Stopping Times of Random Walks on a Hypercube

TL;DR

This paper investigates the stopping times of random walks on high-dimensional hypercubes, focusing on self-intersection, return times, and visitation times, providing asymptotic distributions and bounds using coupling methods.

Contribution

It presents new results on the asymptotic behavior and bounds of stopping times for hypercube random walks, including visits to sets and self-intersections, with coupling techniques.

Findings

Asymptotic distributions for return and visitation times

Bounds on stopping times for various events

Coupling methods effectively used for proofs

Abstract

A random walk on a $N$-dimensional hypercube is a discrete time stochastic process whose state space is the set $\{-1,+1\}^{N}$, which has uniform probability of reaching any neighbour state, and probability zero of reaching a non-neighbour state, in one step. This random walk is often studied as a process associated with the Ehrenfest Urn Model. This paper aims to present results about the time that such random walk takes to self-intersect and to return to a set of states. We also present results about the time that the random walk on a hypercube takes to visit a given set and a random set of states. Asymptotic distributions and bounds are presented for these times. The coupling of random walks is widely used as a tool to prove the results.