Many events in biology are triggered when a diffusing searcher finds a target, which is called a first passage time (FPT). The overwhelming majority of FPT studies have analyzed the time it takes a single searcher to find a target. However, the more relevant timescale in many biological systems is the time it takes the fastest searcher(s) out of many searchers to find a target, which is called an extreme FPT. In this paper, we apply extreme value theory to find a tractable approximation for the full probability distribution of extreme FPTs of diffusion. This approximation can be easily applied in many diverse scenarios, as it depends on only a few properties of the short time behavior of the survival probability of a single FPT. We find this distribution by proving that a careful rescaling of extreme FPTs converges in distribution as the number of searchers grows. This limiting distribution is a type of Gumbel distribution and involves the LambertW function. This analysis yields new explicit formulas for approximations of statistics of extreme FPTs (mean, variance, moments, etc.) which are highly accurate and are accompanied by rigorous error estimates.

中文翻译：

---

扩散的首次通过时间的分布。

当一个分散的搜索者找到一个目标时，就会触发生物学中的许多事件，这称为首次通过时间（FPT）。绝大多数FPT研究已经分析了单个搜索者找到目标所花费的时间。但是，在许多生物系统中，更相关的时间尺度是许多搜索者中最快的搜索者找到目标的时间，这称为极限FPT。在本文中，我们应用极值理论来为扩散的极限FPT的全概率分布找到一个易于处理的近似值。这种近似可以轻松地应用于许多不同的场景，因为它仅取决于单个FPT的生存概率的短时行为的一些属性。我们发现这种分布是通过证明随着搜索者数量的增加，对极端FPT的仔细调整会收敛于分布。此极限分布是Gumbel分布的一种，涉及LambertW函数。该分析为极端FPT（均值，方差，力矩等）的统计近似值提供了新的显式公式，这些公式非常准确，并伴随严格的误差估计。