Asymptotic Enumeration of Convex Polygons

A polygon is an elementary (self-avoiding) cycle in the hypercubic lattice Zdtaking at least one step in every dimension. A polygon on Zdis said to be convex if its length is exactly twice the sum of the side lengths of the smallest hypercube containing it. The number ofd-dimensional convex polygonspn, dof length 2nwithd(n)→∞ is asymptoticallypn, d∼exp −2(2n−d)2n−1 (2n−1)! (2πb(r))−1/2 r−2n+d sinhd r,wherer=r(n, d) is the unique solution ofr coth r=2n/d−1andb(r)=d(r coth r−r2/sinh2 r). The convergence is uniform over alld⩾ω(n) for any functionω(n)→∞. Whendis constant the exponential is replaced with (1−d−1)2d. These results are proved by asymptotically enumerating a larger class of combinatorial objects calledconvex proto-polygonsby the saddle-point method and then finding the asymptotic probability a randomly chosen convex proto-polygon is a convex polygon.