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Bethe lattice solution of a model of SAW's with up to 3 monomers per site and no restriction

In the multiple monomers per site (MMS) model, polymeric chains are represented by walks on a lattice which may visit each site up to K times. We have solved the unrestricted version of this model, where immediate reversals of the walks are allowed (RA) for K = 3 on a Bethe lattice with arbitrary coordination number in the grand-canonical formalism. We found transitions between a non-polymerized and two polymerized phases, which may be continuous or discontinuous. In the canonical situation, the transitions between the extended and the collapsed polymeric phases are always continuous. The transition line is partly composed by tricritical points and partially by critical endpoints, both lines meeting at a multicritical point. In the subspace of the parameter space where the model is related to SASAW's (self-attracting self-avoiding walks), the collapse transition is tricritical. We discuss the relation of our results with simulations and previous Bethe and Husimi lattice calculations for the MMS model found in the literature.