Scaling prediction for self-avoiding polygons revisited

TL;DR

This paper provides a detailed analysis of self-avoiding polygons on various lattices, focusing on perimeter moments and the critical curve near the bicritical point, supporting a conjectured scaling function and identifying a new universal amplitude.

Contribution

It extends previous analyses with new enumeration data, supports a conjectured scaling function, and identifies a new universal amplitude in self-avoiding polygons.

Findings

Support for the conjectured scaling function of rooted polygons

Identification of an additional universal amplitude in unrooted polygons

Analysis of the shape of the critical curve near the bicritical point

Abstract

We analyse new exact enumeration data for self-avoiding polygons, counted by perimeter and area on the square, triangular and hexagonal lattices. In extending earlier analyses, we focus on the perimeter moments in the vicinity of the bicritical point. We also consider the shape of the critical curve near the bicritical point, which describes the crossover to the branched polymer phase. Our recently conjectured expression for the scaling function of rooted self-avoiding polygons is further supported. For (unrooted) self-avoiding polygons, the analysis reveals the presence of an additional additive term with a new universal amplitude. We conjecture the exact value of this amplitude.