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 calculate the average area for positive values of the pressure. For large pressures, the area has the asymptotic behaviour [A]/A(max) = 1 - K(J)/(p) over tilde (2) + O(rho(-(p) over tilde)), where (p) over tilde = pN >> 1, and rho <1. 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 not equal 0. We also study polygons where self-intersections are allowed, verifying numerically that the asymptotic behavior described above continues to hold.