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| Format: | Preprint |
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2014
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| Online Access: | https://arxiv.org/abs/1410.8059 |
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| _version_ | 1866910458809679872 |
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| author | Kong, Yong |
| author_facet | Kong, Yong |
| contents | We use computational method to investigate the number of ways to pack dimers on \emph{odd-by-odd} lattices. In this case, there is always a single vacancy in the lattices. We show that the dimer configuration numbers on $(2k+1) \times (2k+1)$ \emph{odd} square lattices have some remarkable number-theoretical properties in parallel to those of close-packed dimers on $2k \times 2k$ \emph{even} square lattices, for which exact solution exists. Furthermore, we demonstrate that there is an unambiguous logarithm term in the finite size correction of free energy of odd-by-odd lattice strips with any width $n \ge 1$. This logarithm term determines the distinct behavior of the free energy of odd square lattices. These findings reveal a deep and previously unexplored connection between statistical physics models and number theory, and indicate the possibility that the monomer-dimer problem might be solvable. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1410_8059 |
| institution | arXiv |
| publishDate | 2014 |
| record_format | arxiv |
| spellingShingle | Packing dimers on $(2p + 1) \times (2q + 1) $ lattices Kong, Yong Statistical Mechanics Combinatorics We use computational method to investigate the number of ways to pack dimers on \emph{odd-by-odd} lattices. In this case, there is always a single vacancy in the lattices. We show that the dimer configuration numbers on $(2k+1) \times (2k+1)$ \emph{odd} square lattices have some remarkable number-theoretical properties in parallel to those of close-packed dimers on $2k \times 2k$ \emph{even} square lattices, for which exact solution exists. Furthermore, we demonstrate that there is an unambiguous logarithm term in the finite size correction of free energy of odd-by-odd lattice strips with any width $n \ge 1$. This logarithm term determines the distinct behavior of the free energy of odd square lattices. These findings reveal a deep and previously unexplored connection between statistical physics models and number theory, and indicate the possibility that the monomer-dimer problem might be solvable. |
| title | Packing dimers on $(2p + 1) \times (2q + 1) $ lattices |
| topic | Statistical Mechanics Combinatorics |
| url | https://arxiv.org/abs/1410.8059 |