[Paper Review] Square lattice site percolation thresholds for complex neighbourhoods

Mariusz Majewski, Krzysztof Malarz|ArXiv.org|Sep 25, 2006

Theoretical and Computational Physics3 references14 citations

TL;DR

This paper computes site percolation thresholds on square lattices with complex neighbourhoods extending up to the 5th coordination shell (including 4th and 5th nearest neighbours), using the Hoshen–Kopelman algorithm and finite-size scaling. The key finding is that percolation thresholds do not monotonically decrease with coordination number or neighbourhood radius, challenging universality assumptions in percolation theory and revealing non-trivial dependencies on lattice geometry beyond coordination number alone.

ABSTRACT

In this paper we compute the square lattice random sites percolation thresholds in case when sites from the 4th and the 5th coordination shells are included for neighbourhood. The obtained results support earlier claims, that (a) the coordination number and the space dimension are insufficient for building universal formulae for percolation thresholds and (b) that percolation threshold may not decrease monotonically with lattice site coordination number.

Motivation & Objective

• To compute site percolation thresholds for square lattices with complex neighbourhoods extending beyond nearest neighbours.
• To investigate whether coordination number and spatial dimension alone suffice to predict percolation thresholds.
• To examine the influence of neighbourhood radius and geometric structure on percolation thresholds.
• To test the universality of percolation threshold formulae by comparing lattices with identical coordination numbers but different neighbourhood configurations.

Proposed method

• The Hoshen–Kopelman algorithm is used to label and identify clusters of occupied sites in finite lattices.
• Finite-size scaling is applied by analyzing percolation probability $ P(p) $ across system sizes $ L = 100, 500, 1000 $ to locate the crossing point indicating $ p_c $.
• Neighbourhoods are systematically defined by including sites within radii $ r = 1, \sqrt{2}, 2, \sqrt{5}, 2\sqrt{2} $, corresponding to 2N to 24N coordination numbers.
• The study evaluates 24 distinct neighbourhood configurations, including combinations of 2N, 3N, 4N, 5N, and 6N neighbours.
• Percolation thresholds are estimated as the occupation probability where $ P(p) $ crosses between finite-size curves, with $ \Delta p = 10^{-3} $ used for precision.
• Symmetry-based labelling ensures consistent identification of cluster connectivity in non-regular neighbourhoods.

Experimental results

Research questions

• RQ1 Does the percolation threshold $ p_c $ decrease monotonically with increasing coordination number $ z $, even when higher-order neighbours are included?
• RQ2 Can coordination number $ z $ and spatial dimension $ d $ alone predict $ p_c $, or are additional geometric parameters necessary?
• RQ3 How does the inclusion of 4th and 5th nearest neighbours affect the percolation threshold compared to standard von Neumann and Moore neighbourhoods?
• RQ4 Are there multiple lattices with identical $ d $, $ z $, and $ r $ but different $ p_c $, indicating that these parameters are insufficient for universal formulae?
• RQ5 What is the role of neighbourhood radius $ r $ in determining $ p_c $, and does it correlate with threshold values independently of $ z $?

Key findings

• The percolation threshold for 5N neighbourhood (5th nearest neighbours) is $ p_c \approx 0.270 $, refined from earlier estimates.
• For 6N+5N+4N+3N+2N (24 neighbours), $ p_c \approx 0.164 $, the lowest threshold observed in the study.
• The threshold for 5N+2N neighbourhood is $ p_c \approx 0.236 $, showing a significant drop compared to 5N alone.
• The threshold for 6N+5N+4N+3N+2N is lower than for 6N+5N+4N+2N, indicating that inclusion of 3N neighbours further reduces $ p_c $.
• Despite identical coordination number $ z = 12 $, configurations like 6N+5N+4N+3N and 6N+4N+3N+2N have different $ p_c $ values, demonstrating geometric dependence beyond $ z $.
• The study confirms that $ p_c $ does not monotonically decrease with $ z $; for example, $ p_c(6N+5N) \approx 0.199 $ is higher than $ p_c(6N+5N+4N) \approx 0.179 $, showing non-monotonic behavior.