Consider N independently diffusing particles that reversibly bind to a target. We study a problem recently introduced by Grebenkov of finding the first passage time (FPT) for K of the N particles to be simultaneously bound to the target. Since binding is reversible, bound particles may unbind before the requisite K particles bind to the target. This so-called “impatience” leads to a delicate temporal coupling between particles. Recent work found the mean of this FPT in the case that N = K = 2 in a one-dimensional spatial domain. In this paper, we approximate the full distribution of the FPT for any N ≥ K ≥ 1 in a broad class of domains in any space dimension. We prove that our approximation (i) is exact in the limit that the target and/or binding rate is small and (ii) is an upper bound in any parameter regime. Our approximation is analytically tractable and we give explicit formulas for its mean and distribution. These results reveal that the FPT can depend sensitively and nonlinearly on both K and N. The analysis is accompanied by detailed numerical simulations.

First passage time (FPT) theory is often used to estimate timescales in cellular and molecular biology. While the overwhelming majority of studies have focused on the time it takes a given single Brownian searcher to reach a target, cellular processes are instead often triggered by the arrival of the first molecule out of many molecules. In these scenarios, the more relevant timescale is the FPT of the first Brownian searcher to reach a target from a large group of independent and identical Brownian searchers. Though the searchers are identically distributed, one searcher will reach the target before the others and will thus have the fastest FPT. This fastest FPT depends on extremely rare events and its mean can be orders of magnitude faster than the mean FPT of a given single searcher. In this paper, we use rigorous probabilistic methods to study this fastest FPT. We determine the asymptotic behavior of all the moments of this fastest FPT in the limit of many searchers in a general class of two and three dimensional domains. Our results correct and extend recent results on this problem.

Many physical, chemical, and biological systems depend on the first passage time (FPT) of a diffusive searcher to a target. Typically, this FPT is much slower than the characteristic diffusion timescale. For example, this is the case if the target is small (the narrow escape problem) or if the searcher must escape a potential well. However, many systems depend on the first time a searcher finds the target out of a large group of searchers, which is the so-called extreme FPT. Since this extreme FPT vanishes in the limit of many searchers, the prohibitively slow FPTs of diffusive search can be negated by deploying enough searchers. However, the notion of ‘enough searchers’ is poorly understood. How can one determine if a system is in the slow regime (dominated by small targets or a deep potential, for example) or the fast regime (dominated by many searchers)? How can one estimate the extreme FPT in these different regimes? In this paper, we answer these questions by deriving conditions which ensure that a system is in either regime and finding approximations of the full distribution and all the moments of the extreme FPT in these regimes. Our analysis reveals the critical effect that initial searcher distribution and target reactivity can have on extreme FPTs.