Loop correlations in random wire models

TL;DR

This paper introduces a family of loop soup models on hypercubic lattices, conjectures their correlation behavior in higher dimensions, and proves a specific case related to the XY spin system.

Contribution

It proposes new loop models with conjectured correlation structures and proves a key probabilistic property for a model related to the XY spin system.

Findings

Loop correlations in higher dimensions are conjectured to follow Poisson-Dirichlet distributions.

Proved that in a specific wire model, the probability of an even partition matches Poisson-Dirichlet predictions.

Introduces a novel framework connecting loop models to classical spin systems.

Abstract

We introduce a family of loop soup models on the hypercubic lattice. The models involve links on the edges, and random pairings of the link endpoints on the sites. We conjecture that loop correlations of distant points are given by Poisson-Dirichlet correlations in dimensions three and higher. We prove that, in a specific random wire model that is related to the classical XY spin system, the probability that distant sites form an even partition is given by the Poisson-Dirichlet counterpart.