Asymptotic enumeration of convex polygons (Q1374182)

A polygon is a self-avoiding cycle in the hypercubic lattice \(\mathbb{Z}^d\) taking at least one step in every dimension. It is convex if its length is exactly twice the sum of the side lengths of the smallest hypercube containing it. An asymptotic expression is given for the number \(p_{n,d}\) of \(d\)-dimensional convex polygons of length \(2n\) with \(d(n)\to\infty\). Results are proved by asymptotically enumerating a larger class of objects called convex proto-polygons by the saddle-point method and then finding the asymptotic probability that a randomly chosen proto-polygon is a polygon.