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 mu(t) exp[-Jb] is associated with a polygon with perimeter t and b bends. The mean perimeter is calculated as a function of the fugacity mu and the bending rigidity J. In the limit mu -> 0, the mean perimeter has the asymptotic behaviour < t > / 4 root A similar or equal to 1 - K(J)/(ln mu)(2) + O(mu/ln mu). The constant K(J) is found to be the same for both kinds of polygons, suggesting that self-avoiding polygons may also exhibit the same asymptotic behaviour.