Extreme hitting probabilities for diffusion

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.

37 pages, 7 figures

References in corpus (9)

• Paradigm shift in diffusion-mediated surface phenomena
• Occupation times of random walks in confined geometries: From random trap model to diffusion limited reactions
• Mortality, Redundancy, and Diversity in Stochastic Search
• Extreme value statistics of ergodic Markov processes from first passage times in the large deviation limit
• Partially Reflected Brownian Motion: A Stochastic Approach to Transport Phenomena
• Diversity Improves Speed and Accuracy in Social Networks
• Extreme first passage times for random walks on networks
• Maximum of N Independent Brownian Walkers till the First Exit From the Half Space
• Depletion of Resources by a Population of Diffusing Species

Cited by in corpus (1)

• First Passage Time for Many Particle Diffusion in Space-Time Random Environments