Asymptotic enumeration of convex polygons

A polygon is an elementary (self-avoiding) cycle in the hypercubic lattice Z(d) taking at least one step in every dimension. A polygon on Z(d) is said to be convex if its length is exactly tu ice the sum of the side lengths of the smallest hypercube containing it. The number of d-dimensional convex polygons p(n,d) of length 2n with d(n) --> infinity is asymptotically P-n,P-d similar to exp (2(2n - d)/-2n - 1) (2n - 1)! (2 pi b(r))(-1/2) r(-2n+d) sinh(d)r, where r = r(n, d) is the unique solution of r coth r = 2n/d - 1 and b(r) = d( r coth r - r(2)/sinh(2) r). The convergence is uniform over all d greater than or equal to omega(n) for any function omega(n) --> infinity. When d is constant the exponential is replaced with (1 - d(-1))(2d). These results are proved by asymptotically enumerating a larger class of combinatorial objects called convex proto-polygons by the saddle-point method and then finding the asymptotic probability a randomly chosen convex proto-polygon is a convex polygon. (C) 1997 Academic Press.