Continuous Kasteleyn theory for the bead model
Samuel G. G. Johnston
King’s College London, UK

Abstract
Consider the semidiscrete torus representing unit length strings running in parallel. A bead configuration on is a point process on 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 may be expressed in terms of Fredholm determinants of certain operators on . We obtain an explicit formula for the volumes of bead configurations on . The asymptotics of this formula confirm a recent prediction in the free probability literature. Thereafter we study random bead configurations on , 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.
Cite this article
Samuel G. G. Johnston, Continuous Kasteleyn theory for the bead model. J. Eur. Math. Soc. (2025), published online first
DOI 10.4171/JEMS/1728