Method of images for one-dimensional discrete random walk under a reflecting barrier

TL;DR

This paper revisits the method of images for a one-dimensional discrete symmetric random walk with a reflecting barrier, clarifying its correct application by proposing a shifted image position to resolve previous inconsistencies.

Contribution

It demonstrates that the method of images can be correctly applied to the reflecting barrier case by shifting the image position, resolving prior inconsistencies.

Findings

Method of images is valid with a shifted image position.

Clarifies previous inconsistencies in applying the method.

Provides a corrected approach for transition probability calculations.

Abstract

The transition probability for a one-dimensional discrete symmetric random walk under a reflecting barrier was once given by the method of images. [S. Chandrasekhar, Rev. Mod. Phys. 15, 1 (1943).] However, several inconsistencies have been reported when the method of images is applied in cases where a reflecting barrier is considered, even after the exact solution has been obtained. Here, we explicitly show that the method of images becomes applicable if the image position is shifted.