Probabilistic and analytical properties of the last passage percolation constant in a weighted random directed graph

Abstract

To each edge (i,j), i<j, of the complete directed graph on the integers we assign unit weight with probability p or weight x with probability 1−p, independently from edge to edge, and give to each path weight equal to the sum of its edge weights. If W0,nx is the maximum weight of all paths from 0 to n then W0,nx/n→Cp(x), as n→∞, almost surely, where Cp(x) is positive and deterministic. We study Cp(x) as a function of x, for fixed 0<p<1, and show that it is a strictly increasing convex function that is not differentiable if and only if x is a nonpositive rational or a positive integer except 1 or the reciprocal of it. We allow x to be any real number, even negative, or, possibly, −∞. The case x=−∞ corresponds to the well-studied directed version of the Erdős–Rényi random graph (known as Barak–Erdős graph) for which Cp(−∞)=limx→−∞Cp(x) has been studied as a function of p in a number of papers.