  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.
