On empty balls of critical 2-dimensional branching random walks

Abstract: Let ${Z_n}{n\geq 0 }$ be a critical $d$-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 \cite{reves02} , R\'ev\'esz shows that if $d=1$, then $R_n/n$ converges in law to an exponential distribution as $n\to\infty$. Moreover, R\'ev\'esz (2002) conjectured that $$\lim{n\to\infty}\frac{R_n}{\sqrt n}\overset{\text{law}}=\text{non-degenerate~for~} d=2,\lim_{n\to\infty}{R_n}\overset{\text{law}}=\text{non-degenerate~for~} d\geq3.$$ Later, Hu (2005) \cite{hu05} confirmed the case of $d\geq3$. In this work, we have completely confirmed the case of $d=2$. It turns out that the limit distribution can be precisely characterized through the super-Brownian motion. Moreover, we also give complete results of the branching random walk without finite second moment. As a by-product, this article also improves the assumption of maximal displacements of branching random walks \cite[Theorem 1]{lalley2015} to optimal.