A one-dimensional Poisson growth model with non-overlapping intervals.
Dynamic lilypond model [\textit{O. Häggström} and \textit{R. Meester}, Random Struct. Algorithms 9, 295--315 (1996; Zbl 0866.60088)] and Poisson germ model with maximally non-overlapping spherical grains (PMS) [\textit{D. J. Daley}, \textit{D. Stoyan} and \textit{H. Stoyan}, Adv. Appl. Probab. 31, 610--624 (1999; Zbl 0944.60023)] are two names of the germ-grain model \(\Xi \), the one-dimensional version of which is examined in the present paper. In the model, spherical grains grow at an equal rate from Poisson distributed germs and stop whenever two neighbour grains touch. Whereas computer simulations have been used in the above cited paper by \textit{D. J. Daley} et al. concerned with \(\Xi \subset \mathbb R^d,\;d=1,2,3,\) several analytic results based mainly on the properties of the uncovered complement \(\mathfrak A=\Xi ^c\) are established here. The main results are as follows (for the unit intensity of germs). First, the grain lengths \(V\) have distribution Pr\({(V>y)={\text e}^{-y}\exp ({\text e}^{-y}-1)}\), hence the line fraction of \(\Xi \) is \(\overline {w_1}=E(V)=\text{Pr}(x\in \Xi \mid x\in \mathbb R^1)=1-{\text e}^{-1}\) and Pr\((0\in \Xi )=1-{\text e}^{-1/2}\). Further, it is shown that the uncovered components of \(\mathfrak A\) (more precisely: of its realizations) constitute the regeneration sets of a standard regenerative process; the Kingman \(p\)-function [\textit{J. F. C. Kingman}, ``Regenerative phenomena'' (1972; Zbl 0236.60040)] is then derived and the distribution of the number \(N\) of contiguous grains in a covered interval is found: Pr\((N=k)=(k-1)/k!,\;k=2,3,\dots ,\) and \(E(N)=\) e. The length of a randomly chosen set of contiguous grains has the distribution function \(G(x)=1-{\text e}^{-y}I_1(2\sqrt x)/\sqrt x\), where \(I_1(.)\) is the modified first order Bessel function; hence the mean length of covered intervals composing \(\Xi \) is \(\alpha ={\text e}-1\), that one of uncovered intervals forming \(\mathfrak A\) is \(\beta =1\). The mean length of a chain of \(k\) contiguous grains is \(1-k^{-1}\). The distribution of the end-grain lengths \(V_e\) (such grains have only one neighbour grain) is Pr\((V_e\leq y)=(1-{\text e}^{-y})\exp (\frac {1}{2}{\text e}^{-y})\) and \(E(V_e)=0.727291\), var~\(V_e=0.684176\). On the other hand, the average length of totally interior grains is much smaller, namely 0.367125. Besides these quantitative results, also the ingenious proofs of 15 theorems and seven lemmas are of particular value. Finally, the relations of the PMS model to the Stienen model are sketched.
scientific article; zbMATH DE number 2102350
scientific article