Site percolation thresholds on triangular lattice with complex neighborhoods

TL;DR

This paper calculates site percolation thresholds for a triangular lattice with various complex neighborhoods using Monte Carlo simulations and threshold estimation methods, providing precise values for different neighborhood combinations.

Contribution

It introduces a combined Monte Carlo and statistical estimation approach to accurately determine percolation thresholds for complex neighborhoods on a triangular lattice.

Findings

Thresholds for various neighborhoods are precisely estimated.

The method accurately recovers known thresholds for simple neighborhoods.

Results demonstrate the effectiveness of the combined Monte Carlo and estimation approach.

Abstract

We determine thresholds $p_c$ for random site percolation on a triangular lattice for neighbourhoods containing nearest (NN), next-nearest (2NN), next-next-nearest (3NN), next-next-next-nearest (4NN) and next-next-next-next-nearest (5NN) neighbours, and their combinations forming regular hexagons (3NN+2NN+NN, 5NN+4NN+NN, 5NN+4NN+3NN+2NN, 5NN+4NN+3NN+2NN+NN). We use a fast Monte Carlo algorithm, by Newman and Ziff [M. E. J. Newman and R. M. Ziff, Physical Review 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. [N. Bastas, K. Kosmidis, P. Giazitzidis, and M. Maragakis, Physical Review E 90, 062101 (2014)], of estimating thresholds from low statistics data. The estimated values of percolation thresholds are $p_c(\text{4NN})=0.192410(43)$, $p_c(\text{3NN+2NN})=0.232008(38)$, $p_c(\text{5NN+4NN})=0.140286(5)$, $p_c(\text{3NN+2NN+NN})=0.215484(19)$, $p_c(\text{5NN+4NN+NN})=0.131792(58)$, $p_c(\text{5NN+4NN+3NN+2NN})=0.117579(41)$, $p_c(\text{5NN+4NN+3NN+2NN+NN})=0.115847(21)$. The method is tested on the standard case of site percolation on triangular lattice, where $p_c(\text{NN})=p_c(\text{2NN})=p_c(\text{3NN})=p_c(\text{5NN})=\frac{1}{2}$ is recovered with five digits accuracy $p_c(\text{NN})=0.500029(46)$ by averaging over one thousand lattice realisations only.