Geometrical description of phase transitions in terms of diagrams and their growth function

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Abstract
Many statistical mechanical models lend themselves to a geometrical description in terms of diagrams built out of elemental units such as bonds, plaquettes, etc. In some cases, these diagrams represent physical systems such as polymers, surfaces, etc., and can be identified occasionally with certain unphysical limits of some spin models. These diagrams exhibit additional transitions, such as percolation, which may not be easily detectable in the original spin model they correspond to. Thus a system of diagrams forms an important statistical mechanical model in its own right, requiring a direct study. We introduce an entropy function S for such a system, which possesses all the thermodynamic properties of an entropy. This entropy function is not the same as the usual entropy of the original statistical model to which the diagram system is related. In particular, the equation of state can be recast in a form so that it can be easily integrated to yield the entropy function, something that may not be easily done for the original model. A knowledge of S allows us to obtain the free energy ω by a Legendre transform. Using this approach, we calculate S and ω for various geometrical objects on a Bethe or a Bethe-like lattice. This then yields a ‘‘mean-field’’ approximation for S and ω for realistic lattices. These include branched polymers and random surfaces among others. The entropy function σ per elemental unit from which diagrams are built gives rise to the growth function μ=exp(σ), which plays an important role in locating the singularities in the force energy and, hence, the phase transition. We also discuss the relevance of μ for the dilute limit. We illustrate our results by various examples.