Convergence of directed random graphs to the Poisson-weighted infinite tree

Weconsider a directed graph on the integerswith a directed edge fromvertex i to j present with probability n−1, whenever i < j , independently of all other edges. Moreover, to each edge (i, j) we assign weight n−1(j − i). We show that the closure of vertex 0 in such a weighted random graph converges in distribution to the Poisson-weighted infinite tree as n → ∞. In addition, we derive limit theorems for the length of the longest path in the subgraph of the Poisson-weighted infinite tree which has all vertices at weighted distance of at most ρ from the root.