{"title":"On the empty balls of a critical or subcritical branching random walk","authors":"Shuxiong Zhang, Jie Xiong","doi":"10.1007/s10473-024-0525-0","DOIUrl":null,"url":null,"abstract":"<p>Let {<i>Z</i><sub><i>n</i></sub>}<sub><i>n</i>≥0</sub> be a critical or subcritical <i>d</i>-dimensional branching random walk started from a Poisson random measure whose intensity measure is the Lebesugue measure on ℝ<sup><i>d</i></sup>. Denote by <i>R</i><sub><i>n</i></sub>:= sup{<i>u</i> > 0: <i>Z</i><sub><i>n</i></sub>({<i>x</i> ∈ ℝ<sup><i>d</i></sup>: ∣<i>x</i>∣ < <i>u</i>}) = 0} the radius of the largest empty ball centered at the origin of <i>Z</i><sub><i>n</i></sub>. In this work, we prove that after suitable renormalization, <i>R</i><sub><i>n</i></sub> converges in law to some non-degenerate distribution as <i>n</i> → ∈. 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.</p>","PeriodicalId":50998,"journal":{"name":"Acta Mathematica Scientia","volume":"369 1","pages":""},"PeriodicalIF":1.2000,"publicationDate":"2024-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Acta Mathematica Scientia","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s10473-024-0525-0","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
Let {Zn}n≥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 ℝd. Denote by Rn:= sup{u > 0: Zn({x ∈ ℝd: ∣x∣ < u}) = 0} the radius of the largest empty ball centered at the origin of Zn. In this work, we prove that after suitable renormalization, Rn converges in law to some non-degenerate distribution as n → ∈. 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.
期刊介绍:
Acta Mathematica Scientia was founded by Prof. Li Guoping (Lee Kwok Ping) in April 1981.
The aim of Acta Mathematica Scientia is to present to the specialized readers important new achievements in the areas of mathematical sciences. The journal considers for publication of original research papers in all areas related to the frontier branches of mathematics with other science and technology.