Intersecting loop models on Zd: rigorous results Origina

We consider a general class of (intersecting) loop models in d dimensions, including those related to high-temperature expansions of well-known spin models. We find that the loop models exhibit some interesting features – often in the “unphysical” region of parameter space where all connection with the original spin Hamiltonian is apparently lost. For a particular n=2, d=2 model, we establish the existence of a phase transition, possibly associated with divergent loops. However, for n≫1 and arbitrary d there is no phase transition marked by the appearance of large loops. Furthermore, at least for d=2 (and n large) we find a phase transition characterised by broken translational symmetry.