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

[摘要] 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. (C) 2000 Elsevier Science B.V. All rights reserved. MSC: primary 60D05; secondary 62M30; 60G55.

[发布日期] 2000-12-01

[关键词] MSC : primary 60D05;secondary 62M30;60G55