The Annals of Probability

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

Eric Cator and Piet 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 $2\mathbb{E}(t-X(t))_{+}$, 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 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].

Article information

Ann. Probab., Volume 34, Number 4 (2006), 1273-1295.

First available in Project Euclid: 19 September 2006

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60C05: Combinatorial probability 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 60F05: Central limit and other weak theorems

Longest increasing subsequence Ulam’s problem Hammersley’s process cube root convergence second class particles Burke’s theorem


Cator, Eric; Groeneboom, Piet. Second class particles and cube root asymptotics for Hammersley’s process. Ann. Probab. 34 (2006), no. 4, 1273--1295. doi:10.1214/009117906000000089.

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