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On empty balls of critical 2-dimensional branching random walks

Shuxiong Zhang

TL;DR

This work resolves the empty-ball problem for critical branching random walks on $\mathbb{R}^d$, with a complete treatment in the challenging case $d=2$ by linking the BRW to a two-dimensional super-Brownian motion. It establishes explicit limit laws for the radius of the largest empty ball under both finite-variance and infinite-variance offspring regimes, revealing distinct phase transitions in the latter and expressing the limits through local extinction probabilities of SBM. The analysis leverages spine decomposition and size-biased BRWs, together with SBM scaling limits, to connect spatial extinction events to geometric gaps in the BRW. Additionally, the paper strengthens maximal displacement estimates in one dimension, improving prior results and clarifying the connection between displacement tails and branching variance. Overall, the results advance the understanding of spatial branching processes and their geometric empty regions through rigorous probabilistic and scaling-limit techniques.

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}, 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 $$\lim_{n\to\infty}\frac{R_n}{\sqrt n}\overset{\text{law}}=\text{non-trival~distri.,}~d=2; \lim_{n\to\infty}{R_n}\overset{\text{law}}=\text{non-trival~distri.,}~d\geq3.$$ 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}.

On empty balls of critical 2-dimensional branching random walks

TL;DR

This work resolves the empty-ball problem for critical branching random walks on , with a complete treatment in the challenging case by linking the BRW to a two-dimensional super-Brownian motion. It establishes explicit limit laws for the radius of the largest empty ball under both finite-variance and infinite-variance offspring regimes, revealing distinct phase transitions in the latter and expressing the limits through local extinction probabilities of SBM. The analysis leverages spine decomposition and size-biased BRWs, together with SBM scaling limits, to connect spatial extinction events to geometric gaps in the BRW. Additionally, the paper strengthens maximal displacement estimates in one dimension, improving prior results and clarifying the connection between displacement tails and branching variance. Overall, the results advance the understanding of spatial branching processes and their geometric empty regions through rigorous probabilistic and scaling-limit techniques.

Abstract

Let be a critical -dimensional branching random walk started from a Poisson random measure whose intensity measure is the Lebesgue measure on . Denote by the radius of the largest empty ball centered at the origin of . In \cite{reves02}, Révész shows that if , then converges in law to an exponential random variable as . Moreover, Révész (2002) conjectured that Later, Hu (2005) \cite{hu05} confirmed the case of . This work confirms the case of . 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}.

Paper Structure

This paper contains 6 sections, 10 theorems, 167 equations.

Table of Contents

  1. Introduction and Main results
  2. Introduction
  3. Main results
  4. Proof of Theorem \ref{['thdimmaxplace']}: Maximal Displacement
  5. Proof of Theorem \ref{['thdim2']}: Empty Ball With Finite Variance
  6. Proof of Theorem \ref{['thdim4']}: Empty Ball With Infinite Variance

Key Result

Theorem 1.1

Assume $d=2$ and $\sigma^2<\infty$. If $\mathbb{E}[((X^{(1)})^+)^4+((X^{(2)})^+)^4]<\infty$, then for any $r>0$, where $\{X_t\}_{t\geq0}$ is a two-dimensional super-Brownian motion with branching mechanism $\psi(u)=\sigma^2u^2$.

Theorems & Definitions (25)

  • Theorem 1.1
  • Remark 1.1
  • Theorem 1.2
  • Remark 1.2
  • Remark 1.3
  • Theorem 1.3
  • Remark 1.4
  • Remark 1.5
  • Lemma 2.1
  • proof
  • ...and 15 more