The paper discusses two models for non-overlapping finite line-segmentsconstructed via the lilypond protocol, operating here on a given array ofpoints in the plane with which are associated directions. At time 0, eachline-segment starts growing at unit rate around its center in the givendirection; each line-segment, under Model 1, ceases growth when one of its endshits another line, while under Model 2, its growth ceases either when one ofits ends hits another line, or when it is hit by the growing end of some otherline. The paper shows that these procedures are well-defined and givesconstructive algorithms to compute the lengths of the segments. Moreover itspecifies assumptions under which stochastic versions, i.e. models based onpoint processes, exist. Afterwards it deals with the question as to whetherthere is percolation in Model 1. The paper concludes with a section containingseveral conjectures and final remarks.
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