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

Abstract

We consider a directed graph on the integers with a directed edge from vertex $i$ to $j$ present with probability $n^{-1}$, whenever $i\lt j$, 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 $n\rightarrow \infty$. 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 $\rho$ from the root.