Polynomial sequences for bond percolation critical thresholds
In this paper, I compute the inhomogeneous (multi-probability) 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.
- Publication Date:
- Report Number(s):
- LLNL-JRNL-470576
Journal ID: ISSN 1742-5468
- Grant/Contract Number:
- AC52-07NA27344
- 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
- Research Org:
- Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)
- Sponsoring Org:
- USDOE
- Country of Publication:
- United States
- Language:
- English
- Subject:
- 97 MATHEMATICS AND COMPUTING
- OSTI Identifier:
- 1399718
Scullard, Christian R. Polynomial sequences for bond percolation critical thresholds. United States: N. p.,
Web. doi:10.1088/1742-5468/2011/09/P09022.
Scullard, Christian R. Polynomial sequences for bond percolation critical thresholds. United States. doi:10.1088/1742-5468/2011/09/P09022.
Scullard, Christian R. 2011.
"Polynomial sequences for bond percolation critical thresholds". United States.
doi:10.1088/1742-5468/2011/09/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 (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 = {2011},
month = {9}
}