Limit theorems for a random directed slab graph

D. Denisov, S. Foss, T. Konstantopoulos

Research output: Contribution to journalArticle

6 Citations (Scopus)

Abstract

We consider a stochastic directed graph on the integers whereby a directed edge between i and a larger integer j exists with probability pj-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 × 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| × |I| random matrix in the Gaussian unitary ensemble (GUE).

Original languageEnglish
Pages (from-to)702-733
JournalAnnals of Applied Probability
Volume22
Issue number2
DOIs
Publication statusPublished - Apr 2012

Keywords

  • Random graph
  • partial order
  • functional central limit theorem
  • GUE
  • last passage percolation

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