Bounds on Multiple Self-avoiding Polygons

Abstract

A self-avoiding polygon is a lattice polygon consisting of a closed self-avoiding walk on a square lattice. Surprisingly little is known rigorously about the enumeration of self-avoiding polygons, although there are numerous conjectures that are believed to be true and strongly supported by numerical simulations. As an analogous problem to this study, we consider multiple self-avoiding polygons in a confined region as a model for multiple ring polymers in physics. We find rigorous lower and upper bounds for the number p(mxn) of distinct multiple self-avoiding polygons in the m x n rectangular grid on the square lattice. For m = 2, p(2xn) = 2(n-1) - 1. And for integers m, n >= 3, 2(m+n-3) (17/10)((m -2) (n-2)) <= p(mxn) <= 2(m+n-3) (13/16)((m-2) (n-2)).