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

2012/10/01 by Pierre‐Loïc Méliot, Pierre-Loïc Méliot, Méliot, Pierre-Loïc
Mathematics · #Advanced Algebra and Geometry #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Probability (math.PR) #math.PR

paper · pdf · doi:10.48550/arxiv.1210.0480

Revised version, shortened and with certain computations moved into appendices at the end of the paper. 58 pages, 3 figures

openalex publication_date 2012/10/01 · arxiv created 2013/02/04 · arxiv updated 2013/02/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

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