Extreme escape from a cusp: when does geometry matter for the fastest Brownian particles moving in crowded cellular environments?

Abstract: We study here the extreme statistics of Brownian particles escaping from a cusp funnel: the fastest Brownian particles among $n$ follow an ensemble of optimal trajectories located near the shortest path from the source to the target. For the time of such first arrivers, we derive an asymptotic formula that differs from the classical narrow escape and dire strait obtained for the mean first passage time. Consequently, when particles are initially distributed at a given distance from a cusp, the fastest do see some properties characterizing the cusp geometry. Therefore, when many particles diffuse around impermeable obstacles, the geometry plays a role in the time to reach a target. In the biological context of cellular transduction with signalling molecules, having to escape such cusp-like domains slows down fast signaling. To conclude, generating multiple copies of the same molecule helps bypass a crowded environment to transmit a molecular signal quickly.