Site percolation thresholds on triangular lattice with complex neighborhoods.

We determine thresholds p 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(4NN)=0.192410(43), p(3NN+2NN)=0.232008(38), p(5NN+4NN)=0.140286(5), p(3NN+2NN+NN)=0.215484(19), p(5NN+4NN+NN)=0.131792(58), p(5NN+4NN+3NN+2NN)=0.117579(41), and p(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(NN)=p(2NN)=p(3NN)=p(5NN)=12 is recovered with five digits accuracy p(NN)=0.500029(46) by averaging over one thousand lattice realizations only.