Simple cubic random-site percolation thresholds for neighborhoods containing fourth-nearest neighbors

2015/01/31 by Krzysztof Malarz, K. Malarz
Mathematics · Physics and Astronomy · #Combinatorics #Computer science #Condensed matter physics #Cubic crystal system #Electrical resistivity and conductivity #Lattice (music) #Mathematics #Monte Carlo method #Percolation (cognitive psychology) #Percolation threshold #Physics #Quantum mechanics #Random Matrices and Applications #Simple cubic lattice #Simple random sample #Statistical physics #Statistics #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #cond-mat.stat-mech #k-nearest neighbors algorithm #math-ph #math.MP

paper · pdf · doi:10.1103/physreve.91.043301

published as Phys. Rev. E 91, 043301 (2015) · 5 pages, 3 figures

arxiv created 2015/03/21 · openalex publication_date 2015/04/07 · arxiv updated 2015/04/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

In this paper, random-site percolation thresholds for a simple cubic (SC) lattice with site neighborhoods containing next-next-next-nearest neighbors (4NN) are evaluated with Monte Carlo simulations. A recently proposed algorithm with low sampling for percolation thresholds estimation (Bastas et al., arXiv:1411.5834) is implemented for the studies of the top-bottom wrapping probability. The obtained percolation thresholds are p_C(4NN)=0.311\phantom\rule0.16em0ex60(12),p_C(4NN+NN)=0.150\phantom\rule0.16em0ex40(12),p_C(4NN+2NN)=0.159\phantom\rule0.16em0ex50(12),p_C(4NN+3NN)=0.204\phantom\rule0.16em0ex90(12),p_C(4NN+2NN+NN)=0.114\phantom\rule0.16em0ex40(12),p_C(4NN+3NN+NN)=0.119\phantom\rule0.16em0ex20(12),p_C(4NN+3NN+2NN)=0.113\phantom\rule0.16em0ex30(12), and p_C(4NN+3NN+2NN+NN)=0.100\phantom\rule0.16em0ex00(12), where 3NN, 2NN, and NN stand for next-next-nearest neighbors, next-nearest neighbors, and nearest neighbors, respectively. As an SC lattice with 4NN neighbors may be mapped onto two independent interpenetrated SC lattices but with a lattice constant that is twice as large, the percolation threshold p_C(4NN) is exactly equal to p_C(NN). The simplified method of Bastas et al. allows for uncertainty of the percolation threshold value p_C to be reached, similar to that obtained with the classical method but ten times faster.

Citations

• Method for estimating critical exponents in percolation processes with low sampling
• Square-lattice site percolation at increasing ranges of neighbor bonds
• Scaling corrections: site-percolation and Ising model in three dimensions
• A fast Monte Carlo algorithm for site or bond percolation
• Percolation
• Percolation Theory for Mathematicians