Second class particles and cube root asymptotics for Hammersley's process

@article{Cator2006SecondCP,
  title={Second class particles and cube root asymptotics for Hammersley's process},
  author={Eric A. Cator and Piet Groeneboom},
  journal={Annals of Probability},
  year={2006},
  volume={34},
  pages={1273-1295}
}
We show that, for a stationary version of Hammersley’s process, with Poisson sources on the positive x-axis and Poisson sinks on the positive y-axis, the variance of the length of a longest weakly North–East path L(t, t) from (0, 0) to (t, t) is equal to 2E(t − X(t))+, where X(t) is the location of a second class particle at time t . This implies that both E(t −X(t))+ and the variance of L(t, t) are of order t2/3. Proofs are based on the relation between the flux and the path of a second class… 

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