Poisson-Dirichlet branching random walks

TL;DR

This paper analyzes the expected minimal position in specific branching random walks with complex displacement structures, including Poisson-Dirichlet and Poisson-weighted trees, also determining the expected height of a recursive tree.

Contribution

It provides precise asymptotic estimates for the minimal position in certain branching random walks and applies these results to recursive trees.

Findings

Expected minimal position determined within O(1) accuracy

Results apply to Poisson-Dirichlet branching random walk

Expected height of a random recursive tree also determined

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,...), where each v_j is the sum of j independent 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).