On empty balls of critical 2-dimensional branching random walks

Let $\{Z_n{\}}_{n\ge 0}$ be a critical $d$-dimensional branching random walk started from a Poisson random measure whose intensity measure is the Lebesgue measure on ${\mathbb{R}}^d$. Denote by $R_n:=\sup \{u>0:Z_n(\{x\in {\mathbb{R}}^d:\mathrm{\mid }x\mathrm{\mid }<u\})=0\}$ the radius of the largest empty ball centered at the origin of $Z_n$. In \cite{reves02}, Révész shows that if $d=1$, then $R_n\mathrm{/}n$ converges in law to an exponential random variable as $n\to \mathrm{\infty }$. Moreover, Révész (2002) conjectured that $$\underset{n\to \mathrm{\infty }}{\lim }\frac{R_n}{\sqrt{n}}\text{non-trival distri., }d=2;\underset{n\to \mathrm{\infty }}{\lim }{R}_n\text{non-trival distri., }d\ge 3.$$ Later, Hu (2005) \cite{hu05} confirmed the case of $d\ge 3$. This work confirms 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 empty balls of the branching random walk with infinite second moment offspring law. As a by-product, this article also improves the assumption of maximal displacements of branching random walks \cite[Theorem 1]{lalley2015}.