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 W^x_{0,n} is the maximum weight of all paths from 0 to n then W^x_{0,n}/n \to C_p(x), as n\to\infty, almost surely, where C_p(x) is positive and deterministic. We study C_p(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, -\infty. The case x=-\infty corresponds to the well-studied directed version of the Erd"os-R'enyi random graph (known as Barak-Erd"os graph) for which C_p(-\infty) = lim_{x\to -\infty} C_p(x) has been studied as a function of p in a number of papers.