Takis Konstantopoulos: Limit theorems for a random directed slab graph

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. 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 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 random matrix.