Site and bond percolation thresholds of all k-uniform tilings with k≤ 10
Abstract
We determine the site and bond percolation thresholds of all 23,396 k-uniform tilings of the plane by regular polygons with k≤ 10, the complete set according to the known enumeration. We use Monte Carlo simulation on tori and the condition that no wrapping and a cross-wrapping cluster are equally probable, the Monte Carlo form of the critical-polynomial criterion, which needs no knowledge of the large-scale anisotropy that most of these lattices have. The 486 thresholds with k≤4 have uncertainties of a few units in the sixth decimal and reproduce the 22 Archimedean reference values (rms deviation 0.71 standard errors) and unpublished bond thresholds of the 20 two-uniform lattices (χ2=19.5 for 20 values); the others have median statistical uncertainties of 2×10−5 (k≤7) to 7×10−5 (k≥8) and no detectable finite-size bias. The vertex-type fractions alone fix the thresholds for k≥5 to about 2×10−3 (linear rule fitted on k≤4), while tilings with equal composition differ by up to 6×10−3 (site) and 2×10−2 (bond). In the range of domain sizes covered, descriptors of radius one account for more than half of this difference, and stackings of identical strips have thresholds linear in the fraction of each boundary type. For tilings that differ only by the registry of strips separated by h rows of squares, the bond threshold difference decreases by a factor of 6-8 per row, comparable to the leading momentum-π eigenvalue of the square-band transfer matrix.
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BibTeX
@misc{elisov2026site,
title = {{Site and bond percolation thresholds of all $k$-uniform tilings with $k\le 10$}},
author = {Maksim Elisov},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.31190},
eprint = {2610.31190},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.31190},
}
RIS
TY - UNPB
TI - Site and bond percolation thresholds of all k-uniform tilings with k≤ 10
AU - Maksim Elisov
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2610.31190
UR - https://arxiv.org/abs/2610.31190
AB - We determine the site and bond percolation thresholds of all 23,396 k-uniform tilings of the plane by regular polygons with k≤ 10, the complete set according to the known enumeration. We use Monte Carlo simulation on tori and the condition that no wrapping and a cross-wrapping cluster are equally probable, the Monte Carlo form of the critical-polynomial criterion, which needs no knowledge of the large-scale anisotropy that most of these lattices have. The 486 thresholds with k≤4 have uncertainties of a few units in the sixth decimal and reproduce the 22 Archimedean reference values (rms deviation 0.71 standard errors) and unpublished bond thresholds of the 20 two-uniform lattices (χ^2=19.5 for 20 values); the others have median statistical uncertainties of 2×10^−5 (k≤7) to 7×10^−5 (k≥8) and no detectable finite-size bias. The vertex-type fractions alone fix the thresholds for k≥5 to about 2×10^−3 (linear rule fitted on k≤4), while tilings with equal composition differ by up to 6×10^−3 (site) and 2×10^−2 (bond). In the range of domain sizes covered, descriptors of radius one account for more than half of this difference, and stackings of identical strips have thresholds linear in the fraction of each boundary type. For tilings that differ only by the registry of strips separated by h rows of squares, the bond threshold difference decreases by a factor of 6-8 per row, comparable to the leading momentum-π eigenvalue of the square-band transfer matrix.
ER -