Asymptotic Behavior of Inflated Lattice Polygons

Details Asymptotic Behavior of Inflated Lattice Polygons Asymptotic Behavior of Inflated Lattice Polygons

2007-10-08

arxiv.org/abs/0710.1509v1

J. Stat. Phys. Vol 131, pg 393 (2008).

Physics

Condensed Matter

Statistical Mechanics

7 pages

Scientific paper

10.1007/s10955-008-9512-4

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 $/A_{max} = 1 - K(J)/\tilde{p}^2 + \mathcal{O}(\rho^{-\tilde{p}})$, 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.

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