Suppose given a realization of a Poisson process on the line: call the poin
ts 'germs' because at a given instant 'grains' start growing around every g
erm, 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 germ
s be thrown uniformly and independently onto the circumference of a circle,
and let grains grow under a similar protocol. Then the expected fraction o
f the circle remaining uncovered is the nth partial sum of the usual series
for e(-1). These results, which sharpen inequalities obtained earlier, hav
e one-sided analogues: the grains on the positive axis alone do not cover t
he origin with probability e(-1/2), and the conditional probability that th
e 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 t
he 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 numbe
r of grains in such a block, and for the length of a grain. The points of t
he line not covered by any grain constitute a Kingman-type regenerative phe
nomenon for which the associated p-function p(t) gives the conditional prob
ability that a point at distance t from an uncovered point is also uncovere
d. 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 62
M30; 60G55.