TWO LILYPOND SYSTEMS OF FINITE LINE-SEGMENTS
Daryl J. Daley
Sven Ebert
Günter Last
Abstract: line-segments constructed via the lilypond protocol, operating here on a given array of
points in with which are associated directions . At time zero, for each
and every , a line-segment starts growing at unit rate around the point in the
direction , the point remaining at the centre of ; 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 of all . 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.
2010 AMS Mathematics Subject Classification: Primary: 60D05; Secondary:
60G55.
Keywords and phrases: Hardcore model, point process, cluster, percolation, lilypond
growth protocol.