On the analyticity properties of scaling functions in models of polymer collapse

Abstract

We consider the mathematical properties of the generating and partition functions in the two-variable scaling region about the tricritical point in some models of polymer collapse. We concentrate on models that have a similar behaviour to that of interacting partially-directed self-avoiding walks (IPDSAW) in two dimensions. However, we do not restrict the discussion to that model. After describing the properties for a general class of models, and stating exactly what we mean by scaling, we prove the following theorem: If the generating function of finite-size partition functions has a tricritical cross-over scaling form around the theta -point, and the associated tricritical scaling function, g, has a finite radius of convergence, then the partition function has a finite-size scaling form and importantly the finite-size scaling function, f, is an entire function. In the IPDSAW case we have an explicit representation of the finite-size scaling function. We point out that given our description of tricritical scaling this theorem should apply mutatis mutandis to a wider class of theta -point models. We discuss the result in relation to the Edwards model of polymer collapse for which it has recently been argued that the finite-size scaling functions are not entire.