Bounds on multiple self-avoiding polygons

TL;DR

This paper establishes rigorous bounds on the number of multiple self-avoiding polygons within rectangular grids, advancing understanding of their enumeration and modeling multiple ring polymers.

Contribution

It provides the first rigorous lower and upper bounds for the count of multiple self-avoiding polygons in rectangular lattice regions.

Findings

Exact count for 2 x n polygons: 2^{n-1}-1.

Bounds for m,n ≥ 3: lower bound involves (17/10)^{(m-2)(n-2)}.

Bounds for m,n ≥ 3: upper bound involves (31/16)^{(m-2)(n-2)}.

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 $m,n \geq 3$, $$2^{m+n-3} \left(\frac{17}{10}\right)^{(m-2)(n-2)} \ \leq \ p_{m \times n} \ \leq \ 2^{m+n-3} \left(\frac{31}{16}\right)^{(m-2)(n-2)}.$$