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

TL;DR

This paper studies the asymptotic behavior of the largest empty ball radius in critical or subcritical branching random walks in multiple dimensions, showing convergence in distribution after renormalization.

Contribution

It extends previous results by establishing the limiting distribution of the largest empty ball radius, depending on offspring law and dimension, for general critical/subcritical branching random walks.

Findings

Renormalized radius converges in law to a non-degenerate distribution.

Scaling depends on offspring law and dimension.

Completes previous work for binary branching Wiener process.

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.