Convex lattice polygons of fixed area with perimeter-dependent weights

Abstract

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^-θconve^K(t)?s for large s and t less than a critical threshold t_c, where K(t) is a t-dependent constant, and θ_conv is a critical exponent which does not change with t. Using heuristic arguments, we find that θ_conv is 1⁄4 for the square lattice, but -1⁄4 for the hexagonal lattice. The reason for this unexpected nonuniversality of θ_conv is traced to existence of sharp corners in the asymptotic shape of these polygons.