Percolation critical polynomial as a graph invariant
Abstract
Every lattice for which the bond percolation critical probability can be found exactly possesses a critical polynomial, with the root in [0; 1] providing the threshold. Recent work has demonstrated that this polynomial may be generalized through a definition that can be applied on any periodic lattice. The polynomial depends on the lattice and on its decomposition into identical finite subgraphs, but once these are specified, the polynomial is essentially unique. On lattices for which the exact percolation threshold is unknown, the polynomials provide approximations for the critical probability with the estimates appearing to converge to the exact answer with increasing subgraph size. In this paper, I show how the critical polynomial can be viewed as a graph invariant like the Tutte polynomial. In particular, the critical polynomial is computed on a finite graph and may be found using the deletioncontraction algorithm. This allows calculation on a computer, and I present such results for the kagome lattice using subgraphs of up to 36 bonds. For one of these, I find the prediction p_{c} = 0:52440572:::, which differs from the numerical value, p_{c} = 0:52440503(5), by only 6:9 X 10^{7}.
 Authors:

 Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)
 Publication Date:
 Research Org.:
 Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)
 Sponsoring Org.:
 USDOE
 OSTI Identifier:
 1399716
 Alternate Identifier(s):
 OSTI ID: 1101627
 Report Number(s):
 LLNLJRNL520672
Journal ID: ISSN 15393755; PLEEE8
 Grant/Contract Number:
 AC5207NA27344
 Resource Type:
 Accepted Manuscript
 Journal Name:
 Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
 Additional Journal Information:
 Journal Volume: 86; Journal Issue: 4; Journal ID: ISSN 15393755
 Publisher:
 American Physical Society (APS)
 Country of Publication:
 United States
 Language:
 English
 Subject:
 97 MATHEMATICS AND COMPUTING
Citation Formats
Scullard, Christian R. Percolation critical polynomial as a graph invariant. United States: N. p., 2012.
Web. doi:10.1103/PhysRevE.86.041131.
Scullard, Christian R. Percolation critical polynomial as a graph invariant. United States. doi:10.1103/PhysRevE.86.041131.
Scullard, Christian R. Thu .
"Percolation critical polynomial as a graph invariant". United States. doi:10.1103/PhysRevE.86.041131. https://www.osti.gov/servlets/purl/1399716.
@article{osti_1399716,
title = {Percolation critical polynomial as a graph invariant},
author = {Scullard, Christian R.},
abstractNote = {Every lattice for which the bond percolation critical probability can be found exactly possesses a critical polynomial, with the root in [0; 1] providing the threshold. Recent work has demonstrated that this polynomial may be generalized through a definition that can be applied on any periodic lattice. The polynomial depends on the lattice and on its decomposition into identical finite subgraphs, but once these are specified, the polynomial is essentially unique. On lattices for which the exact percolation threshold is unknown, the polynomials provide approximations for the critical probability with the estimates appearing to converge to the exact answer with increasing subgraph size. In this paper, I show how the critical polynomial can be viewed as a graph invariant like the Tutte polynomial. In particular, the critical polynomial is computed on a finite graph and may be found using the deletioncontraction algorithm. This allows calculation on a computer, and I present such results for the kagome lattice using subgraphs of up to 36 bonds. For one of these, I find the prediction pc = 0:52440572:::, which differs from the numerical value, pc = 0:52440503(5), by only 6:9 X 107.},
doi = {10.1103/PhysRevE.86.041131},
journal = {Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics},
number = 4,
volume = 86,
place = {United States},
year = {2012},
month = {10}
}
Web of Science
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Works referencing / citing this record:
Critical manifold of the kagomelattice Potts model
journal, November 2012
 Jacobsen, Jesper Lykke; Scullard, Christian R.
 Journal of Physics A: Mathematical and Theoretical, Vol. 45, Issue 49
Transfer matrix computation of generalized critical polynomials in percolation
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 Scullard, Christian R.; Jacobsen, Jesper Lykke
 Journal of Physics A: Mathematical and Theoretical, Vol. 45, Issue 49
Transfer matrix computation of critical polynomials for twodimensional Potts models
journal, February 2013
 Jacobsen, Jesper Lykke; Scullard, Christian R.
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Highprecision percolation thresholds and Pottsmodel critical manifolds from graph polynomials
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 Jacobsen, Jesper Lykke
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Pottsmodel critical manifolds revisited
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 Journal of Physics A: Mathematical and Theoretical, Vol. 49, Issue 12
On the growth constant for squarelattice selfavoiding walks
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 Journal of Physics A: Mathematical and Theoretical, Vol. 49, Issue 49