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Title: Transfer matrix computation of generalized critical polynomials in percolation

Abstract

Percolation thresholds have recently been studied by means of a graph polynomial PB(p), henceforth referred to as the critical polynomial, that may be defined on any periodic lattice. The polynomial depends on a finite subgraph B, called the basis, and the way in which the basis is tiled to form the lattice. The unique root of P B(p) in [0, 1] either gives the exact percolation threshold for the lattice, or provides an approximation that becomes more accurate with appropriately increasing size of B. Initially P B(p) was defined by a contraction-deletion identity, similar to that satisfied by the Tutte polynomial. Here, we give an alternative probabilistic definition of P B(p), which allows for much more efficient computations, by using the transfer matrix, than was previously possible with contraction-deletion. We present bond percolation polynomials for the (4, 82), kagome, and (3, 122) lattices for bases of up to respectively 96, 162, and 243 edges, much larger than the previous limit of 36 edges using contraction-deletion. We discuss in detail the role of the symmetries and the embedding of B. For the largest bases, we obtain the thresholds p c(4, 82) = 0.676 803 329 · · ·, p c(kagome) =more » 0.524 404 998 · · ·, p c(3, 122) = 0.740 420 798 · · ·, comparable to the best simulation results. We also show that the alternative definition of P B(p) can be applied to study site percolation problems.« less

Authors:
 [1];  [2]
  1. Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)
  2. Ecole Normale Superieure, Paris (France). LPTENS; Univ. Pierre et Marie Curie, Paris (France)
Publication Date:
Research Org.:
Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)
Sponsoring Org.:
USDOE
OSTI Identifier:
1399712
Report Number(s):
LLNL-JRNL-593972
Journal ID: ISSN 1751-8113
Grant/Contract Number:  
AC52-07NA27344
Resource Type:
Journal Article: Accepted Manuscript
Journal Name:
Journal of Physics. A, Mathematical and Theoretical
Additional Journal Information:
Journal Volume: 45; Journal Issue: 49; Journal ID: ISSN 1751-8113
Publisher:
IOP Publishing
Country of Publication:
United States
Language:
English
Subject:
97 MATHEMATICS AND COMPUTING

Citation Formats

Scullard, Christian R., and Jacobsen, Jesper Lykke. Transfer matrix computation of generalized critical polynomials in percolation. United States: N. p., 2012. Web. doi:10.1088/1751-8113/45/49/494004.
Scullard, Christian R., & Jacobsen, Jesper Lykke. Transfer matrix computation of generalized critical polynomials in percolation. United States. https://doi.org/10.1088/1751-8113/45/49/494004
Scullard, Christian R., and Jacobsen, Jesper Lykke. Thu . "Transfer matrix computation of generalized critical polynomials in percolation". United States. https://doi.org/10.1088/1751-8113/45/49/494004. https://www.osti.gov/servlets/purl/1399712.
@article{osti_1399712,
title = {Transfer matrix computation of generalized critical polynomials in percolation},
author = {Scullard, Christian R. and Jacobsen, Jesper Lykke},
abstractNote = {Percolation thresholds have recently been studied by means of a graph polynomial PB(p), henceforth referred to as the critical polynomial, that may be defined on any periodic lattice. The polynomial depends on a finite subgraph B, called the basis, and the way in which the basis is tiled to form the lattice. The unique root of PB(p) in [0, 1] either gives the exact percolation threshold for the lattice, or provides an approximation that becomes more accurate with appropriately increasing size of B. Initially PB(p) was defined by a contraction-deletion identity, similar to that satisfied by the Tutte polynomial. Here, we give an alternative probabilistic definition of PB(p), which allows for much more efficient computations, by using the transfer matrix, than was previously possible with contraction-deletion. We present bond percolation polynomials for the (4, 82), kagome, and (3, 122) lattices for bases of up to respectively 96, 162, and 243 edges, much larger than the previous limit of 36 edges using contraction-deletion. We discuss in detail the role of the symmetries and the embedding of B. For the largest bases, we obtain the thresholds pc(4, 82) = 0.676 803 329 · · ·, pc(kagome) = 0.524 404 998 · · ·, pc(3, 122) = 0.740 420 798 · · ·, comparable to the best simulation results. We also show that the alternative definition of PB(p) can be applied to study site percolation problems.},
doi = {10.1088/1751-8113/45/49/494004},
url = {https://www.osti.gov/biblio/1399712}, journal = {Journal of Physics. A, Mathematical and Theoretical},
issn = {1751-8113},
number = 49,
volume = 45,
place = {United States},
year = {2012},
month = {9}
}

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Cited by: 18 works
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Works referenced in this record:

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Universal condition for critical percolation thresholds of kagomé-like lattices
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The Application of Non-Crossing Partitions to Improving Percolation Threshold Bounds
journal, September 2006


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journal, June 1994


Shape-dependent universality in percolation
journal, April 1999


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journal, August 2009


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journal, August 2001


Critical behaviour of the two-dimensional Potts model with a continuous number of states; A finite size scaling analysis
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Critical behaviour of random-bond Potts models: a transfer matrix study
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Rigorous confidence intervals for critical probabilities
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journal, September 2008


Spanning probability in 2D percolation
journal, November 1992


Efficient Monte Carlo Algorithm and High-Precision Results for Percolation
journal, November 2000


Convergence of threshold estimates for two-dimensional percolation
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Monte Carlo study of the site-percolation model in two and three dimensions
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Complementary algorithms for graphs and percolation
journal, August 2007


Pseudo-random-number generators and the square site percolation threshold
journal, September 2008


Bond percolation on honeycomb and triangular lattices
journal, June 1981


Site percolation thresholds for Archimedean lattices
journal, July 1999


    Works referencing / citing this record:

    Classical phase transitions in a one-dimensional short-range spin model
    journal, November 2018


    Percolation in finite matching lattices
    journal, December 2016


    The three-state Potts antiferromagnet on plane quadrangulations
    journal, July 2018


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


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


    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


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