Length Distributions in Loop Soups

TL;DR

This paper derives the distribution of loop lengths in high-dimensional lattice ensembles, showing it follows a Poisson-Dirichlet distribution for macroscopic loops, supported by analytical calculations and numerical simulations.

Contribution

It introduces a method to calculate loop length distributions using sigma models and replica techniques, revealing the Poisson-Dirichlet form for macroscopic loops.

Findings

Loop length distribution is Poisson-Dirichlet for macroscopic loops.

Analytical methods connect loop distributions to sigma models.

Numerical simulations support the theoretical predictions.

Abstract

Statistical lattice ensembles of loops in three or more dimensions typically have phases in which the longest loops fill a finite fraction of the system. In such phases it is natural to ask about the distribution of loop lengths. We show how to calculate moments of these distributions using $CP^{n-1}$ or $RP^{n-1}$ and O(n) $\sigma$ models together with replica techniques. The resulting joint length distribution for macroscopic loops is Poisson-Dirichlet with a parameter $\theta$ fixed by the loop fugacity and by symmetries of the ensemble. We also discuss features of the length distribution for shorter loops, and use numerical simulations to test and illustrate our conclusions.