Macroscopic Loops in the Bose Gas, Spin O(N) and Related Models

Abstract

AbstractWe consider a general system of interacting random loops which includes several models of interest, such as theSpin O(N) model,random lattice permutations, a version of theinteracting Bose gasin discrete space and of theloop O(N) model.We consider the system in$${\mathbb {Z}}^d$$Zd,$$d \ge 3$$d≥3, and prove the occurrence of macroscopic loops whose length is proportional to the volume of the system. More precisely, we approximate$${\mathbb {Z}}^d$$Zdby 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.