An elementary derivation of first and last return times of 1D random walks

TL;DR

This paper provides an elementary derivation of the distributions of first and last return times for one-dimensional random walks, highlighting their heavy-tailed and arcsine law characteristics, with accessible methods for physics students.

Contribution

It introduces simple, direct enumeration methods to derive key return time distributions in 1D random walks, making the concepts accessible to undergraduates.

Findings

First return time distribution is heavy-tailed with infinite mean.

Last return time follows the arcsine law.

Derivation method is accessible to physics undergraduates.

Abstract

Random walks, and in particular, their first passage times, are ubiquitous in nature. Using direct enumeration of paths, we find the first return time distribution of a 1D random walker, which is a heavy-tailed distribution with infinite mean. Using the same method we find the last return time distribution, which follows the arcsine law. Both results have a broad range of applications in physics and other disciplines. The derivation presented here is readily accessible to physics undergraduates, and provides an elementary introduction into random walks and their intriguing properties.