  A variety of systems in physics, chemistry, biology, and psychology are
modeled in terms of diffusing "searchers" looking for "targets." Examples range
from gene regulation, to cell sensing, to human decision-making. A commonly
studied statistic in these models is the so-called hitting probability for each
target, which is the probability that a given single searcher finds that
particular target. However, the decisive event in many systems is not the
arrival of a given single searcher to a target, but rather the arrival of the
fastest searcher to a target out of many searchers. In this paper, we study the
probability that the fastest diffusive searcher hits a given target in the many
searcher limit, which we call the extreme hitting probability. We first prove
an upper bound for the decay of the probability that the searcher finds a
target other than the closest target. This upper bound applies in very general
settings and depends only on the relative distances to the targets.
Furthermore, we find the exact asymptotics of the extreme hitting probabilities
in terms of the short-time distribution of when a single searcher hits a
target. These results show that the fastest searcher always hits the closest
target in the many searcher limit. While this fact is intuitive in light of
recent results on the time it takes the fastest searcher to find a target, our
results give rigorous, quantitative estimates for the extreme hitting
probabilities. We illustrate our results in several examples and numerical
simulations.
