ON THE EMPTY BALLS OF A CRITICAL OR SUBCRITICAL BRANCHING RANDOM WALK*

doi:10.1007/s10473-024-0525-0

Acta mathematica scientia,Series B ›› 2024, Vol. 44 ›› Issue (5): 2051-2072.doi: 10.1007/s10473-024-0525-0

ON THE EMPTY BALLS OF A CRITICAL OR SUBCRITICAL BRANCHING RANDOM WALK^*

Shuxiong Zhang^1,†, Jie Xiong²

1. School of Mathematics and Statistics, Anhui Normal University, Wuhu 241002, China;
2. Department of Mathematics and SUSTech International Center for Mathematics, Southern University of Science and Technology, Shenzhen 518055, China

Received:2022-05-09 Revised:2024-05-21 Online:2024-10-25 Published:2024-10-22
Contact: †Shuxiong Zhang, E-mail,: shuxiong.zhang@mail.bnu.edu.cn
About author:Jie Xiong, E-mail,: xiongj@sustech.edu.cn
Supported by:
Jie Xiong's research was supported by the National Key R&D Program of China (2022YFA1006102).

Abstract

Abstract: Let $\{Z_n\}_{n\geq 0 }$ be a critical or subcritical $d$-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 this work, we prove that after suitable renormalization, $R_n$ converges in law to some non-degenerate distribution as $n\to\infty$. Furthermore, our work shows that the renormalization scales depend on the offspring law and the dimension of the branching random walk. This completes the results of Révész [13] for the critical binary branching Wiener process.

Key words: empty ball, dimension, branching random walk, super-Brownian motion

CLC Number:

60J68

Shuxiong Zhang, Jie Xiong. ON THE EMPTY BALLS OF A CRITICAL OR SUBCRITICAL BRANCHING RANDOM WALK^*[J].Acta mathematica scientia,Series B, 2024, 44(5): 2051-2072.

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ON THE EMPTY BALLS OF A CRITICAL OR SUBCRITICAL BRANCHING RANDOM WALK^*

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