LIMIT THEOREMS FOR A RANDOM DIRECTED SLAB GRAPH

• 成果类型: Article
• 署名作者: Denisov, D.; Foss, S.; Konstantopoulos, T.
• 署名单位: Cardiff University; Heriot Watt University; Uppsala University
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/11-AAP783
• 发表日期: 2012
• 页码: 702-733
• 关键词: level-spacing distributions; chain lengths; fluctuations; ergodicity; shape

摘要

We consider a stochastic directed graph on the integers whereby a directed edge between i and a larger integer j exists with probability p(j-i) depending solely on the distance between the two integers. Under broad conditions, we identify a regenerative structure that enables us to prove limit theorems for the maximal path length in a long chunk of the graph. The model is an extension of a special case of graphs studied in [Markov Process. Related Fields 9 (2003) 413-468]. We then consider a similar type of graph but on the slab Z x I, where I is a finite partially ordered set. We extend the techniques introduced in the first part of the paper to obtain a central limit theorem for the longest path. When I is linearly ordered, the limiting distribution can be seen to be that of the largest eigenvalue of a |I| x |I| random matrix in the Gaussian unitary ensemble (GUE).