Poisson–Dirichlet branching random walks

Abstract

We determine, to within $O(1)$, the expected minimal position at level $n$ in certain branching random walks. The walks under consideration have displacement vector $(v_{1},v_{2},\ldots)$, where each $v_{j}$ is the sum of $j$ independent $\operatorname{Exponential}(1)$ random variables and the different $v_{i}$ need not be independent. In particular, our analysis applies to the Poisson–Dirichlet branching random walk and to the Poisson-weighted infinite tree. As a corollary, we also determine the expected height of a random recursive tree to within $O(1)$.