Clustering properties of d -dimensional overlapping spheres

Various clustering properties of d-dimensional overlapping (i.e., Poisson distributed) spheres are investigated. We evaluate n_k, the average number of connected clusters of k particles (called k-mers) per unit particle, for k = 2,3,4 and v_k, 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 n_k and v_k. 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.