Exclusion processes in higher dimensions: Stationary measures and convergence

• 成果类型: Article
• 署名作者: Bramson, M; Liggett, TM
• 署名单位: University of Minnesota System; University of Minnesota Twin Cities; University of California System; University of California Los Angeles
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/009117905000000341
• 发表日期: 2005
• 页码: 2255-2313
• 关键词: attractive particle-systems; blocking measures

摘要

There has been significant progress recently in our understanding of the stationary measures of the exclusion process on Z. The corresponding situation in higher dimensions remains largely a mystery. In this paper we give necessary and sufficient conditions for a product measure to be stationary for the exclusion process on an arbitrary set, and apply this result to find examples on Z(d) and on homogeneous trees in which product measures are stationary even when they are neither homogeneous nor reversible. We then begin the task of narrowing down the possibilities for existence of other stationary measures for the process on Zd. In particular, we study stationary measures that are invariant under translations in all directions orthogonal to a fixed nonzero vector. We then prove a number of convergence results as t -> infinity for the measure of the exclusion process. Under appropriate initial conditions, we show convergence of such measures to the above stationary measures. We also employ hydrodynamics to provide further examples of convergence.