Asymptotic Behavior of Inflated Lattice Polygons

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