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

Abstract

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].

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

References in corpus (2)

• Hammersley's process with sources and sinks
• Second-order fluctuations and current across characteristic for a one-dimensional growth model of independent random walks

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