2012/08/16 by Salvatore Torquato, Yang Jiao · 62 citations
Mathematics · Physics and Astronomy · #Combinatorics #Complex Network Analysis Techniques #Curse of dimensionality #Dimension (graph theory) #Euclidean space #Function (biology) #Hypercube #Mathematical analysis #Mathematics #Percolation (cognitive psychology) #Percolation critical exponents #Percolation threshold #Physics #Quantum mechanics #Statistical physics #Statistics #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #Upper and lower bounds #cond-mat.mtrl-sci #cond-mat.stat-mech
paper · pdf · doi:10.1063/1.4742750
published in The Journal of Chemical Physics 137(7), 074106 (American Institute of Physics) · 24 pages, 10 figures
openalex publication_date 2012/08/16 · arxiv created 2012/08/18 · arxiv updated 2012/08/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06
In the first paper of this series [S. Torquato, J. Chem. Phys. 136, 054106 (2012)], analytical results concerning the continuum percolation of overlapping hyperparticles in d-dimensional Euclidean space 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 η(c). We utilize the tightest lower bound obtained in the first paper to formulate an efficient simulation method, called the 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 η(c) and associated mean number of overlaps per particle N(c) for both overlapping hyperspheres and oriented hypercubes for 3 ≤ d ≤ 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 η(c) and 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.