R. Brak et Al. Owczarek, ON THE ANALYTICITY PROPERTIES OF SCALING FUNCTIONS IN MODELS OF POLYMER COLLAPSE, Journal of physics. A, mathematical and general, 28(17), 1995, pp. 4709-4725
We consider the mathematical properties of the generating and partitio
n functions in the two-variable scaling region about the tricritical p
oint in some models of polymer collapse. We concentrate-on models that
have a similar behaviour to that of interacting partially-directed se
lf-avoiding walks (IPDSAW) in two dimensions. However, we do not restr
ict 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-ever scaling form ar
ound the theta-point, and the associated tricritical scaling function,
(g) over cap, has a finite radius of convergence, then the partition
function has a finite-size scaling form and importantly the finite-siz
e scaling function, (f) over cap, is an entire function. In the IPDSAW
case we have an explicit representation of the finite-size scaling fu
nction. 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 o
f polymer collapse for which it has recently been argued that the fini
te-size scaling functions are not entire.