Continuous Kasteleyn theory for the bead model

TL;DR

This paper develops a continuous Kasteleyn theory for bead configurations on a semi-discrete torus, deriving explicit formulas for their volumes, confirming predictions in free probability, and connecting the structure to exclusion processes and TASEP.

Contribution

It introduces a continuous Kasteleyn framework for bead models, providing explicit volume formulas, asymptotic analysis, and a novel probabilistic representation of TASEP on the ring.

Findings

Partition functions expressed as Fredholm determinants.

Explicit volume formulas for bead configurations.

Determinantal structure linked to exclusion processes.

Abstract

Consider the semi-discrete torus $\mathbb{T}_n := [0,1) \times \{0,1,\ldots,n-1\}$ representing $n$ unit length strings running in parallel. A bead configuration on $\mathbb{T}_n$ is a point process on $\mathbb{T}_n$ with the property that between every two consecutive points on the same string, there lies a point on each of the neighbouring strings. In this article we develop a continuous version of Kasteleyn theory to show that partition functions for bead configurations on $\mathbb{T}_n$ may be expressed in terms of Fredholm determinants of certain operators on $\mathbb{T}_n$. We obtain an explicit formula for the volumes of bead configurations on $\mathbb{T}_n$. The asymptotics of this formula confirm a recent prediction in the free probability literature. Thereafter we study random bead configurations on $\mathbb{T}_n$, showing that they have a determinantal structure which can be connected with exclusion processes. We use this machinery to construct a new probabilistic representation of TASEP on the ring.