Distribution of extreme first passage times of diffusion

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.

Distribution of first-reaction times with target regions on boundaries of shell-like domains

“…The impact of such an extreme event on the FRTs and their PDF has been rather extensively discussed within the recent years, and several analytical analyses have been proposed (see, e.g. [84,[91][92][93][94] and references therein). An extension of such analyses for the geometrical setting studied here should be of interest.…”

Reversible Target-Binding Kinetics of Multiple Impatient Particles

“…They also briefly considered higher-order moments and argued the universality of the logarithmic decay for other diffusive processes. This seminal work was further extended by several authors [44][45][46][47][48][49][50] . For instance, Basnayake et al as well as Lawley and Madrid gave rigorous mathematical proofs for the asymptotic behavior of these moments [44][45][46] (see also Appendix A 1 for new results concerning the behavior of the mean of the slowest FPT T 0 N,N ).…”

Moth Mating: Modeling Female Pheromone Calling and Male Navigational Strategies to Optimize Reproductive Success

“…However, competition arises when multiple males attempt to reach the same female and only the first one can mate. In this section, we consider N competing moths and adapt some known results on first arrival times (Lawley [68]) to give analytical estimates for the average time it takes the first male to reach the female. These results are valid in the N 1 limit.…”

“…In accordance with our previous discussion, we identify the male's first arrival time to the female-generated plume as the first mating time. Lawley [68] analyzes the probability P a (t a < t) that the first arrival time t a out of a cohort of N independent random walkers is less than t and shows that if P a (t a < t) can be approximated in the short time limit as…”