On the empty balls of a critical or subcritical branching random walk (2212.12833v1)

Abstract: 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.