CLUSTERING PROPERTIES OF D-DIMENSIONAL OVERLAPPING SPHERES

Various clustering properties of d-dimensional overlapping (i.e., Pois son distributed) spheres are investigated. We evaluate n(k), the avera ge number of connected clusters of k particles (called k-mers) per uni t particle, for k=2,3,4 and nu(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 nu(k). We study the behavior of these cluster stati stics as the percolation threshold is approached from below, and we ri gorously show that two of these averaged quantities do not diverge for d greater than or equal to 2.