# Universal behaviour of 3D loop soup models

## TL;DR

This paper explores the universal properties of 3D loop soup models, revealing their phase transitions, the role of Poisson-Dirichlet distributions in loop structures, and implications for symmetry breaking in quantum systems.

## Contribution

It introduces the Poisson-Dirichlet distribution as a key tool for understanding loop structures and phase transitions in 3D loop soup models, connecting statistical physics with partition theory.

## Key findings

- Long-range order corresponds to macroscopic loops.
- Loop lengths follow a Poisson-Dirichlet distribution.
- Heuristic arguments lead to exact calculations of distribution parameters.

## Abstract

These notes describe several loop soup models and their {\it universal behaviour} in dimensions greater or equal to 3. These loop models represent certain classical or quantum statistical mechanical systems. These systems undergo phase transitions that are characterised by changes in the structures of the loops. Namely, long-range order is equivalent to the occurrence of macroscopic loops. There are many such loops, and the joint distribution of their lengths is always given by a {\it Poisson-Dirichlet distribution}. This distribution concerns random partitions and it is not widely known in statistical physics. We introduce it explicitly, and we explain that it is the invariant measure of a mean-field split-merge process. It is relevant to spatial models because the macroscopic loops are so intertwined that they behave effectively in mean-field fashion. This heuristics can be made exact and it allows to calculate the parameter of the Poisson-Dirichlet distribution. We discuss consequences about symmetry breaking in certain quantum spin systems.