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

We study the inflated phase of two dimensional lattice polygons, both convex and column-convex, with fixed area A and variable perimeter, when a weight μ^t \exp[- Jb] is associated to a polygon with perimeter t and b bends. The mean perimeter is calculated as a function of the fugacity μand the bending rigidity J. In the limit μ-> 0, the mean perimeter has the asymptotic behaviour \avg{t}/4 \sqrt{A} \simeq 1 - K(J)/(\ln μ)^2 + O (μ/ \ln μ) . The constant K(J) is found to be the same for both types of polygons, suggesting that self-avoiding polygons should also exhibit the same asymptotic behaviour.