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Title: Polynomial sequences for bond percolation critical thresholds

Journal Article · · Journal of Statistical Mechanics
 [1]
  1. Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)
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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.

Research Organization:
Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)
Sponsoring Organization:
USDOE
Grant/Contract Number:
AC52-07NA27344
OSTI ID:
1399718
Report Number(s):
LLNL-JRNL-470576
Journal Information:
Journal of Statistical Mechanics, Vol. 2011, Issue 09; ISSN 1742-5468
Publisher:
IOP PublishingCopyright Statement
Country of Publication:
United States
Language:
English
Citation Metrics:
Cited by: 14 works
Citation information provided by
Web of Science

References (31)

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Cited By (7)

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

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97 MATHEMATICS AND COMPUTING