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# Bounds on multiple self-avoiding polygons

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Published:2018-01-16
Printed: Sep 2018
• Kyungpyo Hong,
National Institute for Mathematical Sciences, Daejeon 34047, Korea
• Seungsang Oh,
Department of Mathematics, Korea University, Seoul 02841, Korea
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## 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 $m,n \geq 3$, $$2^{m+n-3} \left(\tfrac{17}{10} \right)^{(m-2)(n-2)} \ \leq \ p_{m \times n} \ \leq \ 2^{m+n-3} \left(\tfrac{31}{16} \right)^{(m-2)(n-2)}.$$
 Keywords: ring polymer, self-avoiding polygon
 MSC Classifications: 57M25 - Knots and links in $S^3$ {For higher dimensions, see 57Q45} 82B20 - Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs 82B41 - Random walks, random surfaces, lattice animals, etc. [See also 60G50, 82C41] 82D60 - Polymers

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