Simple cubic random-site percolation thresholds for complex neighbourhoods

TL;DR

This paper uses computer simulations and finite-size scaling to estimate percolation thresholds for various complex neighbourhoods in a simple cubic lattice, revealing a monotonic decrease with increasing coordination number.

Contribution

It provides new percolation threshold estimates for complex neighbourhoods in simple cubic lattices using simulation and scaling analysis.

Findings

Thresholds decrease monotonically with coordination number.

Threshold values are precisely estimated for multiple neighbourhood combinations.

Simulation results differ from square lattice behavior.

Abstract

In this communication with computer simulation we evaluate simple cubic random-site percolation thresholds for neighbourhoods including the nearest neighbours (NN), the next-nearest neighbours (2NN) and the next-next-nearest neighbours (3NN). Our estimations base on finite-size scaling analysis of the percolation probability vs. site occupation probability plots. The Hoshen--Kopelman algorithm has been applied for cluster labelling. The calculated thresholds are 0.1372(1), 0.1420(1), 0.0976(1), 0.1991(1), 0.1036(1), 0.2455(1) for (NN + 2NN), (NN + 3NN), (NN + 2NN + 3NN), 2NN, (2NN + 3NN), 3NN neighbourhoods, respectively. In contrast to the results obtained for a square lattice the calculated percolation thresholds decrease monotonically with the site coordination number z, at least for our inspected neighbourhoods.