A note on random walks in a hypercube

TL;DR

This paper analyzes the behavior of simple random walks on an n-dimensional hypercube, deriving probabilities of hitting specific vertices, which generalizes previous models in electric network theory.

Contribution

It extends existing models by calculating hitting probabilities for vertices sharing an edge in hypercube random walks, broadening understanding of such stochastic processes.

Findings

Derived hitting probabilities for vertices sharing an edge in hypercube random walks

Generalized previous models in electric network theory to higher dimensions

Provided analytical tools for studying random walks in complex structures

Abstract

We study a simple random walk on an n-dimensional hypercube. For any starting position we find the probability of hitting vertex a before hitting vertex b, whenever a and b share the same edge. This generalizes the model in Doyle, P., and Snell, J., "Random Walks and Electric Networks", Mathematical Association of America, 1984 (see Exercise 1.3.7 there).