Giacomo Nasuti — University of Parma #
Rare Events and Redundancy in Random Walkers Target Search in a Finite Domain #
Finding a target in a complex environment is a fundamental challenge across natural systems, from chemical reactions to sperm cells reaching an egg. One powerful strategy to reduce search times is redundancy: deploying multiple independent searchers increases the probability of success, particularly when this is driven by rare events. When the underlying stochastic motion features broadly distributed step lengths, rare long relocations dominate the dynamics, making redundancy especially effective.
In this work, we investigate the statistics of extreme events for the mean first passage time in a system of $N$ independent walkers performing power-law-distributed jumps with finite velocity $v$, where target-reaching events are governed by single large fluctuations. We show that the mean first passage time of the fastest walker scales as $1/N$, representing a dramatic speed-up compared to classical Brownian motion, and saturates at the minimum value $X/v$. The model is further extended to include random velocities.
For a fixed $N$, we identify a crossover governed by a critical tail exponent $\alpha_c$, which separates a regime dominated by a single large fluctuation ("big jump") from a regime characterized by Gaussian extreme-value statistics arising from finite sampling effects. From these results, a scaling law linking the number of searchers $N$ to the size $X$ of the search region is derived.
Ultimately, these findings demonstrate how redundancy, combined with rare-event statistics, can efficiently organize target-search processes in complex biological environments. As a prototypical application, we consider mammalian fertilization, deriving a cross-species scaling relation between the number of spermatozoa and the typical uterine size within a coarse-grained description.