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Search: MSC category 82B20 ( Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs )

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1. CMB Online first

Hong, Kyungpyo; Oh, Seungsang
 Bounds on multiple self-avoiding polygons 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 polygonCategories:57M25, 82B20, 82B41, 82D60

2. CMB 2009 (vol 52 pp. 9)

 On the Spectrum of an $n!\times n!$ Matrix Originating from Statistical Mechanics Let $R_n(\alpha)$ be the $n!\times n!$ matrix whose matrix elements $[R_n(\alpha)]_{\sigma\rho}$, with $\sigma$ and $\rho$ in the symmetric group $\sn$, are $\alpha^{\ell(\sigma\rho^{-1})}$ with $0<\alpha<1$, where $\ell(\pi)$ denotes the number of cycles in $\pi\in \sn$. We give the spectrum of $R_n$ and show that the ratio of the largest eigenvalue $\lambda_0$ to the second largest one (in absolute value) increases as a positive power of $n$ as $n\rightarrow \infty$. Keywords:symmetric group, representation theory, eigenvalue, statistical physicsCategories:20B30, 20C30, 15A18, 82B20, 82B28