Site percolation on square and simple cubic lattices with extended neighborhoods and their continuum limit

Abstract

By means of Monte Carlo simulations, we study long-range site percolation on square and simple cubic lattices with various combinations of nearest neighbors, up to the eighth neighbors for the square lattice and the ninth neighbors for the simple cubic lattice. We find precise thresholds for 23 systems using a single-cluster growth algorithm. Site percolation on lattices with compact neighborhoods can be mapped to problems of lattice percolation of extended shapes, such as disks and spheres, and the thresholds can be related to the continuum thresholds ${\eta }_c$ for objects of those shapes. This mapping implies $z{p}_c\sim 4{\eta }_c=4.51235$ in 2D and $z{p}_c\sim 8{\eta }_c=2.73512$ in 3D for large $z$ for circular and spherical neighborhoods respectively, where $z$ is the coordination number. Fitting our data to the form $p_c=c/(z+b)$ we find good agreement with $c=2^d{\eta }_c$; the constant $b$ represents a finite-$z$ correction term. We also study power-law fits of the thresholds.

References in corpus (32)

• Epidemics and percolation in small-world networks
• Precise determination of the bond percolation thresholds and finite-size scaling corrections for the s.c., f.c.c., and b.c.c. lattices
• Continuum Percolation Thresholds in Two Dimensions
• Scaling corrections: site-percolation and Ising model in three dimensions
• The continuum percolation threshold for interpenetrating squares and cubes
• Universal Formulae for Percolation Thresholds
• Simultaneous analysis of three-dimensional percolation models
• Square lattice site percolation at increasing ranges of neighbor interactions
• Epidemics, disorder, and percolation
• Measures of critical exponents in the four dimensional site percolation
• Simple cubic random-site percolation thresholds for complex neighbourhoods
• Effect of Dimensionality on the Continuum Percolation of Overlapping Hyperspheres and Hypercubes: II. Simulation Results and Analyses
• An Analysis of the Gel Point of Polymer Model Networks by Computer Simulations
• Calculation of percolation thresholds in high dimensions for fcc, bcc, and diamond lattices
• Percolation and the pandemic
• From discrete to continuous percolation in dimensions 3 to 7
• Correction-to-scaling exponent for two-dimensional percolation
• Simple cubic random-site percolation thresholds for neighborhoods containing fourth-nearest neighbors
• Percolation of overlapping squares or cubes on a lattice
• Simulation of percolation on massively-parallel computers
• Critical polynomials in the nonplanar and continuum percolation models
• Bond percolation on simple cubic lattices with extended neighborhoods
• Efficient space virtualisation for Hoshen--Kopelman algorithm
• Equivalent-neighbor percolation models in two dimensions: crossover between mean-field and short-range behavior
• Precise bond percolation thresholds on several four-dimensional lattices
• Restoring site percolation on a damaged square lattice
• Site percolation thresholds on triangular lattice with complex neighborhoods
• Conformal invariance in three dimensional percolation
• Square lattice site percolation thresholds for complex neighbourhoods
• Percolation thresholds for discorectangles: numerical estimation for a range of aspect ratios
• Critical points of Potts and O($N$) models from eigenvalue identities in periodic Temperley-Lieb algebras
• Critical p=1/2 in percolation on semi-infinite strips

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• Critical polynomials in the nonplanar and continuum percolation models
• Percolation thresholds on triangular lattice for neighbourhoods containing sites up-to the fifth coordination zone
• Extended-range percolation in complex networks
• Site and bond percolation on four-dimensional simple hypercubic lattices with extended neighborhoods
• Random site percolation on honeycomb lattices with complex neighborhoods
• General theory for extended-range percolation on simple and multiplex networks
• Percolation in a simple cubic lattice with distortion
• Random site percolation thresholds on square lattice for complex neighborhoods containing sites up to the sixth coordination zone
• Site percolation in distorted square and simple cubic lattices with flexible number of neighbors
• Possible new phase transition in the 3D Ising Model associated with boundary percolation
• Effect of shape anisotropy on percolation of aligned and overlapping objects on lattices
• Universality of random-site percolation thresholds for two-dimensional complex non-compact neighborhoods
• Cumulative Merging Percolation: A long-range percolation process in networks
• Bond percolation in distorted square and triangular lattices
• Percolation in semicontinuum geometries
• Mixed-wet percolation on a dual square lattice
• Site and bond percolation in linearly distorted triangular and square lattices
• Bond percolation in distorted simple cubic and body-centered cubic lattices
• Extended-range percolation in five dimensions