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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

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.

Citation

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

Information

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

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

Subjects:
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

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Vol.33 • No. 3 • May 2005
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