On empty balls of critical 2-dimensional branching random walks

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 Lebesgue 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}, Révész shows that if $d=1$, then $R_n/n$ converges in law to an exponential random variable as $n\to\infty$. Moreover, Révész (2002) conjectured that $$\lim{n\to\infty}\frac{R_n}{\sqrt n}\overset{\text{law}}=\text{non-trival~distri.,}d=2; \lim_{n\to\infty}{R_n}\overset{\text{law}}=\text{non-trivaldistri.,}~d\geq3.$$ Later, Hu (2005) \cite{hu05} confirmed the case of $d\geq3$. 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}.

💡 Research Summary

This paper studies the “empty‑ball” problem for critical branching random walks (BRWs) in two dimensions and beyond. Starting from a Poisson random measure with Lebesgue intensity, each particle reproduces according to a critical offspring distribution (mean 1) and then moves independently according to a centered step‑distribution (X) with non‑degenerate covariance matrix (C). For generation (n) the radius of the largest ball centred at the origin that contains no particles is denoted by
\