Polynomial sequences for bond percolation critical thresholds
Abstract
In this paper, I compute the inhomogeneous (multiprobability) bond critical surfaces for the (4, 6, 12) and (3 ^{4}, 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}(3 ^{4}, 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:

 Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)
 Publication Date:
 Research Org.:
 Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)
 Sponsoring Org.:
 USDOE
 OSTI Identifier:
 1399718
 Report Number(s):
 LLNLJRNL470576
Journal ID: ISSN 17425468
 Grant/Contract Number:
 AC5207NA27344
 Resource Type:
 Accepted Manuscript
 Journal Name:
 Journal of Statistical Mechanics
 Additional Journal Information:
 Journal Volume: 2011; Journal Issue: 09; Journal ID: ISSN 17425468
 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:10.1088/17425468/2011/09/P09022.
Scullard, Christian R. Polynomial sequences for bond percolation critical thresholds. United States. doi:10.1088/17425468/2011/09/P09022.
Scullard, Christian R. Thu .
"Polynomial sequences for bond percolation critical thresholds". United States. doi:10.1088/17425468/2011/09/P09022. https://www.osti.gov/servlets/purl/1399718.
@article{osti_1399718,
title = {Polynomial sequences for bond percolation critical thresholds},
author = {Scullard, Christian R.},
abstractNote = {In this paper, I compute the inhomogeneous (multiprobability) 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 105, 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 = {10.1088/17425468/2011/09/P09022},
journal = {Journal of Statistical Mechanics},
number = 09,
volume = 2011,
place = {United States},
year = {2011},
month = {9}
}
Web of Science