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

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