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

2022/07/13 by Arnaudon, Marc, Coulibaly-Pasquier, Koléhé Abdoulaye, Miclo, Laurent
#FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2207.06052

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.

Related

• Extremal points of high dimensional random walks and mixing times of a Brownian motion on the sphere
• Dimensionality-induced dynamical phase transition in the large deviation of local time density for Brownian motion
• Hitting times to spheres of Brownian motions with and without drifts
• On the separation cut-off phenomenon for Brownian motions on high dimensional rotationally symmetric compact manifolds
• The Brownian motion as the limit of a deterministic system of\n hard-spheres