  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.
