Hammersley's process with sources and sinks

• 成果类型: Article
• 署名作者: Cator, E; Groeneboom, P
• 署名单位: Delft University of Technology
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/009117905000000053
• 发表日期: 2005
• 页码: 879-903
• 关键词: longest increasing subsequences

摘要

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 oil 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.