A SCALING THEORY OF THE COLLAPSE TRANSITION IN GEOMETRIC CLUSTER-MODELS OF POLYMERS AND VESICLES

Much effort has been expended in the past decade to calculate numerica lly the exponents at the collapse transition point in walk, polygon an d animal models. The crossover exponent phi has been of special intere st and sometimes is assumed to obey the relation 2 - alpha = 1/phi wit h the alpha the canonical (thermodynamic) exponent that characterizes the divergence of the specific heat. The reasons for the validity of t his relation are not widely known. We present a scaling theory of coll apse transitions in such models. The free energy and canonical partiti on functions have finite-length scaling forms whilst the grand partiti on function has a tricritical scaling form. The link between the grand and canonical ensembles leads to the above scaling relation. We then comment on the validity of current estimates of the crossover exponent for interacting self-avoiding walks in two dimensions and propose a t est involving the scaling relation which may be used to check these va lues.