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 t^m 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^{-theta_{conv}} exp[K s^(1/2)] for large s and t less than a critical threshold t_c, where K is a t-dependent constant, and theta_{conv} is a critical exponent which does not change with t. We find theta_{conv} is 1/4 for the square lattice, but -1/4 for the hexagonal lattice. The reason for this unexpected non-universality of theta_{conv} is traced to existence of sharp corners in the asymptotic shape of these polygons.