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We study here the extreme statistics of Brownian particles escaping from acusp funnel: the fastest Brownian particles among $n$ follow an ensemble ofoptimal trajectories located near the shortest path from the source to thetarget. For the time of such first arrivers, we derive an asymptotic formulathat differs from the classical narrow escape and dire strait obtained for themean first passage time. Consequently, when particles are initially distributedat a given distance from a cusp, the fastest do see some propertiescharacterizing the cusp geometry. Therefore, when many particles diffuse aroundimpermeable 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 crowdedenvironment to transmit a molecular signal quickly.

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