Extreme hitting probabilities for diffusion

TL;DR

This paper analyzes the probability that the fastest among many diffusive searchers hits a target, establishing bounds and asymptotics that show the fastest searcher always targets the closest one, with implications across various scientific fields.

Contribution

It provides rigorous bounds and exact asymptotics for extreme hitting probabilities in diffusive search models, emphasizing the dominance of the closest target in the many searcher limit.

Findings

Fastest searcher always hits the closest target.

Derived upper bounds depend only on relative distances.

Established asymptotics based on short-time hitting distributions.

Abstract

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.