On the growth constant for square-lattice self-avoiding walks

Jacobsen, Jesper Lykke; Scullard, Christian R.; Guttmann, Anthony J.

doi:10.1088/1751-8113/49/49/494004

Title: On the growth constant for square-lattice self-avoiding walks

Abstract

The growth constant for two-dimensional self-avoiding walks on the honeycomb lattice was conjectured by Nienhuis in 1982, and since that time the corresponding results for the square and triangular lattices have been sought. For the square lattice, a possible conjecture was advanced by one of us (AJG) more than 20 years ago, based on the six significant digit estimate available at the time. This estimate has improved by a further six digits over the intervening decades, and the conjectured value continued to agree with the increasingly precise estimates. Here, we discuss the three most successful methods for estimating the growth constant, including the most recently developed topological transfer-matrix method, due to another of us (JLJ). We show this to be the most computationally efficient of the three methods, and by parallelising the algorithm we have estimated the growth constant significantly more precisely, incidentally ruling out the conjecture, which fails in the 12th digit. Our new estimate of the growth constant is μ(square) = 2.63815853032790 (3).

Authors:

^[1]; Scullard, Christian R. ^[2]; Guttmann, Anthony J. ^[3]

Normal Superior School (ENS)-PSL Research Univ., Paris (France). Lab. for Theoretical Physics (LPTENS); Sorbonne Univ., CNRS, Paris (France); Atomic Energy Commission (CEA), Saclay (France). Inst. of Theoretical Physics
Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)
Univ. of Melbourne (Australia). ARC Center of Excellence for Mathematics and Statistics of Complex Systems and Dept. of Mathematics and Statistics

Publication Date:: Thu Nov 17 00:00:00 EST 2016

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

Sponsoring Org.:: USDOE National Nuclear Security Administration (NNSA); Univ. Inst. of France (IUF); European Research Council (ERC); Australian Research Council (ARC)

OSTI Identifier:: 1459139

Alternate Identifier(s):: OSTI ID: 1332598

Report Number(s):: LLNL-JRNL-747715
Journal ID: ISSN 1751-8113; 932555; TRN: US1901543

Grant/Contract Number:: AC52-07NA27344; NuQFT; DP120100939

Resource Type:: Accepted Manuscript

Journal Name:: Journal of Physics. A, Mathematical and Theoretical

Additional Journal Information:: Journal Volume: 49; Journal Issue: 49; Journal ID: ISSN 1751-8113

Publisher:: IOP Publishing

Country of Publication:: United States

Language:: English

Subject:: 97 MATHEMATICS AND COMPUTING; 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS

Citation Formats


                    Jacobsen, Jesper Lykke, Scullard, Christian R., and Guttmann, Anthony J. On the growth constant for square-lattice self-avoiding walks.  United States: N. p., 2016. 
Web.  doi:10.1088/1751-8113/49/49/494004.

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                    Jacobsen, Jesper Lykke, Scullard, Christian R., & Guttmann, Anthony J. On the growth constant for square-lattice self-avoiding walks.  United States.  https://doi.org/10.1088/1751-8113/49/49/494004

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                    Jacobsen, Jesper Lykke, Scullard, Christian R., and Guttmann, Anthony J. Thu .  
"On the growth constant for square-lattice self-avoiding walks".  United States.  https://doi.org/10.1088/1751-8113/49/49/494004.  https://www.osti.gov/servlets/purl/1459139.

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@article{osti_1459139,

  title        = {On the growth constant for square-lattice self-avoiding walks},

  author       = {Jacobsen, Jesper Lykke and Scullard, Christian R. and Guttmann, Anthony J.},

  abstractNote = {The growth constant for two-dimensional self-avoiding walks on the honeycomb lattice was conjectured by Nienhuis in 1982, and since that time the corresponding results for the square and triangular lattices have been sought. For the square lattice, a possible conjecture was advanced by one of us (AJG) more than 20 years ago, based on the six significant digit estimate available at the time. This estimate has improved by a further six digits over the intervening decades, and the conjectured value continued to agree with the increasingly precise estimates. Here, we discuss the three most successful methods for estimating the growth constant, including the most recently developed topological transfer-matrix method, due to another of us (JLJ). We show this to be the most computationally efficient of the three methods, and by parallelising the algorithm we have estimated the growth constant significantly more precisely, incidentally ruling out the conjecture, which fails in the 12th digit. Our new estimate of the growth constant is μ(square) = 2.63815853032790 (3).},

  doi          = {10.1088/1751-8113/49/49/494004},

  journal      = {Journal of Physics. A, Mathematical and Theoretical},

  number       = 49,

  volume       = 49,

  place        = {United States},

  year         = {Thu Nov 17 00:00:00 EST 2016},

  month        = {Thu Nov 17 00:00:00 EST 2016}

}

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Works referenced in this record:

Critical points of Potts and O( N ) models from eigenvalue identities in periodic Temperley–Lieb algebras
journal, October 2015

Jacobsen, Jesper Lykke
Journal of Physics A: Mathematical and Theoretical, Vol. 48, Issue 45
DOI: 10.1088/1751-8113/48/45/454003

Fractal Geometry and Applications: A Jubilee of Benoît Mandelbrot, Part 2: Multifractals, Probability and Statistical Mechanics, Applications
book, January 2004

Lapidus, Michel L.; van Frankenhuijsen, Machiel
Proceedings of Symposia in Pure Mathematics
DOI: 10.1090/pspum/072.2/2112127

The connective constant of the honeycomb lattice equals sqrt(2+sqrt 2)
journal, January 2012

Duminil-Copin, Hugo; Smirnov, Stanislav
Annals of Mathematics, Vol. 175, Issue 3
DOI: 10.4007/annals.2012.175.3.14

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

High-precision phase diagram of spin glasses from duality analysis with real-space renormalization and graph polynomials
journal, February 2015

Ohzeki, Masayuki; Jacobsen, Jesper Lykke
Journal of Physics A: Mathematical and Theoretical, Vol. 48, Issue 9
DOI: 10.1088/1751-8113/48/9/095001

A new transfer-matrix algorithm for exact enumerations: self-avoiding polygons on the square lattice
journal, February 2012

Clisby, Nathan; Jensen, Iwan
Journal of Physics A: Mathematical and Theoretical, Vol. 45, Issue 11
DOI: 10.1088/1751-8113/45/11/115202

Self-avoiding polygons on the square lattice
journal, January 1999

Jensen, Iwan; Guttmann, Anthony J.
Journal of Physics A: Mathematical and General, Vol. 32, Issue 26
DOI: 10.1088/0305-4470/32/26/305

Exact Enumerations
book, January 2009

Enting, Ian G.; Jensen, Iwan
Polygons, Polyominoes and Polycubes
DOI: 10.1007/978-1-4020-9927-4_7

Generating functions for enumerating self-avoiding rings on the square lattice
journal, December 1980

Enting, I. G.
Journal of Physics A: Mathematical and General, Vol. 13, Issue 12
DOI: 10.1088/0305-4470/13/12/021

A parallel algorithm for the enumeration of self-avoiding polygons on the square lattice
journal, May 2003

Jensen, Iwan
Journal of Physics A: Mathematical and General, Vol. 36, Issue 21
DOI: 10.1088/0305-4470/36/21/304

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

Exact Critical Point and Critical Exponents of $O (n)$ Models in Two Dimensions
journal, October 1982

Nienhuis, Bernard
Physical Review Letters, Vol. 49, Issue 15
DOI: 10.1103/PhysRevLett.49.1062

Square Lattice Self-Avoiding Walks and Polygons
journal, December 2001

Guttmann, A. J.; Conway, A. R.
Annals of Combinatorics, Vol. 5, Issue 3
DOI: 10.1007/PL00013842

The number of polygons on a lattice
journal, July 1961

Hammersley, J. M.
Mathematical Proceedings of the Cambridge Philosophical Society, Vol. 57, Issue 3
DOI: 10.1017/S030500410003557X

Percolation critical polynomial as a graph invariant
journal, October 2012

Scullard, Christian R.
Physical Review E, Vol. 86, Issue 4
DOI: 10.1103/PhysRevE.86.041131

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 phase diagram of spin glasses from duality analysis with real-space renormalization and graph polynomials
text, January 2014

Ohzeki, Masayuki; Jacobsen, Jesper Lykke
arXiv
DOI: 10.48550/arxiv.1410.0166

Self-avoiding polygons on the square lattice
text, January 1999

Jensen, Iwan; Guttmann, Anthony J.
arXiv
DOI: 10.48550/arxiv.cond-mat/9905291

Exact Enumerations
book, January 2009

Enting, Ian G.; Jensen, Iwan
Polygons, Polyominoes and Polycubes
DOI: 10.1007/978-1-4020-9927-4_7

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

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

Works referencing / citing this record:

Two-dimensional interacting self-avoiding walks: new estimates for critical temperatures and exponents
text, January 2019

Beaton, Nicholas R.; Guttmann, Anthony J.; Jensen, Iwan
arXiv
DOI: 10.48550/arxiv.1911.05852

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Similar Records

Title: On the growth constant for square-lattice self-avoiding walks

Abstract

Citation Formats

Critical points of Potts and O( N ) models from eigenvalue identities in periodic Temperley–Lieb algebras journal, October 2015

Fractal Geometry and Applications: A Jubilee of Benoît Mandelbrot, Part 2: Multifractals, Probability and Statistical Mechanics, Applications book, January 2004

The connective constant of the honeycomb lattice equals sqrt(2+sqrt 2) journal, January 2012

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

High-precision phase diagram of spin glasses from duality analysis with real-space renormalization and graph polynomials journal, February 2015

A new transfer-matrix algorithm for exact enumerations: self-avoiding polygons on the square lattice journal, February 2012

Self-avoiding polygons on the square lattice journal, January 1999

Exact Enumerations book, January 2009

Generating functions for enumerating self-avoiding rings on the square lattice journal, December 1980

A parallel algorithm for the enumeration of self-avoiding polygons on the square lattice journal, May 2003

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

Exact Critical Point and Critical Exponents of O ( n ) Models in Two Dimensions journal, October 1982

Square Lattice Self-Avoiding Walks and Polygons journal, December 2001

The number of polygons on a lattice journal, July 1961

Percolation critical polynomial as a graph invariant journal, October 2012

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 phase diagram of spin glasses from duality analysis with real-space renormalization and graph polynomials text, January 2014

Self-avoiding polygons on the square lattice text, January 1999

Exact Enumerations book, January 2009

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

Two-dimensional interacting self-avoiding walks: new estimates for critical temperatures and exponents text, January 2019

Critical points of Potts and O( N ) models from eigenvalue identities in periodic Temperley–Lieb algebras
journal, October 2015

Fractal Geometry and Applications: A Jubilee of Benoît Mandelbrot, Part 2: Multifractals, Probability and Statistical Mechanics, Applications
book, January 2004

The connective constant of the honeycomb lattice equals sqrt(2+sqrt 2)
journal, January 2012

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

High-precision phase diagram of spin glasses from duality analysis with real-space renormalization and graph polynomials
journal, February 2015

A new transfer-matrix algorithm for exact enumerations: self-avoiding polygons on the square lattice
journal, February 2012

Self-avoiding polygons on the square lattice
journal, January 1999

Exact Enumerations
book, January 2009

Generating functions for enumerating self-avoiding rings on the square lattice
journal, December 1980

A parallel algorithm for the enumeration of self-avoiding polygons on the square lattice
journal, May 2003

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

Exact Critical Point and Critical Exponents of $O (n)$ Models in Two Dimensions
journal, October 1982

Square Lattice Self-Avoiding Walks and Polygons
journal, December 2001

The number of polygons on a lattice
journal, July 1961

Percolation critical polynomial as a graph invariant
journal, October 2012

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 phase diagram of spin glasses from duality analysis with real-space renormalization and graph polynomials
text, January 2014

Self-avoiding polygons on the square lattice
text, January 1999

Exact Enumerations
book, January 2009

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

Two-dimensional interacting self-avoiding walks: new estimates for critical temperatures and exponents
text, January 2019