## Abstract

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 twice the sum of the side lengths of the smallest hypercube containing it. The number ofd-dimensional convex polygonsp_{n,d}of length 2nwithd(n)→∞ is asymptoticallyp_{n,d}~exp-2(2n-d)2n-1(2n-1)!(2πb(r)) ^{-1/2}r^{-2n+d}sinh^{d}r,wherer=r(n,d) is the unique solution ofrcothr=2n/d-1andb(r)=d(rcothr-r^{2}/sinh^{2}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.

Original language | English |
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Pages (from-to) | 196-217 |

Number of pages | 22 |

Journal | Journal of Combinatorial Theory. Series A |

Volume | 80 |

Issue number | 2 |

DOIs | |

Publication status | Published - Nov 1997 |

Externally published | Yes |