Limit theorems for a random directed slab graph

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 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).