Denisov, Denis, Foss, Sergey and Konstantopoulos, Takis
(2010)
Limit theorems for a random directed slab graph.
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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 $p_{j-i}$ 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 by Foss and Konstantopoulos, Markov Process and Related Fields, 9, 413-468. We then consider a similar type of graph but on the `slab' $\Z \times I$, where $I$ is a finite partially ordered set. We extend the techniques introduced in the 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 $|I| \times |I|$ random matrix in the Gaussian unitary ensemble (GUE).
| Item Type: | Article |
|---|---|
| Additional Information: | 26 pages, 3 figures |
| Uncontrolled Keywords: | math.PR, math.PR, 05C80, 60F17 (Primary) 60K35, 06A06 (Secondary) |
| Depositing User: | Symplectic Admin |
| Date Deposited: | 23 Aug 2018 06:18 |
| Last Modified: | 19 Jan 2023 01:26 |
| Related URLs: | |
| URI: | https://livrepository.liverpool.ac.uk/id/eprint/3025412 |
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