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Behavior of a second class particle in Hammersley’s process. (English) Zbl 1112.60076
Summary: In the case of a rarefaction fan in a nonstationary Hammersley process, we explicitly calculate the asymptotic behavior of the process as we move out along a ray, and the asymptotic distribution of the angle within the rarefaction fan of a second class particle and a dual second class particle. Furthermore, we consider a stationary Hammersley process and use the previous results to show that trajectories of a second class particle and a dual second class particle touch with probability one, and we give some information on the area enclosed by the two trajectories, up until the first intersection point. This is linked to the area of influence of an added Poisson point in the plane.

MSC:
60K35 Interacting random processes; statistical mechanics type models; percolation theory
60C05 Combinatorial probability
60F05 Central limit and other weak theorems
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