Asymptotic Behavior of Inflated Lattice Polygons

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 $J\ne 0$. We also study polygons where self-intersections are allowed, verifying numerically that the asymptotic behavior described above continues to hold.

7 pages