Barak-Erdős graphs and the infinite-bin model

A Barak-Erdős graph is a directed acyclic version of an Erdős-Rényi graph. It is obtained by performing independent bond percolation with parameter p on the complete graph with vertices {1, ..., n}, where the edge between two vertices i < j is directed from i to j. The length of the longest path in this graph grows linearly with the number of vertices, at rate C(p). In this article, we use a coupling between Barak-Erdős graphs and infinite-bin models to provide explicit estimates on C(p). For p > 1/2, we prove the analyticity of C(p) and we compute its power series expansion. We also show that C(p) has a first derivative but no second derivative at p = 0, providing a two-term asymptotic expansion using a coupling with branching random walks.