The Cut-off Phenomenon for Brownian Motions on Compact Symmetric Spaces

Abstract

In this paper, 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 SO(n), special unitary groups SU(n) and compact symplectic groups USp(n);

• 2. the real, complex and quaternionic Grassmannian varieties (including the real spheres, and the complex or quaternionic projective spaces when q = 1): SO(p + q)/(SO(p)×SO(q)), SU(p + q)/S(U(p)×U(q)) and USp(p + q)/(USp(p)×USp(q));

• 3. the spaces of real, complex and quaternionic structures: SU(n)/SO(n), SO(2n)/ U(n), SU(2n)/USp(n) and USp(n)/UU(n).

Denoting μ _t the law of the Brownian motion at time t, we give explicit lower bounds for d _TV(μ _t ,Haar) if $t < t_{\text{cut-of\/f}}=\alpha \log n$, and explicit upper bounds if $t > t_{\text{cut-of\/f}}$. This provides in particular an answer to some questions raised in recent papers by Chen and Saloff-Coste. Our proofs are inspired by those given by Rosenthal and Porod for products of random rotations in SO(n), and by Diaconis and Shahshahani for products of random transpositions in $\mathfrak{S}_{n}$.