Thermal and percolative transitions and the need for independent symmetry breakings in branched polymers on a Bethe lattice

Abstract

We consider a very general model of equilibrium polymerization of branched polymers. Our model contains, as a special case, the ‘‘a priori equal probability’’ model considered by Flory and Stockmayer. In this limit, the model exhibits only percolation transition. We solve our general model in the interior of a Bethe lattice. There are thermal as well as percolation transitions in the model. Each of the two transitions requires an independent spontaneous symmetry breaking; neither implies the other. Without spontaneous symmetry breaking, the transitions do not manifest themselves. Thermal transitions correspond to singularities in the equation of state. Percolation transitions, on the other hand, do not correspond to any singularity in the equation of state. We also discuss the failure of a topological identity, valid for any finite Cayley tree, in the interior of the Bethe lattice. We consider various different cases to show the usefulness of our model. In particular, we argue that one must distinguish between the ‘‘tree approximation’’ of Flory on a general lattice and our exact solution on the Bethe lattice. The former, in general, allows for loop formation, whereas there are no loops allowed in the latter solution.