A scaling theory of the collapse transition in geometric cluster models of polymers and vesicles

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Abstract
Much effort has been expended in the past decade to calculate numerically the exponents at the collapse transition point in walk, polygon and animal models. The crossover exponent phi has been of special interest and sometimes is assumed to obey the relation 2- alpha =1/ phi with the alpha the canonical (thermodynamic) exponent that characterizes the divergence of the specific heat. The reasons for the validity of this relation are not widely known. The authors present a scaling theory of collapse transitions in such models. The free energy and canonical partition functions have finite-length scaling forms whilst the grand partition function has a tricritical scaling form. The link between the grand and canonical ensembles leads to the above scaling relation. They then comment on the validity of current estimates of the crossover exponent for interacting self-avoiding walks in two dimensions and propose a test involving the scaling relation which may be used to check these values.

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