A one-dimensional Poisson growth model with non-overlapping intervals

Research Report SRR97-010

A one-dimensional Poisson growth model with non-overlapping intervals

D.J. Daley , C.L. Mallows and L.A. Shepp

Abstract: Suppose given a realization of a Poisson process on the line: call the points `germs' because at a given instant `grains' start growing around every germ, stopping for any particular grain when it touches another grain. When all growth stops a fraction $e^{-1}$ of the line remains uncovered. Let n germs be thrown uniformly and independently onto the circumference of a circle, and let grains grow under a similar protocol. Then the expected fraction of the circle remaining uncovered is the $n\,$ th partial sum of the usual series for $e^{-1}$ . These results, which sharpen inequalities obtained earlier, have one-sided analogues: the grains on the positive axis alone do not cover the origin with probability $e^{-1/2}$ , and the conditional probability that the origin is uncovered by these positive grains, given that the germs n and n+1 coincide, is the $n\,$ th partial sum of the series for $e^{-1/2}$ . Despite the close similarity of these results to the rencontre, or matching, problem, we have no inclusion-exclusion derivation of them. We give explicitly the distributions for the length of a contiguous block of grains and the number of grains in such a block, and for the length of a grain. The points of the line not covered by any grain constitute a Kingman-type regenerative phenomenon for which the associated p-function p(t) gives the conditional probability that a point at distance t from an uncovered point is also uncovered. These functions enable us to identify a continuous time Markov chain on the integers for which p(t) is a diagonal transition probability.

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