Polynomial sequences for bond percolation critical thresholds

Scullard, Christian R.

doi:10.1088/1742-5468/2011/09/P09022

Title: Polynomial sequences for bond percolation critical thresholds

Abstract

In this paper, I compute the inhomogeneous (multi-probability) bond critical surfaces for the (4, 6, 12) and (3⁴, 6) using the linearity approximation described in (Scullard and Ziff, J. Stat. Mech. 03021), implemented as a branching process of lattices. I find the estimates for the bond percolation thresholds, pc(4, 6, 12) = 0.69377849... and p_c(3⁴, 6) = 0.43437077..., compared with Parviainen’s numerical results of p_c = 0.69373383... and p_c = 0.43430621... . These deviations are of the order 10^-5, as is standard for this method. Deriving thresholds in this way for a given lattice leads to a polynomial with integer coefficients, the root in [0, 1] of which gives the estimate for the bond threshold and I show how the method can be refined, leading to a series of higher order polynomials making predictions that likely converge to the exact answer. Finally, I discuss how this fact hints that for certain graphs, such as the kagome lattice, the exact bond threshold may not be the root of any polynomial with integer coefficients.

Authors:

Scullard, Christian R. ^[1]

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

Publication Date:: Thu Sep 22 00:00:00 EDT 2011

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

Sponsoring Org.:: USDOE

OSTI Identifier:: 1399718

Report Number(s):: LLNL-JRNL-470576
Journal ID: ISSN 1742-5468

Grant/Contract Number:: AC52-07NA27344

Resource Type:: Accepted Manuscript

Journal Name:: Journal of Statistical Mechanics

Additional Journal Information:: Journal Volume: 2011; Journal Issue: 09; Journal ID: ISSN 1742-5468

Publisher:: IOP Publishing

Country of Publication:: United States

Language:: English

Subject:: 97 MATHEMATICS AND COMPUTING

Citation Formats


                    Scullard, Christian R. Polynomial sequences for bond percolation critical thresholds.  United States: N. p., 2011. 
Web.  doi:10.1088/1742-5468/2011/09/P09022.

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                    Scullard, Christian R. Polynomial sequences for bond percolation critical thresholds.  United States.  https://doi.org/10.1088/1742-5468/2011/09/P09022

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                    Scullard, Christian R. Thu .  
"Polynomial sequences for bond percolation critical thresholds".  United States.  https://doi.org/10.1088/1742-5468/2011/09/P09022.  https://www.osti.gov/servlets/purl/1399718.

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@article{osti_1399718,

  title        = {Polynomial sequences for bond percolation critical thresholds},

  author       = {Scullard, Christian R.},

  abstractNote = {In this paper, I compute the inhomogeneous (multi-probability) bond critical surfaces for the (4, 6, 12) and (34, 6) using the linearity approximation described in (Scullard and Ziff, J. Stat. Mech. 03021), implemented as a branching process of lattices. I find the estimates for the bond percolation thresholds, pc(4, 6, 12) = 0.69377849... and pc(34, 6) = 0.43437077..., compared with Parviainen’s numerical results of pc = 0.69373383... and pc = 0.43430621... . These deviations are of the order 10-5, as is standard for this method. Deriving thresholds in this way for a given lattice leads to a polynomial with integer coefficients, the root in [0, 1] of which gives the estimate for the bond threshold and I show how the method can be refined, leading to a series of higher order polynomials making predictions that likely converge to the exact answer. Finally, I discuss how this fact hints that for certain graphs, such as the kagome lattice, the exact bond threshold may not be the root of any polynomial with integer coefficients.},

  doi          = {10.1088/1742-5468/2011/09/P09022},

  journal      = {Journal of Statistical Mechanics},

  number       = 09,

  volume       = 2011,

  place        = {United States},

  year         = {Thu Sep 22 00:00:00 EDT 2011},

  month        = {Thu Sep 22 00:00:00 EDT 2011}

}

Copy to clipboard

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Works referenced in this record:

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Critical surfaces for general inhomogeneous bond percolation problems
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Works referencing / citing this record:

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

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

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

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

On bond percolation threshold bounds for Archimedean lattices with degree three
journal, June 2017

Wierman, John C.
Journal of Physics A: Mathematical and Theoretical, Vol. 50, Issue 29
DOI: 10.1088/1751-8121/aa76f0

Critical manifold of the kagome-lattice Potts model
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Jacobsen, Jesper Lykke; Scullard, Christian R.
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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

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Similar Records

Title: Polynomial sequences for bond percolation critical thresholds

Abstract

Citation Formats

Estimation of bond percolation thresholds on the Archimedean lattices journal, July 2007

Critical frontier of the Potts and percolation models on triangular-type and kagome-type lattices. I. Closed-form expressions journal, June 2010

Percolation thresholds on two-dimensional Voronoi networks and Delaunay triangulations journal, October 2009

Factorization of percolation density correlation functions for clusters touching the sides of a rectangle journal, February 2009

A bond percolation critical probability determination based on the star-triangle transformation journal, May 1984

Universal crossing probability in anisotropic systems journal, October 2002

Infinite conformal symmetry in two-dimensional quantum field theory journal, July 1984

Critical percolation in finite geometries journal, February 1992

Stochastic geometry of critical curves, Schramm–Loewner evolutions and conformal field theory journal, September 2006

Critical point of planar Potts models journal, September 1979

Equality of bond-percolation critical exponents for pairs of dual lattices journal, May 2009

Percolation in networks with voids and bottlenecks journal, February 2009

Predictions of bond percolation thresholds for the kagomé and Archimedean ( 3 , 12 2 ) lattices journal, April 2006

Exact factorization of correlation functions in two-dimensional critical percolation journal, October 2007

Universal condition for critical percolation thresholds of kagomé-like lattices journal, February 2009

The critical probability for random Voronoi percolation in the plane is 1/2 journal, December 2005

Critical percolation of virtually free groups and other tree-like graphs journal, November 2009

Percolation processes: I. Crystals and mazes journal, July 1957

Exact site percolation thresholds using a site-to-bond transformation and the star-triangle transformation journal, January 2006

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

Rigorous confidence intervals for critical probabilities journal, July 2007

Critical percolation in the plane: conformal invariance, Cardy's formula, scaling limits journal, August 2001

Percolation transitions in two dimensions journal, September 2008

Critical Surfaces for General Bond Percolation Problems journal, May 2008

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

Exact Results for Strongly Correlated Fermions in 2 + 1 Dimensions journal, July 2005

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 text, January 2009

Equality of bond percolation critical exponents for pairs of dual lattices text, January 2009

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

Rigorous confidence intervals for critical probabilities text, January 2007

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

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

Potts-model critical manifolds revisited journal, February 2016

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

On bond percolation threshold bounds for Archimedean lattices with degree three journal, June 2017

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

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

Estimation of bond percolation thresholds on the Archimedean lattices
journal, July 2007

Critical frontier of the Potts and percolation models on triangular-type and kagome-type lattices. I. Closed-form expressions
journal, June 2010

Percolation thresholds on two-dimensional Voronoi networks and Delaunay triangulations
journal, October 2009

Factorization of percolation density correlation functions for clusters touching the sides of a rectangle
journal, February 2009

A bond percolation critical probability determination based on the star-triangle transformation
journal, May 1984

Universal crossing probability in anisotropic systems
journal, October 2002

Infinite conformal symmetry in two-dimensional quantum field theory
journal, July 1984

Critical percolation in finite geometries
journal, February 1992

Stochastic geometry of critical curves, Schramm–Loewner evolutions and conformal field theory
journal, September 2006

Critical point of planar Potts models
journal, September 1979

Equality of bond-percolation critical exponents for pairs of dual lattices
journal, May 2009

Percolation in networks with voids and bottlenecks
journal, February 2009

Predictions of bond percolation thresholds for the kagomé and Archimedean $(3, 12^{2})$ lattices
journal, April 2006

Exact factorization of correlation functions in two-dimensional critical percolation
journal, October 2007

Universal condition for critical percolation thresholds of kagomé-like lattices
journal, February 2009

The critical probability for random Voronoi percolation in the plane is 1/2
journal, December 2005

Critical percolation of virtually free groups and other tree-like graphs
journal, November 2009

Percolation processes: I. Crystals and mazes
journal, July 1957

Exact site percolation thresholds using a site-to-bond transformation and the star-triangle transformation
journal, January 2006

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

Rigorous confidence intervals for critical probabilities
journal, July 2007

Critical percolation in the plane: conformal invariance, Cardy's formula, scaling limits
journal, August 2001

Percolation transitions in two dimensions
journal, September 2008

Critical Surfaces for General Bond Percolation Problems
journal, May 2008

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

Exact Results for Strongly Correlated Fermions in $2 + 1$ Dimensions
journal, July 2005

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
text, January 2009

Equality of bond percolation critical exponents for pairs of dual lattices
text, January 2009

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

Rigorous confidence intervals for critical probabilities
text, January 2007

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

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

Potts-model critical manifolds revisited
journal, February 2016

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

On bond percolation threshold bounds for Archimedean lattices with degree three
journal, June 2017

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

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