The Annals of Probability

Hammersley’s process with sources and sinks

Eric Cator and Piet Groeneboom

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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, an isolated second-class particle, located at the origin at time zero, moves asymptotically, with probability 1, along the characteristic of a conservation equation for Hammersley’s process. This allows us to show that Hammersley’s process without sinks or sources, as defined by Aldous and Diaconis [Probab. Theory Related Fields 10 (1995) 199–213] converges locally in distribution to a Poisson process, a result first proved in Aldous and Diaconis (1995) by using the ergodic decomposition theorem and a construction of Hammersley’s process as a one-dimensional point process, developing as a function of (continuous) time on the whole real line. As a corollary we get the result that EL(t,t)/t converges to 2, as t→∞, where L(t,t) is the length of a longest North-East path from (0,0) to (t,t). The proofs of these facts need neither the ergodic decomposition theorem nor the subadditive ergodic theorem. We also prove a version of Burke’s theorem for the stationary process with sources and sinks and briefly discuss the relation of these results with the theory of longest increasing subsequences of random permutations.

Article information

Source
Ann. Probab., Volume 33, Number 3 (2005), 879-903.

Dates
First available in Project Euclid: 6 May 2005

Permanent link to this document
https://projecteuclid.org/euclid.aop/1115386713

Digital Object Identifier
doi:10.1214/009117905000000053

Mathematical Reviews number (MathSciNet)
MR2135307

Zentralblatt MATH identifier
1066.60011

Subjects
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

Keywords
Longest increasing subsequence Ulam’s problem Hammersley’s process local Poisson convergence totally asymmetric simple exclusion processes (TASEP) second-class particles Burke’s theorem

Citation

Cator, Eric; Groeneboom, Piet. Hammersley’s process with sources and sinks. Ann. Probab. 33 (2005), no. 3, 879--903. doi:10.1214/009117905000000053. https://projecteuclid.org/euclid.aop/1115386713


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References

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