Effect of Dimensionality on the Continuum Percolation of Overlapping Hyperspheres and Hypercubes: II. Simulation Results and Analyses

Abstract

In the first paper of this series [S. Torquato, J. Chem. Phys. {\bf 136}, 054106 (2012)], analytical results concerning the continuum percolation of overlapping hyperparticles in $d$-dimensional Euclidean space ${\mathbb{R}}^d$ were obtained, including lower bounds on the percolation threshold. In the present investigation, we provide additional analytical results for certain cluster statistics, such as the concentration of $k$-mers and related quantities, and obtain an upper bound on the percolation threshold ${\eta }_c$. We utilize the tightest lower bound obtained in the first paper to formulate an efficient simulation method, called the {\it rescaled-particle} algorithm, to estimate continuum percolation properties across many space dimensions with heretofore unattained accuracy. This simulation procedure is applied to compute the threshold ${\eta }_c$ and associated mean number of overlaps per particle ${\mathcal{N}}_c$ for both overlapping hyperspheres and oriented hypercubes for $3\le d\le 11$. These simulations results are compared to corresponding upper and lower bounds on these percolation properties. We find that the bounds converge to one another as the space dimension increases, but the lower bound provides an excellent estimate of ${\eta }_c$ and ${\mathcal{N}}_c$, even for relatively low dimensions. We confirm a prediction of the first paper in this series that low-dimensional percolation properties encode high-dimensional information. We also show that the concentration of monomers dominate over concentration values for higher-order clusters (dimers, trimers, etc.) as the space dimension becomes large. Finally, we provide accurate analytical estimates of the pair connectedness function and blocking function at their contact values for any $d$ as a function of density.

24 pages, 10 figures

References in corpus (2)

• A fast Monte Carlo algorithm for site or bond percolation
• The continuum percolation threshold for interpenetrating squares and cubes

Cited by in corpus (27)

• Continuum Percolation Thresholds in Two Dimensions
• Precise Algorithm to Generate Random Sequential Addition of Hard Hyperspheres at Saturation
• Hard-core deconfinement and soft-surface delocalization from nuclear to quark matter
• Effect of Dimensionality on the Percolation Threshold of Overlapping Nonspherical Hyperparticles
• From discrete to continuous percolation in dimensions 3 to 7
• Site percolation on square and simple cubic lattices with extended neighborhoods and their continuum limit
• Percolation of overlapping squares or cubes on a lattice
• Universal behavior of site and bond percolation thresholds on regular lattices with compact extended-range neighborhoods in 2 and 3 dimensions
• Polymer Glass-Formation in Variable Dimension
• Bond percolation on simple cubic lattices with extended neighborhoods
• Effect of Dimensionality on the Percolation Thresholds of Various $d$-Dimensional Lattices
• Information Content of Hierarchical n-Point Polytope Functions for Quantifying and Reconstructing Disordered Systems
• Conformal invariance in three dimensional percolation
• Disagreement percolation for the hard-sphere model
• Site and bond percolation on four-dimensional simple hypercubic lattices with extended neighborhoods
• Modeling Multi-Cellular Dynamics Regulated by ECM-Mediated Mechanical Communication via Active Particles with Polarized Effective Attraction
• Critical probability of percolation over bounded region in N-dimensional Euclidean space
• On the uniqueness of Gibbs distributions with a non-negative and subcritical pair potential
• High-dimensional percolation criticality and hints of mean-field-like caging of the random Lorentz gas
• Exclusion Volumes of Convex Bodies in High Space Dimensions: Applications to Virial Coefficients and Continuum Percolation
• Continuum percolation expressed in terms of density distributions
• Continuum percolation of polydisperse hyperspheres in infinite dimensions
• Characterizing spatial point processes by percolation transitions
• Limiting shape for First-Passage Percolation models on Random Geometric Graphs
• Extended-range percolation in five dimensions
• Understanding Degeneracy of Two-Point Correlation Functions via Debye Random Media
• Connectedness percolation of fractal liquids