Let $\{ Z_n\}_{ n\geq 0 }$ 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.
Categories: math.PR