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

References in corpus (7)

• A parallel algorithm for the enumeration of self-avoiding polygons on the square lattice
• Scaling behaviour of two-dimensional polygon models
• Scaling function and universal amplitude combinations for self-avoiding polygons
• Exact Scaling Functions for Self-Avoiding Loops and Branched Polymers
• Convex lattice polygons of fixed area with perimeter dependent weights
• On the Asymptotics of the Finite-Perimeter Partition Function of Two-Dimensional Lattice Vesicles
• Smoothening Transition of a Two-Dimensional Pressurized Polymer Ring

Cited by in corpus (1)

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