Open Access
July 2006 Second class particles and cube root asymptotics for Hammersley’s process
Eric Cator, Piet Groeneboom
Ann. Probab. 34(4): 1273-1295 (July 2006). DOI: 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 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].


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Eric Cator. Piet Groeneboom. "Second class particles and cube root asymptotics for Hammersley’s process." Ann. Probab. 34 (4) 1273 - 1295, July 2006.


Published: July 2006
First available in Project Euclid: 19 September 2006

zbMATH: 1101.60076
MathSciNet: MR2257647
Digital Object Identifier: 10.1214/009117906000000089

Primary: 60C05 , 60K35
Secondary: 60F05

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

Rights: Copyright © 2006 Institute of Mathematical Statistics

Vol.34 • No. 4 • July 2006
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