Clustering properties of d-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 k52,3,4 and vk , the expected volume of a k-mer, for k52,3,4 by using our general expressions for these quantities for d51, 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. @S1063-651X~96!02411-7#