Macroscopic loops in the Bose gas, Spin O(N) and related models

TL;DR

This paper proves the emergence of macroscopic loops in various interacting random loop models on high-dimensional lattices, demonstrating that large loops occupy a positive fraction of the system volume under broad conditions.

Contribution

It establishes the occurrence of macroscopic loops in a general class of models, including Spin O(N), lattice permutations, and Bose gas, under minimal assumptions.

Findings

Macroscopic loops occupy a positive fraction of the volume in high dimensions.

Results hold for models with various interaction potentials, including unbounded and hard-core.

Uniform positivity of the probability of large loops visiting distant vertices.

Abstract

We consider a general system of interacting random loops which includes several models of interest, such as the Spin O(N) model, random lattice permutations, a version of the interacting Bose gas in discrete space and of the loop O(N) model. We consider the system in $\mathbb{Z}^d$, $d \geq 3$, and prove the occurrence of macroscopic loops whose length is proportional to the volume of the system. More precisely, we approximate $\mathbb{Z}^d$ by finite boxes and, given any two vertices whose distance is proportional to the diameter of the box, we prove that the probability of observing a loop visiting both is uniformly positive. Our results hold under general assumptions on the interaction potential, which may have bounded or unbounded support or introduce hard-core constraints.