THE ORIENTED SWAP PROCESS

• 成果类型: Article
• 署名作者: Angel, Omer; Holroyd, Alexander; Romik, Dan
• 署名单位: University of British Columbia; Microsoft; Hebrew University of Jerusalem
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/09-AOP456
• 发表日期: 2009
• 页码: 1970-1998
• 关键词: particle; profile

摘要

Particles labelled 1,..., n are initially arranged in increasing order. Subsequently, each pair of neighboring particles that is currently in increasing order swaps according to a Poisson process of rate 1. We analyze the asymptotic behavior of this process as n -> infinity. We prove that the space-time trajectories of individual particles converge (when suitably scaled) to a certain family of random curves with two points of non-differentiability, and that the permutation matrix at a given time converges to a certain deterministic measure with absolutely continuous and singular parts. The absorbing state (where all particles are in decreasing order) is reached at time (2 + o(1))n. The finishing times of individual particles converge to deterministic limits, with fluctuations asymptotically governed by the Tracy-Widom distribution.