Solution of a model of self-avoiding walks with multiple monomers per   site on the Bethe lattice
  We solve a model of self-avoiding walks with up to two monomers per site on the Bethe lattice. This model, inspired on the Domb-Joyce model, was recently proposed to describe the collapse transition observed in interacting polymers [J. Krawczyk et al, Phys. Rev. Lett. 96, 240603 (2006)]. When immediate self-reversals are allowed (RA model), the solution displays a phase diagram with a polymerized phase and a non-polymerized phase, separated by a phase transition which is of first order for a non-vanishing statistical weight of doubly occupied sites. If the configurations are restricted forbidding immediate self-reversals (RF model), a richer phase diagram is found, displaying a tricritical point and a critical endpoint. 
Pablo Serra and Juergen F. Stilck