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

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 ℤ^d , d ≥ 3 , and prove the occurrence of macroscopic loops whose length is proportional to the volume of the system. More precisely, we approximate 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.