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 $s^{-{\theta }_{\mathit{conv}}}{e}^{K\left(t\right)\sqrt{s}}$ for large $s$ and $t$ less than a critical threshold $t_c$, where $K\left(t\right)$ is a $t$-dependent constant, and ${\theta }_{\mathit{conv}}$ is a critical exponent which does not change with $t$. Using heuristic arguments, we find that ${\theta }_{\mathit{conv}}$ is $1∕4$ for the square lattice, but $-1∕4$ for the hexagonal lattice. The reason for this unexpected nonuniversality of ${\theta }_{\mathit{conv}}$ is traced to existence of sharp corners in the asymptotic shape of these polygons.