Beads on the torus via scaling limits of dimer matchings

TL;DR

This paper explores the bead model on the torus through scaling limits of dimer models, providing an alternative approach to previous continuous Kasteleyn theory methods and aiming to serve as an accessible introduction.

Contribution

It introduces a new method using scaling limits of dimer models to analyze the bead model on the torus, complementing prior continuous Kasteleyn theory approaches.

Findings

Main results obtained via dimer model scaling limits

Provides an alternative to continuous Kasteleyn theory

Aims to serve as an accessible introduction to the topic

Abstract

In a previous article, we develop a continuous version of Kasteleyn theory to study the bead model on the torus. These are the point processes on the semi-discrete torus $\mathbb{T}_n := [0,1) \times \{0,1,\ldots,n-1\}$ (thought of as $n$ unit length strings wrapped around a doughnut) with the property that between every two consecutive points on same string, there lies a point on the neighbouring strings. In this companion article, we obtain the main results of the previous article via an alternative route, using scaling limits of dimer models as opposed to the continuous Kasteleyn theory. In any case, we hope that the article may serve as a gentle introduction to Kasteleyn theory on the torus.