论文标题
在关键或亚临界分支随机行走的空球上
On the empty balls of a critical or subcritical branching random walk
论文作者
论文摘要
令$ \ {z_n \} _ {n \ geq 0} $为关键或亚临界$ d $ - 维二维分支随机步行，从泊松随机度量开始，其强度度量是$ \ mathbb {r}^d $的强度度量。用$ r_n：= \ sup \ {u> 0：z_n（\ {x \ in \ mathbb {r}^d：| x | <u \}）= 0 \} $最大的空球的半径以$ z_n $为中心。在这项工作中，我们证明，经过适当的翻新化，$ r_n $在法律上收集到某些非脱位分布，称为$ n \ to \ infty $。此外，我们的工作表明，重新归一化量表取决于后代定律和分支随机步行的维度，这为关键的二进制分支维也纳过程完成了\ cite {reves02}的结果。
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, which completes the results of \cite{reves02} for the critical binary branching Wiener process.
