On the Analyticity Properties of Scaling Functions in Models of Polymer Collapse

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 θ-point, and the associated tricritical scaling function, ĝ, has a finite radius of convergence, then the partition function has a finite-size scaling form and importantly the finite-size scaling function, f̂ , 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 θ-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. Short title: Scaling functions of Polymer Collapse PACS numbers: 05.50.+q, 05.70.fh, 61.41.+e