Finite clusters in high density Boolean models with balls of varying sizes

In this paper we study finite clusters in a high density Boolean model with balls of two distinct sizes. Alexander (1993) studied the geometric struct ures of finite clusters in a high density Boolean model with balls of fixed size and showed that the only possible structure admitted by such events i s that all Poisson points comprising the cluster are packed tightly inside a small sphere. When the balls are of varying sizes, the event that the clu ster consists of k(1) big balls and k(2) small balls (both k(1), k(2) great er than or equal to 1) occurs only when the centres of all big balls are co mpressed in a small sphere and the centres of the small balls are distribut ed uniformly inside the region formed by the big balls in such a way that t he small balls are totally contained inside the big balls. We also show tha t it is most likely that a finite cluster in a high density Boolean model w ith varying ball sizes is made up only of small balls.