Trajectory-to-trajectory fluctuations in first-passage phenomena in bounded domains

TL;DR

This paper investigates the fluctuations in first-passage times of random walkers in bounded domains, revealing a transition in the distribution shape that impacts the reliability of mean first passage time as a characteristic measure.

Contribution

It introduces a novel diagnostic method using the uniformity index to quantify trajectory fluctuations and demonstrates the universality of the transition in first passage statistics across dimensions.

Findings

The uniformity index distribution transitions from unimodal to bimodal depending on starting position.

Mean First Passage Time is only reliable when starting far from the boundary.

Trajectory-to-trajectory fluctuations are significant near the boundary.

Abstract

We study the statistics of the first passage of a random walker to absorbing subsets of the boundary of compact domains in different spatial dimensions. We describe a novel diagnostic method to quantify the trajectory-to-trajectory fluctuations of the first passage, based on the distribution of the so-called uniformity index $\omega$, measuring the similarity of the first passage times of two independent walkers starting at the same location. We show that the characteristic shape of $P(\omega)$ exhibits a transition from unimodal to bimodal, depending on the starting point of the trajectories. From the study of different geometries in one, two and three dimensions, we conclude that this transition is a generic property of first passage phenomena in bounded domains. Our results show that, in general, the Mean First Passage Time (MFPT) is a meaningful characteristic measure of the first passage behaviour only when the Brownian walkers start sufficiently far from the absorbing boundary. Strikingly, in the opposite case, the first passage statistics exhibit large trajectory-to-trajectory fluctuations and the MFPT is not representative of the actual behaviour.