Effect of dimensionality on the continuum percolation of overlapping hyperspheres and hypercubes.
We show analytically that the [0, 1], [1, 1], and [2, 1] Padé approximants of the mean cluster number S for both overlapping hyperspheres and overlapping oriented hypercubes are upper bounds on this quantity in any Euclidean dimension d. These results lead to lower bounds on the percolation threshold density η(c), which become progressively tighter as d increases and exact asymptotically as d → ∞, i.e., η(c) → 2(-d). Our analysis is aided by a certain remarkable duality between the equilibrium hard-hypersphere (hypercube) fluid system and the continuum percolation model of overlapping hyperspheres (hypercubes). Analogies between these two seemingly different problems are described. We also obtain Percus-Yevick-like approximations for the mean cluster number S in any dimension d that also become asymptotically exact as d → ∞. We infer that as the space dimension increases, finite-sized clusters become more ramified or "branch-like." These analytical estimates are used to assess simulation results for η(c) up to 20 dimensions in the case of hyperspheres and up to 15 dimensions in the case of hypercubes. Our analysis sheds light on the radius of convergence of the density expansion for S and naturally leads to an analytical approximation for η(c) that applies across all dimensions for both hyperspheres and oriented hypercubes. Finally, we describe the extension of our results to the case of overlapping particles of general anisotropic shape in d dimensions with a specified orientational probability distribution.
Department of Chemistry, Princeton Center for Theoretical Science, Princeton Institute for the Science and Technology of Materials, and Program in Applied and Computational Mathematics, Princeton University, Princeton, New Jersey 08544, USA. torquato@electron.princeton.edu
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Torquato