Perturbation of the equilibrium for a totally asymmetric stick process in one dimension

• 成果类型: Article
• 署名作者: Seppäläinen, T
• 署名单位: Iowa State University
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/aop/1008956327
• 发表日期: 2001
• 页码: 176-204
• 关键词: increasing subsequences; random permutations; simple exclusion; fluctuations

摘要

We study the evolution of a small perturbation of the equilibrium of a totally asymmetric one-dimensional interacting system. The model we take as an example is Hammersley's process as seen from a tagged particle, which can be viewed as a process of interacting positive-valued stick heights on the sites of Z. It is known that under Euler scaling (space and time scale n) the empirical stick profile obeys the Burgers equation. We refine this result in two ways. If the process starts close enough to equilibrium, then over times n(v) for 1 less than or equal to v < 3, and up to errors that vanish in hydrodynamic scale, the dynamics merely translates the initial stick configuration. In particular, on the hydrodynamic time scale, diffusive fluctuations are translated rigidly. A time evolution for the perturbation is visible under a particular family of scalings: over times n(v), 1 < v < 3/2, a perturbation of order n(1-v) from equilibrium follows the inviscid Burgers equation. The results for the stick model are derived from asymptotic results for tagged particles in Hammersley's process.