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