The cut-off phenomenon for Brownian motions on symmetric spaces of compact type

TL;DR

This paper establishes the cut-off phenomenon for Brownian motions on various symmetric spaces of compact type, providing explicit bounds and answering a recent open question in the field.

Contribution

It proves the cut-off phenomenon for Brownian motions on classical symmetric spaces of compact type, with explicit bounds and a precise cutoff time.

Findings

Explicit lower bounds for total variation distance before cutoff

Explicit upper bounds for total variation distance after cutoff

Identification of cutoff time as proportional to log n

Abstract

We prove the cut-off phenomenon in total variation distance for the Brownian motions traced on the classical symmetric spaces of compact type, that is to say: (1) the classical simple compact Lie groups: special orthogonal groups, special unitary groups and compact symplectic groups; (2) the real, complex and quaternionic Grassmannian varieties (including the real spheres and complex or quaternionic projective spaces); (3) the spaces of structures: SU(n)/SO(n), SO(2n)/U(n), SU(2n)/USp(n), and USp(n)/U(n). In each case, we give explicit lower bounds for the total variation distance DTV(mu_t,Haar) if t < tcut-off = a log n, and explicit upper bounds if t > tcut-off. This gives in particular an answer to a question raised in recent papers by Chen and Saloff-Coste.