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

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 nth 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 nth 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.