One-Dimensional Poisson Growth Models With Random and Asymmetric Growth

Suppose there is a Poisson process of points Xi on the line. Starting at time zero, a grain begins to grow from each point Xi, growing at rate Ai to the left and rate Bi to the right, with the pairs (Ai, Bi) being i.i.d. A grain stops growing as soon as it touches another grain. When all growth stops, the line consists of covered intervals (made up of contiguous grains) separated by gaps. We show (i) a fraction 1/e of the line remains uncovered, (ii) the fraction of covered intervals which contain exactly k grains is (k−1)/k!, (iii) the length of a covered interval containing k grains has a gamma(k−1) distribution, (iv) the distribution of the grain sizes depends only on the distribution of the total growth rate Ai+Bi, and other results. Similar theorems are obtained for growth processes on a circle; in this case we need only assume the pairs (Ai, Bi) are exchangeable. These results extend those of Daley, et al. (2000) who studied the case where Ai=Bi=1. Simulation results are given to illustrate the various theorems.