Poisson.Dirichlet branching random walks

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 (v1,v2,.), where each vj is the sum of j independent Exponential(1) random variables and the different vi 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).