First-passage on disordered intervals

TL;DR

This paper develops a simplified method using backward equations and generating functions to analyze first-passage times in disordered one-dimensional systems, revealing bimodal distributions and differences across realizations.

Contribution

It introduces a new, simpler approach for calculating moments and distributions of first-passage times in disordered intervals, improving upon previous methods.

Findings

Distribution of first-passage times can be bimodal for certain disorder realizations.

Significant disparities in first-passage times exist between different disorder realizations.

The developed approach efficiently computes all moments and distributions of first-passage times.

Abstract

We investigate the first-passage properties of nearest-neighbor hopping on a finite interval with disordered hopping rates. We develop an approach that relies on the backward equation, in conjunction with probability generating functions, to obtain all moments, as well as the distribution of first-passage times. Our approach is simpler than previous approaches that are based on either the forward equation or recursive method, in which the $m^{\rm th}$ moment requires all preceding moments. For the interval with two absorbing boundaries, we elucidate the disparity in the first-passage times between different realizations of the hopping rates and also unexpectedly find that the distribution of first-passage times can be \emph{bimodal} for certain realizations of the hopping rates.