Distribution of extreme first passage times of diffusion

TL;DR

This paper derives a practical approximation for the distribution of the fastest first passage times among many diffusing searchers, using extreme value theory, with explicit formulas and error estimates.

Contribution

It introduces a novel application of extreme value theory to approximate the distribution of extreme FPTs in diffusion, involving the Gumbel distribution and LambertW function.

Findings

Derived explicit formulas for extreme FPT distribution

Provided rigorous error estimates for the approximation

Validated the approximation's high accuracy in diverse scenarios

Abstract

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.