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


Méliot, Pierre-Loïc (2014). The Cut-off Phenomenon for Brownian Motions on Compact Symmetric Spaces. Potential Analysis, 40(4):427-509.

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 $\mathfrak{S}_{n}$ .

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 $\mathfrak{S}_{n}$ .

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Item Type:Journal Article, refereed, original work
Communities & Collections:National licences > 142-005
Dewey Decimal Classification:510 Mathematics
Scopus Subject Areas:Physical Sciences > Analysis
Uncontrolled Keywords:Analysis
Language:English
Date:1 May 2014
Deposited On:07 Aug 2019 07:43
Last Modified:15 Apr 2021 15:07
Publisher:Springer
ISSN:0926-2601
OA Status:Green
Publisher DOI:https://doi.org/10.1007/s11118-013-9356-7

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