# Bond percolation thresholds on Archimedean lattices from critical polynomial roots

## Abstract

We present percolation thresholds calculated numerically with the eigenvalue formulation of the method of critical polynomials; developed in the last few years, it has already proven to be orders of magnitude more accurate than traditional techniques. Here, we report the result of large parallel calculations to produce what we believe may become the reference values of bond percolation thresholds on the Archimedean lattices for years to come. For example, for the kagome lattice we find p_{c}= 0.524 404 999 167 448 20 (1) , whereas the best estimate using standard techniques is p_{c}= 0.524 404 99 (2) . We further provide strong evidence that there are two classes of lattices: one for which the first three scaling exponents characterizing the finite-size corrections to p_{c} are Δ = 6, 7, 8 , and another for which Δ = 4, 6, 8. We discuss the open questions related to the method, such as the full scaling law, as well as its potential for determining the critical points of other models.

- Authors:

- Publication Date:

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

- Sponsoring Org.:
- USDOE National Nuclear Security Administration (NNSA); USDOE Laboratory Directed Research and Development (LDRD) Program

- OSTI Identifier:
- 1602370

- Alternate Identifier(s):
- OSTI ID: 1670554

- Report Number(s):
- LLNL-JRNL-795621

Journal ID: ISSN 2643-1564; PPRHAI; 012050

- Grant/Contract Number:
- AC52-07NA27344; 19-DR-013

- Resource Type:
- Published Article

- Journal Name:
- Physical Review Research

- Additional Journal Information:
- Journal Name: Physical Review Research Journal Volume: 2 Journal Issue: 1; Journal ID: ISSN 2643-1564

- Publisher:
- American Physical Society (APS)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 98 NUCLEAR DISARMAMENT, SAFEGUARDS, AND PHYSICAL PROTECTION

### Citation Formats

```
Scullard, Christian R., and Jacobsen, Jesper Lykke. Bond percolation thresholds on Archimedean lattices from critical polynomial roots. United States: N. p., 2020.
Web. doi:10.1103/PhysRevResearch.2.012050.
```

```
Scullard, Christian R., & Jacobsen, Jesper Lykke. Bond percolation thresholds on Archimedean lattices from critical polynomial roots. United States. doi:https://doi.org/10.1103/PhysRevResearch.2.012050
```

```
Scullard, Christian R., and Jacobsen, Jesper Lykke. Fri .
"Bond percolation thresholds on Archimedean lattices from critical polynomial roots". United States. doi:https://doi.org/10.1103/PhysRevResearch.2.012050.
```

```
@article{osti_1602370,
```

title = {Bond percolation thresholds on Archimedean lattices from critical polynomial roots},

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

abstractNote = {We present percolation thresholds calculated numerically with the eigenvalue formulation of the method of critical polynomials; developed in the last few years, it has already proven to be orders of magnitude more accurate than traditional techniques. Here, we report the result of large parallel calculations to produce what we believe may become the reference values of bond percolation thresholds on the Archimedean lattices for years to come. For example, for the kagome lattice we find pc= 0.524 404 999 167 448 20 (1) , whereas the best estimate using standard techniques is pc= 0.524 404 99 (2) . We further provide strong evidence that there are two classes of lattices: one for which the first three scaling exponents characterizing the finite-size corrections to pc are Δ = 6, 7, 8 , and another for which Δ = 4, 6, 8. We discuss the open questions related to the method, such as the full scaling law, as well as its potential for determining the critical points of other models.},

doi = {10.1103/PhysRevResearch.2.012050},

journal = {Physical Review Research},

number = 1,

volume = 2,

place = {United States},

year = {2020},

month = {2}

}

DOI: https://doi.org/10.1103/PhysRevResearch.2.012050

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Figures / Tables found in this record:

*Figures/Tables have been extracted from DOE-funded journal article accepted manuscripts.*