Limit theorems for a random directed slab graph

TL;DR

This paper studies the asymptotic behavior of the longest paths in a class of random directed graphs on integers and slabs, establishing limit theorems including a CLT and connections to GUE eigenvalues.

Contribution

It introduces a regenerative approach for analyzing maximal path lengths and extends results to slab graphs, linking the limiting distribution to GUE eigenvalues.

Findings

Proved limit theorems for maximal path length in directed graphs.

Established a central limit theorem for slab graphs.

Connected the distribution of longest paths to GUE eigenvalues.

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).