Asymptotic Behavior of Inflated Lattice Polygons

TL;DR

This paper investigates the asymptotic behavior of inflated lattice polygons with fixed perimeter, revealing a universal correction term for the average area across convex, column-convex, and self-avoiding polygons, supported by enumeration and simulation data.

Contribution

It establishes a universal asymptotic formula for the area of inflated lattice polygons, applicable to various polygon types, including self-avoiding polygons, with validation through enumeration and Monte Carlo methods.

Findings

The average area approaches the maximum with a correction term proportional to 1/tilde{p}^2.

The correction constant K(J) is identical for convex and column-convex polygons.

Predictions match enumeration data for J=0 and Monte Carlo simulations for J≠0.

Abstract

We study the inflated phase of two dimensional lattice polygons with fixed perimeter $N$ and variable area, associating a weight $\exp[pA - Jb ]$ to a polygon with area $A$ and $b$ bends. For convex and column-convex polygons, we show that $<A >/A_{max} = 1 - K(J)/\tilde{p}^2 + \mathcal{O}(\rho^{-\tilde{p}})$, where $\tilde{p}=pN \gg 1$, and $\rho<1$. The constant $K(J)$ is found to be the same for both types of polygons. We argue that self-avoiding polygons should exhibit the same asymptotic behavior. For self-avoiding polygons, our predictions are in good agreement with exact enumeration data for J=0 and Monte Carlo simulations for $J \neq 0$. We also study polygons where self-intersections are allowed, verifying numerically that the asymptotic behavior described above continues to hold.