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Cutoff in separation for compact harmonic manifold

Koléhè Coulibaly-Pasquier, Magalie Bénéfice

Abstract

We show that cutoff in separation occurs for Brownian motion in some families of compact harmonic manifolds. We compute the cutoff time and windows in four families of compact harmonic manifold namely S n , CP n , HP n and RP n (the first three families are the only families of compact simply connected harmonic manifolds, see [19]). The proof is based on sharp strong stationary times and sufficiently accurate asymptotic expansions of their means and variances.

Cutoff in separation for compact harmonic manifold

Abstract

We show that cutoff in separation occurs for Brownian motion in some families of compact harmonic manifolds. We compute the cutoff time and windows in four families of compact harmonic manifold namely S n , CP n , HP n and RP n (the first three families are the only families of compact simply connected harmonic manifolds, see [19]). The proof is based on sharp strong stationary times and sufficiently accurate asymptotic expansions of their means and variances.
Paper Structure (22 sections, 20 theorems, 142 equations, 1 table)

This paper contains 22 sections, 20 theorems, 142 equations, 1 table.

Table of Contents

  1. Overview
  2. Introduction
  3. Main Results
  4. Paper structure
  5. Preliminaries
  6. Dual process for Brownian motion
  7. Harmonic manifolds
  8. Strong stationary time on harmonic manifolds
  9. Intertwined relations on harmonic manifolds
  10. Study of the first covering time
  11. Cutoff for the complex projective space
  12. Computation of the mean, first part of the proof of Proposition \ref{['PR1']}
  13. The computation of the variance, second part of the proof of Proposition \ref{['PR1']}
  14. Cutoff for the quaternionic projective space
  15. Computation of the mean, first part of the proof of Proposition \ref{['PR2']}
  16. ...and 7 more sections

Key Result

Theorem 4

Denote by $M_n$ one of the following families of Riemannian manifolds (with the usual Riemannian metric): The (two-time accelerated) Brownian motion $(X^n)_n$ on $M_n$, admits a cutoff in separation with cutoff time $a_n\sim$ and a window $b_n$ of order $\frac{1}{d_n}$. More details about the cutoff will be detailed in Theorems T1, T2, T3, T4 depending of the chosen families.

Theorems & Definitions (33)

  • Definition 1
  • Remark 2
  • Definition 3
  • Theorem 4
  • Remark 5
  • Remark 6
  • Remark 7
  • Remark 8
  • Definition 9
  • Remark 10
  • ...and 23 more