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 of 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 of the number $p_{m\times n}$ of distinct multiple self-avoiding polygons in the $m\times n$ rectangular grid on the square lattice. For $m=2$, $p_{2\times n}=2^{n-1}-1$. And, for integers $m,n\ge 3$, $$2^{m+n-3}{\left(\frac{17}{10}\right)}^{(m-2)(n-2)}\le p_{m\times n}\le 2^{m+n-3}{\left(\frac{31}{16}\right)}^{(m-2)(n-2)}.$$