Clustering properties ofd-dimensional overlapping spheres

Various clustering properties of d-dimensional overlapping (i.e., Poisson distributed) spheres are investigated. We evaluate nₖ, the average number of connected clusters of k particles (called k-mers) per unit particle, for k=2,3,4 and vₖ, the expected volume of a k-mer, for k=2,3,4 by using our general expressions for these quantities for d=1, 2, or 3. We use these calculations to obtain low-density expansions of various averaged cluster numbers and volumes, which can be obtained from the nₖ and vₖ. We study the behavior of these cluster statistics as the percolation threshold is approached from below, and we rigorously show that two of these averaged quantities do not diverge for d{}2. {} 1996 The American Physical Society.