Upper bounds on the growth rates of hard squares and related models via corner transfer matrices
We study the growth rate of the hard squares lattice gas, equivalent to the number of independent sets on the square lattice, and two related models — non-attacking kings and read-write isolated memory. We use an assortment of techniques from combinatorics, statistical mechanics and linear algebra...
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| Format: | Article |
| Language: | English |
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Discrete Mathematics & Theoretical Computer Science
2015-01-01
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| Series: | Discrete Mathematics & Theoretical Computer Science |
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| Online Access: | https://dmtcs.episciences.org/2486/pdf |
| _version_ | 1797270203280654336 |
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| author | Yao-Ban Chan |
| author_facet | Yao-Ban Chan |
| author_sort | Yao-Ban Chan |
| collection | DOAJ |
| description | We study the growth rate of the hard squares lattice gas, equivalent to the number of independent sets on the square lattice, and two related models — non-attacking kings and read-write isolated memory. We use an assortment of techniques from combinatorics, statistical mechanics and linear algebra to prove upper bounds on these growth rates. We start from Calkin and Wilf’s transfer matrix eigenvalue bound, then bound that with the Collatz-Wielandt formula from linear algebra. To obtain an approximate eigenvector, we use an ansatz from Baxter’s corner transfer matrix formalism, optimised with Nishino and Okunishi’s corner transfer matrix renormalisation group method. This results in an upper bound algorithm which no longer requires exponential memory and so is much faster to calculate than a direct evaluation of the Calkin-Wilf bound. Furthermore, it is extremely parallelisable and so allows us to make dramatic improvements to the previous best known upper bounds. In all cases we reduce the gap between upper and lower bounds by 4-6 orders of magnitude. |
| first_indexed | 2024-04-25T02:00:32Z |
| format | Article |
| id | doaj.art-61a68006be2e4afbaab48946dbad99af |
| institution | Directory Open Access Journal |
| issn | 1365-8050 |
| language | English |
| last_indexed | 2024-04-25T02:00:32Z |
| publishDate | 2015-01-01 |
| publisher | Discrete Mathematics & Theoretical Computer Science |
| record_format | Article |
| series | Discrete Mathematics & Theoretical Computer Science |
| spelling | doaj.art-61a68006be2e4afbaab48946dbad99af2024-03-07T15:01:26ZengDiscrete Mathematics & Theoretical Computer ScienceDiscrete Mathematics & Theoretical Computer Science1365-80502015-01-01DMTCS Proceedings, 27th...Proceedings10.46298/dmtcs.24862486Upper bounds on the growth rates of hard squares and related models via corner transfer matricesYao-Ban Chan0School of Mathematics and Physics [Brisbane]We study the growth rate of the hard squares lattice gas, equivalent to the number of independent sets on the square lattice, and two related models — non-attacking kings and read-write isolated memory. We use an assortment of techniques from combinatorics, statistical mechanics and linear algebra to prove upper bounds on these growth rates. We start from Calkin and Wilf’s transfer matrix eigenvalue bound, then bound that with the Collatz-Wielandt formula from linear algebra. To obtain an approximate eigenvector, we use an ansatz from Baxter’s corner transfer matrix formalism, optimised with Nishino and Okunishi’s corner transfer matrix renormalisation group method. This results in an upper bound algorithm which no longer requires exponential memory and so is much faster to calculate than a direct evaluation of the Calkin-Wilf bound. Furthermore, it is extremely parallelisable and so allows us to make dramatic improvements to the previous best known upper bounds. In all cases we reduce the gap between upper and lower bounds by 4-6 orders of magnitude.https://dmtcs.episciences.org/2486/pdftransfer matrixgrowth ratehard squaresupper bound[info.info-dm] computer science [cs]/discrete mathematics [cs.dm] |
| spellingShingle | Yao-Ban Chan Upper bounds on the growth rates of hard squares and related models via corner transfer matrices Discrete Mathematics & Theoretical Computer Science transfer matrix growth rate hard squares upper bound [info.info-dm] computer science [cs]/discrete mathematics [cs.dm] |
| title | Upper bounds on the growth rates of hard squares and related models via corner transfer matrices |
| title_full | Upper bounds on the growth rates of hard squares and related models via corner transfer matrices |
| title_fullStr | Upper bounds on the growth rates of hard squares and related models via corner transfer matrices |
| title_full_unstemmed | Upper bounds on the growth rates of hard squares and related models via corner transfer matrices |
| title_short | Upper bounds on the growth rates of hard squares and related models via corner transfer matrices |
| title_sort | upper bounds on the growth rates of hard squares and related models via corner transfer matrices |
| topic | transfer matrix growth rate hard squares upper bound [info.info-dm] computer science [cs]/discrete mathematics [cs.dm] |
| url | https://dmtcs.episciences.org/2486/pdf |
| work_keys_str_mv | AT yaobanchan upperboundsonthegrowthratesofhardsquaresandrelatedmodelsviacornertransfermatrices |