied to count tilings with prescribed flux values. Finally, we study the limit distribution of the number of tilings with a given flux value as a uniform scaling dilates the lattice L.