A new two-variable generating function for convex self-avoiding polygons

Abstract

The width of a self-avoiding polygon on the square lattice is defined as the minimal (horizontal or vertical) distance between two of its parallel edges. If the polygons are convex, this distance is internal. The perimeter generating functions for such convex polygons, whose widths exceed a threshold, can be given explicitly. From these expressions, a two-variable (width and perimeter) generating function can be constructed. The corresponding phase diagram shows two types of critical behaviour, which meet at a tricritical point.