Macroscopic loops in the Spin O(N), double dimer 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, the double dimer model, random lattice permutations, and is related to the loop O(N) model and to the interacting Bose gas in discrete space. We introduce some of the main open questions about the size and the geometry of the loops, and present some recent results about the occurrence of macroscopic loops in dimensions d > 2 as the inverse temperature is large enough. More precisely, we consider arbitrary large boxes and prove that, given any two vertices whose distance is proportional to the diameter of the box, the probability of observing a loop visiting both is uniformly positive.