On the separation cut-off phenomenon for Brownian motions on high dimensional spheres

TL;DR

This paper demonstrates that high-dimensional Brownian motions on spheres exhibit an abrupt separation cut-off near a specific time scale, using a new perturbative approach to estimate hitting times and establish the phenomenon.

Contribution

It introduces a novel elementary perturbative method for estimating hitting times, revealing the sharp cut-off in separation convergence for high-dimensional spheres.

Findings

Separation convergence occurs abruptly around ln(n)/n time scale.

The new approach simplifies analysis of hitting times in small noise regimes.

The results confirm the cut-off phenomenon for high-dimensional spherical Brownian motions.

Abstract

This note proves that the separation convergence towards the uniform distribution abruptly occurs at times around ln(n)/n for the (time-accelerated by 2) Brownian motion on the sphere with a high dimension n. The arguments are based on a new and elementary perturbative approach for estimating hitting times in a small noise context. The quantitative estimates thus obtained are applied to the strong stationary times constructed in a privious article by the authors to deduce the wanted cut-off phenomenon.