On empty balls of critical 2-dimensional branching random walks
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911400089092096 |
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| author | Zhang, Shuxiong |
| author_facet | Zhang, Shuxiong |
| contents | 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}. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_16225 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On empty balls of critical 2-dimensional branching random walks Zhang, Shuxiong Probability 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}. |
| title | On empty balls of critical 2-dimensional branching random walks |
| topic | Probability |
| url | https://arxiv.org/abs/2502.16225 |