Site percolation thresholds on triangular lattice with complex neighborhoods

Krzysztof Malarz

doi:10.1063/5.0022336

Site percolation thresholds on triangular lattice with complex neighborhoods

Chaos. 2020 Dec;30(12):123123. doi: 10.1063/5.0022336.

Author

Krzysztof Malarz¹

Affiliation

¹ AGH University of Science and Technology, Faculty of Physics and Applied Computer Science, al. Mickiewicza 30, 30-059 Kraków, Poland.

PMID: 33380057
DOI: 10.1063/5.0022336

Abstract

We determine thresholds p_c for random site percolation on a triangular lattice for neighborhoods containing nearest (NN), next-nearest (2NN), next-next-nearest (3NN), next-next-next-nearest (4NN), and next-next-next-next-nearest (5NN) neighbors, and their combinations forming regular hexagons (3NN+2NN+NN, 5NN+4NN+NN, 5NN+4NN+3NN+2NN, and 5NN+4NN+3NN+2NN+NN). We use a fast Monte Carlo algorithm, by Newman and Ziff [Phys. Rev. E 64, 016706 (2001)], for obtaining the dependence of the largest cluster size on occupation probability. The method is combined with a method, by Bastas et al. [Phys. Rev. E 90, 062101 (2014)], for estimating thresholds from low statistics data. The estimated values of percolation thresholds are p_c(4NN)=0.192410(43), p_c(3NN+2NN)=0.232008(38), p_c(5NN+4NN)=0.140286(5), p_c(3NN+2NN+NN)=0.215484(19), p_c(5NN+4NN+NN)=0.131792(58), p_c(5NN+4NN+3NN+2NN)=0.117579(41), and p_c(5NN+4NN+3NN+2NN+NN)=0.115847(21). The method is tested on the standard case of site percolation on the triangular lattice, where p_c(NN)=p_c(2NN)=p_c(3NN)=p_c(5NN)=12 is recovered with five digits accuracy p_c(NN)=0.500029(46) by averaging over one thousand lattice realizations only.