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

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

2006-03-14

arxiv.org/abs/math/0603345v2

Annals of Probability 2006, Vol. 34, No. 4, 1273-1295

Mathematics

Probability

Published at http://dx.doi.org/10.1214/009117906000000089 in the Annals of Probability (http://www.imstat.org/aop/) by the Ins

Scientific paper

10.1214/009117906000000089

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 $2\mathbb {E}(t-X(t))_+$, where $X(t)$ is the location of a second class particle at time $t$. This implies that both $\mathbb {E}(t-X(t))_+$ and the variance of $L(t,t)$ are of order $t^{2/3}$. 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].

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