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

Abstract

This paper proves that the separation convergence toward 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 (Arnaudon, Coulibaly-Pasquier and Miclo (2020)) to deduce the wanted cut-off phenomenon.