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 In and show that the sum of weights of all polygons with a fixed area s varies as s(-thetaconv)e(K(t)roots) for large s and t less than a critical threshold t(c), where K(t) is a t-dependent constant, and theta(conv) is a critical exponent which does not change with I. Using heuristic arguments, we find that theta(conv) is 1/4 for the square lattice, but -1/4 for the hexagonal lattice. The reason for this unexpected nonuniversality of theta(conv) is traced to existence of sharp corners in the asymptotic shape of these polygons.