A numerical study of the 3D random interchange and random loop models

TL;DR

This paper numerically investigates the 3D random interchange and loop models, identifying transition times for long loops and characterizing their length distributions as Poisson-Dirichlet, providing insights into phase transitions in these models.

Contribution

It offers the first detailed numerical analysis of transition times and loop length distributions in 3D random interchange and loop models.

Findings

Transition times for long loops are identified.

Loop length distributions follow Poisson-Dirichlet with parameters 1 or 1/2.

Provides numerical evidence for phase transition behavior.

Abstract

We have studied numerically the random interchange model and related loop models on the three-dimensional cubic lattice. We have determined the transition time for the occurrence of long loops. The joint distribution of the lengths of long loops is Poisson-Dirichlet with parameter 1 or 1/2.