Rare Events, Many Searchers, and Fast Target Reaching in a Finite Domain

Abstract: Finding a target in a complex environment is a fundamental challenge in nature, from chemical reactions to sperm reaching an egg. An effective strategy to reduce the time needed to reach a target is to deploy many searchers, increasing the likelihood that at least one will succeed by using the statistics of rare events. When the underlying stochastic process involves broadly distributed step sizes, rare long jumps dominate the dynamics, making the use of multiple searchers particularly powerful. We investigate the statistics of extreme events for the mean first passage time in a system of $N$ independent walkers moving with jumps distributed according to a power law, where target-reaching is governed by single, large fluctuations. We show that the mean first passage time of the fastest walker scales as $\langle \tau_N \rangle \sim 1/N$, representing a dramatic speed-up compared to classical Brownian search strategies. From this, we derive a scaling law relating the number of walkers required to reach a target within a given time to the size $X$ of the search region. As an application, we model biological fertilization, predicting how the optimal number of spermatozoa scales with uterus size across species. Our predictions match empirical data, suggesting that evolution may have exploited rare-event dynamics and broadly distributed motion to optimize reproductive success. This theory applies broadly to any population of searchers operating within a region of size $X$, providing a universal framework for efficient search in disordered or high-dimensional environments.