We study the statistics of the first passage of a random walker to absorbingsubsets of the boundary of compact domains in different spatial dimensions. Wedescribe a novel diagnostic method to quantify the trajectory-to-trajectoryfluctuations of the first passage, based on the distribution of the so-calleduniformity index $omega$, measuring the similarity of the first passage timesof two independent walkers starting at the same location. We show that thecharacteristic shape of $P(omega)$ exhibits a transition from unimodal tobimodal, depending on the starting point of the trajectories. From the study ofdifferent geometries in one, two and three dimensions, we conclude that thistransition is a generic property of first passage phenomena in bounded domains.Our results show that, in general, the Mean First Passage Time (MFPT) is ameaningful characteristic measure of the first passage behaviour only when theBrownian walkers start sufficiently far from the absorbing boundary.Strikingly, in the opposite case, the first passage statistics exhibit largetrajectory-to-trajectory fluctuations and the MFPT is not representative of theactual behaviour.
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