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Two lilypond systems of finite line-segments

Daley D. J., Ebert S., Last G.

Probability and Mathematical Statistics

2016

Vol. 36, Fasc. 2

221--246

The paper discusses two models for non-overlapping finite line-segments constructed via the lilypond protocol, operating here on a given array of points P = {Pi} in R2 with which are associated directions {θi}. At time zero, for each and every i, a line-segment Li starts growing at unit rate around the point Pi in the direction θi, the point Pi remaining at the centre of Li; each line-segment, under Model 1, ceases growth when one of its ends hits another line, while under Model 2, its growth ceases either when one of its ends hits another line or when it is hit by the growing end of some other line. The paper shows that these procedures are well defined and gives constructive algorithms to compute the half-lengths Ri of all Li. Moreover, it specifies assumptions under which stochastic versions, i.e. models based on point processes, exist. Afterwards, it deals with the question as to whether there is percolation in Model 1. The paper concludes with a section containing several conjectures and final remarks.