Intersecting loop models on Z: rigorous results

Ž . 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 ns2, ds2 model, we establish the existence of a phase transition, possibly associated with divergent loops. However, for n41 and arbitrary d there is no phase transition marked by the appearance of large Ž . loops. Furthermore, at least for ds2 and n large we find a phase transition characterised by broken translational symmetry. q 2000 Elsevier Science B.V. All rights reserved. PACS: 05.50.qq; 64.60.-i; 75.10.Hk