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May 2005 Hammersley’s process with sources and sinks
Eric Cator, Piet Groeneboom
Ann. Probab. 33(3): 879-903 (May 2005). DOI: 10.1214/009117905000000053


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.


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Eric Cator. Piet Groeneboom. "Hammersley’s process with sources and sinks." Ann. Probab. 33 (3) 879 - 903, May 2005.


Published: May 2005
First available in Project Euclid: 6 May 2005

zbMATH: 1066.60011
MathSciNet: MR2135307
Digital Object Identifier: 10.1214/009117905000000053

Primary: 60C05 , 60K35
Secondary: 60F05

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

Rights: Copyright © 2005 Institute of Mathematical Statistics


Vol.33 • No. 3 • May 2005
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