On the empty balls of a critical or subcritical branching random walk

Abstract

Let $\{Z_n{\}}_{n\ge 0}$ be a critical or subcritical $d$-dimensional branching random walk started from a Poisson random measure whose intensity measure is the Lebesugue measure on ${\mathbb{R}}^d$. Denote by $R_n:=\sup \{u>0:Z_n(\{x\in {\mathbb{R}}^d:|x|<u\})=0\}$ the radius of the largest empty ball centered at the origin of $Z_n$. In this work, we prove that after suitable renormalization, $R_n$ converges in law to some non-degenerate distribution as $n\to \infty$. Furthermore, our work shows that the renormalization scales depend on the offspring law and the dimension of the branching random walk, which completes the results of \cite{reves02} for the critical binary branching Wiener process.