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

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 Wx0,n is the maximum weight of all paths from 0 to n then Wx0,n/n -> Cp(x), as n -> infinity, 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, -infinity. The case x = -infinity corresponds to the well-studied directed version of the Erdos-Renyi random graph (known as Barak-Erdos graph) for which Cp(-infinity) = limx ->-infinity Cp(x) has been studied as a function of p in a number of papers.