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Title: 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 deletion-contraction 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 10-7.

Authors:
 [1]
  1. 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):
LLNL-JRNL-520672
Journal ID: ISSN 1539-3755; PLEEE8
Grant/Contract Number:  
AC52-07NA27344
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 1539-3755
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 deletion-contraction 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 10-7.},
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}
}

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Cited by: 9 works
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Works referenced in this record:

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journal, June 2010


Percolation transitions in two dimensions
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Critical surfaces for general inhomogeneous bond percolation problems
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    Works referencing / citing this record:

    Critical manifold of the kagome-lattice Potts model
    journal, November 2012

    • Jacobsen, Jesper Lykke; Scullard, Christian R.
    • Journal of Physics A: Mathematical and Theoretical, Vol. 45, Issue 49
    • DOI: 10.1088/1751-8113/45/49/494003

    Transfer matrix computation of generalized critical polynomials in percolation
    journal, November 2012

    • Scullard, Christian R.; Jacobsen, Jesper Lykke
    • Journal of Physics A: Mathematical and Theoretical, Vol. 45, Issue 49
    • DOI: 10.1088/1751-8113/45/49/494004

    Transfer matrix computation of critical polynomials for two-dimensional Potts models
    journal, February 2013

    • Jacobsen, Jesper Lykke; Scullard, Christian R.
    • Journal of Physics A: Mathematical and Theoretical, Vol. 46, Issue 7
    • DOI: 10.1088/1751-8113/46/7/075001

    High-precision percolation thresholds and Potts-model critical manifolds from graph polynomials
    journal, March 2014


    Potts-model critical manifolds revisited
    journal, February 2016

    • Scullard, Christian R.; Jacobsen, Jesper Lykke
    • Journal of Physics A: Mathematical and Theoretical, Vol. 49, Issue 12
    • DOI: 10.1088/1751-8113/49/12/125003

    On the growth constant for square-lattice self-avoiding walks
    journal, November 2016

    • Jacobsen, Jesper Lykke; Scullard, Christian R.; Guttmann, Anthony J.
    • Journal of Physics A: Mathematical and Theoretical, Vol. 49, Issue 49
    • DOI: 10.1088/1751-8113/49/49/494004