The Annals of Applied Probability

Limit theorems for a random directed slab graph

D. Denisov, S. Foss, and T. Konstantopoulos

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Abstract

We consider a stochastic directed graph on the integers whereby a directed edge between i and a larger integer j exists with probability pji 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).

Article information

Source
Ann. Appl. Probab. Volume 22, Number 2 (2012), 702-733.

Dates
First available in Project Euclid: 2 April 2012

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1333372011

Digital Object Identifier
doi:10.1214/11-AAP783

Mathematical Reviews number (MathSciNet)
MR2953567

Zentralblatt MATH identifier
1241.05132

Subjects
Primary: 05C80: Random graphs [See also 60B20] 60F17: Functional limit theorems; invariance principles
Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 06A06: Partial order, general

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

Citation

Denisov, D.; Foss, S.; Konstantopoulos, T. Limit theorems for a random directed slab graph. Ann. Appl. Probab. 22 (2012), no. 2, 702--733. doi:10.1214/11-AAP783. https://projecteuclid.org/euclid.aoap/1333372011.


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