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Bounds on Multiple Self-avoiding Polygons
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- Journal:
- Canadian Mathematical Bulletin / Volume 61 / Issue 3 / 01 September 2018
- Published online by Cambridge University Press:
- 20 November 2018, pp. 518-530
- Print publication:
- 01 September 2018
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- Article
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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 problemto this study, we considermultiple 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
${{p}_{m\times n}}$ 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 
$m,\,n\,\ge \,3$,
$${{2}^{m+n-3}}\left( \frac{17}{10} \right){{\,}^{\left( m-2 \right)\left( n-2 \right)}}\,\le \,{{p}_{m\times n}}\,\le \,{{2}^{m+n-3}}\left( \frac{31}{16} \right){{\,}^{\left( m-2 \right)\left( n-2 \right)}}.$$
