Expected First Return Times for Random Walks on Bounded Grids

TL;DR

This paper presents a general formula for calculating the expected first return time of a random walk on finite graphs and applies it to bounded rectangular grids with various boundary conditions.

Contribution

It introduces a universal formula for expected return times and demonstrates its application to different grid boundary scenarios.

Findings

Derived a general formula for expected first return times.

Calculated return times for various grid boundary conditions.

Provides a framework applicable to finite graphs and grids.

Abstract

We derive a general formula for computing the expected first return time of a random walk on a finite graph. Using this framework, we calculate the expected first return time in various settings over bounded rectangular grids with different boundary conditions.