Site percolation thresholds on triangular lattice with complex neighborhoods

Krzysztof Malarz

Published 2020-06-28Version 1

We determine thresholds $p_c$ for random site percolation on triangular lattice for neighborhoods containing nearest (NN), next-nearest (2NN) and next-next-nearest (3NN) neighbors, and their combinations (2NN+NN, 3NN+NN, 3NN+2NN, 3NN+2NN+NN). We use a fast Monte Carlo algorithm, by Newman--Ziff, for obtaining the dependence of the largest cluster size on occupation probability. The method is combined with a method, by Bastas et al., of estimating thresholds from low statistics data. The estimated values of percolation thresholds are $\hat p_c(\text{2NN})=0.232020(36)$, $\hat p_c(\text{3NN})=0.140250(36)$, $\hat p_c(\text{2NN+NN})=0.215459(36)$, $\hat p_c(\text{3NN+NN})=0.131660(36)$, $\hat p_c(\text{3NN+2NN})=0.117460(36)$, $\hat p_c(\text{3NN+2NN+NN})= 0.115740(36)$. The method is tested on the standard case of site percolation on triangular lattice, where $p_c(\text{NN})=\frac{1}{2}$ is recovered with five digits accuracy $\hat p_c(\text{NN})=0.499971(36)$ by averaging over thousand lattice realizations only.