On empty balls of critical 2-dimensional branching random walks

TL;DR

This paper investigates the asymptotic behavior of the largest empty ball in a critical 2D branching random walk, confirming a conjecture and linking the limit distribution to super-Brownian motion, also extending results to infinite variance offspring laws.

Contribution

It confirms the conjectured limit distribution for the 2D case and characterizes it via super-Brownian motion, also extending results to infinite second moment offspring laws.

Findings

Limit distribution of the largest empty ball in 2D branching random walk is characterized.

The limit distribution is connected to super-Brownian motion.

Results are extended to branching random walks with infinite variance offspring law.

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 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}, R\'ev\'esz shows that if $d=1$, then $R_n/n$ converges in law to an exponential random variable as $n\to\infty$. Moreover, R\'ev\'esz (2002) conjectured that $$\lim_{n\to\infty}\frac{R_n}{\sqrt n}\overset{\text{law}}=\text{non-trival~distri.,}~d=2; \lim_{n\to\infty}{R_n}\overset{\text{law}}=\text{non-trival~distri.,}~d\geq3.$$ 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}.