Asymptotic behaviour of convex and column-convex lattice polygons with fixed area and varying perimeter

TL;DR

This paper analyzes the asymptotic behavior of convex and column-convex lattice polygons with fixed area and variable perimeter, revealing a universal constant K(J) influencing their mean perimeter as the fugacity approaches zero.

Contribution

It derives the asymptotic form of the mean perimeter for convex and column-convex polygons, showing the same constant K(J) for both, indicating universality and potential extension to self-avoiding polygons.

Findings

Mean perimeter scales as 1 - K(J)/( ln a0)^2 for small a0a0

Constant K(J) is identical for convex and column-convex polygons

Suggests similar asymptotic behavior for self-avoiding polygons

Abstract

We study the inflated phase of two dimensional lattice polygons, both convex and column-convex, with fixed area A and variable perimeter, when a weight \mu^t \exp[- Jb] is associated to a polygon with perimeter t and b bends. The mean perimeter is calculated as a function of the fugacity \mu and the bending rigidity J. In the limit \mu -> 0, the mean perimeter has the asymptotic behaviour \avg{t}/4 \sqrt{A} \simeq 1 - K(J)/(\ln \mu)^2 + O (\mu/ \ln \mu) . The constant K(J) is found to be the same for both types of polygons, suggesting that self-avoiding polygons should also exhibit the same asymptotic behaviour.