Random Walk with Shrinking Steps: First Passage Characteristics

TL;DR

This paper investigates the mean first passage time of a one-dimensional random walk with exponentially shrinking steps, revealing complex behaviors, fractal structures, and phase transitions depending on the shrinking parameter.

Contribution

It introduces a detailed analysis of first passage times for a discrete random walk with exponentially decreasing step sizes, highlighting rich mathematical structures and transition phenomena.

Findings

Support of distribution evolves in a complex, time-dependent manner.

Critical values of the shrinking parameter induce phase transitions in mean first passage time.

Discrete system exhibits fractal-like and self-repetitive structures.

Abstract

We study the mean first passage time of a one-dimensional random walker with step sizes decaying exponentially in discrete time. That is step sizes go like $\lambda^{n}$ with $\lambda\leq1$ . We also present, for pedagogical purposes, a continuum system with a diffusion constant decaying exponentially in continuous time. Qualitatively both systems are alike in their global properties. However, the discrete case shows very rich mathematical structure, depending on the value of the shrinking parameter, such as self-repetitive and fractal-like structure for the first passage characteristics. The results we present show that the most important quantitative behavior of the discrete case is that the support of the distribution function evolves in time in a rather complicated way in contrast to the time independent lattice structure of the ordinary random walker. We also show that there are critical values of $\lambda$ defined by the equation $\lambda^{K}+2\lambda^{P}-2=0$ with $\{K,N\}\in{\mathcal N}$ where the mean first passage time undergo transitions.