Percolation critical polynomial as a graph invariant

Scullard, Christian R.

doi:10.1103/PhysRevE.86.041131

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 p_c = 0:52440572:::, which differs from the numerical value, p_c = 0:52440503(5), by only 6:9 X 10^-7.

Authors:

Scullard, Christian R. ^[1]

Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)

Publication Date:: Thu Oct 18 00:00:00 EDT 2012

Research Org.:: Lawrence Livermore National Laboratory (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

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                    Scullard, Christian R. Percolation critical polynomial as a graph invariant.  United States: N. p., 2012. 
Web.  doi:10.1103/PhysRevE.86.041131.

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                    Scullard, Christian R. Percolation critical polynomial as a graph invariant.  United States.  https://doi.org/10.1103/PhysRevE.86.041131

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                    Scullard, Christian R. Thu .  
"Percolation critical polynomial as a graph invariant".  United States.  https://doi.org/10.1103/PhysRevE.86.041131.  https://www.osti.gov/servlets/purl/1399716.

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@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         = {Thu Oct 18 00:00:00 EDT 2012},

  month        = {Thu Oct 18 00:00:00 EDT 2012}

}

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Works referenced in this record:

Critical frontier of the Potts and percolation models on triangular-type and kagome-type lattices. II. Numerical analysis
journal, June 2010

Ding, Chengxiang; Fu, Zhe; Guo, Wenan
Physical Review E, Vol. 81, Issue 6
DOI: 10.1103/PhysRevE.81.061111

Percolation transitions in two dimensions
journal, September 2008

Feng, Xiaomei; Deng, Youjin; Blöte, Henk W. J.
Physical Review E, Vol. 78, Issue 3
DOI: 10.1103/PhysRevE.78.031136

Critical Surfaces for General Bond Percolation Problems
journal, May 2008

Scullard, Christian R.; Ziff, Robert M.
Physical Review Letters, Vol. 100, Issue 18
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Polynomial sequences for bond percolation critical thresholds
journal, September 2011

Scullard, Christian R.
Journal of Statistical Mechanics: Theory and Experiment, Vol. 2011, Issue 09
DOI: 10.1088/1742-5468/2011/09/P09022

Generalized cell–dual-cell transformation and exact thresholds for percolation
journal, January 2006

Ziff, Robert M.
Physical Review E, Vol. 73, Issue 1
DOI: 10.1103/PhysRevE.73.016134

Critical surfaces for general inhomogeneous bond percolation problems
journal, March 2010

Scullard, Christian R.; Ziff, Robert M.
Journal of Statistical Mechanics: Theory and Experiment, Vol. 2010, Issue 03
DOI: 10.1088/1742-5468/2010/03/P03021

Critical point of planar Potts models
journal, September 1979

Wu, F. Y.
Journal of Physics C: Solid State Physics, Vol. 12, Issue 17
DOI: 10.1088/0022-3719/12/17/002

Phase diagram of anisotropic planar Potts ferromagnets: a new conjecture
journal, August 1982

Tsallis, C.
Journal of Physics C: Solid State Physics, Vol. 15, Issue 23
DOI: 10.1088/0022-3719/15/23/007

Modern Graph Theory
journal, December 2000

Diaconis, Persi; Bollobas, Bela
Journal of the American Statistical Association, Vol. 95, Issue 452
DOI: 10.2307/2669801

Percolation transitions in two dimensions
text, January 2009

Feng, Xiaomei; Deng, Youjin; Blote, Henk W. J.
arXiv
DOI: 10.48550/arxiv.0901.1370

Critical frontier for the Potts and percolation models on triangular-type and kagome-type lattices II: Numerical analysis
text, January 2010

Ding, Chengxiang; Fu, Zhe; Guo, Wenan
arXiv
DOI: 10.48550/arxiv.1001.1488

Polynomial sequences for bond percolation critical thresholds
text, January 2011

Scullard, Christian R.
arXiv
DOI: 10.48550/arxiv.1103.3540

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

Jacobsen, Jesper Lykke
Journal of Physics A: Mathematical and Theoretical, Vol. 47, Issue 13
DOI: 10.1088/1751-8113/47/13/135001

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

Critical manifold of the kagome-lattice Potts model
text, January 2012

Jacobsen, Jesper Lykke; Scullard, Christian R.
arXiv
DOI: 10.48550/arxiv.1204.0622

Transfer matrix computation of critical polynomials for two-dimensional Potts models
text, January 2012

Jacobsen, Jesper Lykke; Scullard, Christian R.
arXiv
DOI: 10.48550/arxiv.1211.4335

High-precision percolation thresholds and Potts-model critical manifolds from graph polynomials
preprint, January 2014

Jacobsen, Jesper Lykke
arXiv
DOI: 10.48550/arxiv.1401.7847

On the growth constant for square-lattice self-avoiding walks
text, January 2016

Jacobsen, Jesper Lykke; Scullard, Christian R.; Guttmann, Anthony J.
arXiv
DOI: 10.48550/arxiv.1607.02984

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Similar Records

Title: Percolation critical polynomial as a graph invariant

Abstract

Citation Formats

Critical frontier of the Potts and percolation models on triangular-type and kagome-type lattices. II. Numerical analysis journal, June 2010

Percolation transitions in two dimensions journal, September 2008

Critical Surfaces for General Bond Percolation Problems journal, May 2008

Polynomial sequences for bond percolation critical thresholds journal, September 2011

Generalized cell–dual-cell transformation and exact thresholds for percolation journal, January 2006

Critical surfaces for general inhomogeneous bond percolation problems journal, March 2010

Critical point of planar Potts models journal, September 1979

Phase diagram of anisotropic planar Potts ferromagnets: a new conjecture journal, August 1982

Modern Graph Theory journal, December 2000

Percolation transitions in two dimensions text, January 2009

Critical frontier for the Potts and percolation models on triangular-type and kagome-type lattices II: Numerical analysis text, January 2010

Polynomial sequences for bond percolation critical thresholds text, January 2011

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

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

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

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

Potts-model critical manifolds revisited journal, February 2016

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

Critical manifold of the kagome-lattice Potts model text, January 2012

Transfer matrix computation of critical polynomials for two-dimensional Potts models text, January 2012

High-precision percolation thresholds and Potts-model critical manifolds from graph polynomials preprint, January 2014

On the growth constant for square-lattice self-avoiding walks text, January 2016

Critical frontier of the Potts and percolation models on triangular-type and kagome-type lattices. II. Numerical analysis
journal, June 2010

Percolation transitions in two dimensions
journal, September 2008

Critical Surfaces for General Bond Percolation Problems
journal, May 2008

Polynomial sequences for bond percolation critical thresholds
journal, September 2011

Generalized cell–dual-cell transformation and exact thresholds for percolation
journal, January 2006

Critical surfaces for general inhomogeneous bond percolation problems
journal, March 2010

Critical point of planar Potts models
journal, September 1979

Phase diagram of anisotropic planar Potts ferromagnets: a new conjecture
journal, August 1982

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journal, December 2000

Percolation transitions in two dimensions
text, January 2009

Critical frontier for the Potts and percolation models on triangular-type and kagome-type lattices II: Numerical analysis
text, January 2010

Polynomial sequences for bond percolation critical thresholds
text, January 2011

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

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

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

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

Potts-model critical manifolds revisited
journal, February 2016

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

Critical manifold of the kagome-lattice Potts model
text, January 2012

Transfer matrix computation of critical polynomials for two-dimensional Potts models
text, January 2012

High-precision percolation thresholds and Potts-model critical manifolds from graph polynomials
preprint, January 2014

On the growth constant for square-lattice self-avoiding walks
text, January 2016