Convex lattice polygons of fixed area with perimeter-dependent weights

We study fully convex polygons with a given area, and variable perimeter length on square and hexagonal lattices. We attach a weight tm to a convex polygon of perimeter m and show that the sum of weights of all polygons with a fixed area s varies as s-θconveK(t)?s for large s and t less than a critical threshold tc, where K(t) is a t-dependent constant, and θconv is a critical exponent which does not change with t. Using heuristic arguments, we find that θconv is 1⁄4 for the square lattice, but -1⁄4 for the hexagonal lattice. The reason for this unexpected nonuniversality of θconv is traced to existence of sharp corners in the asymptotic shape of these polygons