### 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 language | English |
---|---|

Pages (from-to) | 702-733 |

Journal | Annals of Applied Probability |

Volume | 22 |

Issue number | 2 |

DOIs | |

Publication status | Published - Apr 2012 |

### Keywords

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

## Fingerprint Dive into the research topics of 'Limit theorems for a random directed slab graph'. Together they form a unique fingerprint.

## Cite this

Denisov, D., Foss, S., & Konstantopoulos, T. (2012). Limit theorems for a random directed slab graph.

*Annals of Applied Probability*,*22*(2), 702-733. https://doi.org/10.1214/11-AAP783