Extremal points of high dimensional random walks and mixing times of a Brownian motion on the sphere

TL;DR

This paper analyzes the probability of the origin being an extremal point in high-dimensional random walks and derives bounds on the mixing times of spherical Brownian motion, revealing asymptotic behaviors in high dimensions.

Contribution

It provides new asymptotic estimates for extremal points in high-dimensional random walks and bounds on the mixing times of spherical Brownian motion.

Findings

Probability of origin as extremal point transitions around 1/2 within specific step ranges.

Derived bounds for the ?pi/2-covering time of spherical Brownian motion.

Established asymptotic behavior of extremal points in high-dimensional settings.

Abstract

We derive asymptotics for the probability of the origin to be an extremal point of a random walk in R^n. We show that in order for the probability to be roughly 1/2, the number of steps of the random walk should be between e^{c n / log n}$ and e^{C n log n}. As a result, we attain a bound for the ?pi/2-covering time of a spherical brownian motion.