q-linear approximants: Scaling functions for polygon models

Abstract

The perimeter and area generating functions of exactly solvable polygon models satisfy q-functional equations, where q is the area variable. The behaviour in the vicinity of the point where the perimeter generating function diverges can often be described by a scaling function. We develop the method of q-linear approximants in order to extract the approximate scaling behaviour of polygon models when an exact solution is not known. We test the validity of our method by approximating exactly solvable q-linear polygon models. This leads to scaling functions for a number of q-linear polygon models, notably generalized rectangles, Ferrers diagrams, and stacks.