AI 中文总结
该论文研究简单与临界分支随机游走香肠的容量及击中时间，通过环形分解临界分支随机游走的方法，得到相关容量与击中概率的估计，该分解或有进一步应用。
AI 中文摘要
对于有限集$A\subseteq\mathbb Z^d$，我们研究将$A$沿简单分支随机游走或临界分支随机游走的闵可夫斯基和运输得到的香肠的容量。我们首先建立随机游走香肠和分支香肠的定量容量估计，进而得到高维中多种游走的闵可夫斯基和击中有限集的概率估计。我们的方法基于将临界分支随机游走分解为连续波的环形分解，该分解可能具有进一步的应用。
英文摘要
For a finite set $A\subseteq\mathbb Z^d$, we study the capacities of a sausage obtained by transporting $A$ along the Minkowski sum of simple or critical branching random walks. We first establish quantitative capacity estimates for the random walk sausage and the branching sausage. This in turn, gives us estimates for the probability that Minkowski sums of a mixture of walks hit a finite set in high dimensions. Our approach is based on an annular decomposition of a critical branching random walk into successive waves, which may have further applications.
