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

E. Cator, P. Groeneboom

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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 2double-struck E sign(t - X(t))+, where X(t) is the location of a second class particle at time t. This implies that both double-struck E sign(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 particle, continuing the approach of Cator and Groeneboom [Ann. Probab. 33 (2005) 879-903]. © Institute of Mathematical Statistics, 2006.
Original languageEnglish
Pages (from-to)1273-1295
Number of pages24
JournalAnnals of probability
Issue number4
Publication statusPublished - 2006

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