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 _c, which become progressively tighter as d increases and exact asymptotically as d → , i.e., _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 _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 _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.