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A spectral dominance approach to large random matrices: part II

Charles Bertucci, Jean-Michel Lasry, Pierre Louis Lions

Abstract

This paper is the second of a series devoted to the study of the dynamics of the spectrum of large random matrices. We study general extensions of the partial differential equation arising to characterize the limit spectral measure of the Dyson Brownian motion. We provide a regularizing result for those generalizations. We also show that several results of part I extend to cases in which there is no spectral dominance property. We then provide several modeling extensions of such models as well as several identities for the Dyson Brownian motion.

A spectral dominance approach to large random matrices: part II

Abstract

This paper is the second of a series devoted to the study of the dynamics of the spectrum of large random matrices. We study general extensions of the partial differential equation arising to characterize the limit spectral measure of the Dyson Brownian motion. We provide a regularizing result for those generalizations. We also show that several results of part I extend to cases in which there is no spectral dominance property. We then provide several modeling extensions of such models as well as several identities for the Dyson Brownian motion.
Paper Structure (21 sections, 24 theorems, 127 equations)

This paper contains 21 sections, 24 theorems, 127 equations.

Table of Contents

  1. Introduction
  2. Notation and results of the first part
  3. Notation
  4. Characterization of solutions of the Dyson equation
  5. Derivation of the Dyson equation
  6. Corrections on the first part
  7. A regularizing result
  8. Other extensions of the Dyson equation
  9. A stability result for non-parabolic equations
  10. Time regularity of solutions
  11. Comments on the cases in which $c(\cdot)$ vanishes
  12. The case of a singular drift
  13. Some identities for the Dyson flow
  14. Decay of $L^p$ norms
  15. Around the free entropy
  16. ...and 6 more sections

Key Result

Proposition 4.1

Consider $m$ a smooth Lipschitz solution of dysong such that $m_0 \in \mathcal{P}(\mathbb{R})$. Assume that there exists $C_0 >0$ such that Then, for any $T > 0$, there exists $C > 0$ depending only $C_0$, $b$ and $T$ such that

Theorems & Definitions (52)

  • Definition 2.1
  • Definition 2.2
  • Proposition 4.1
  • proof
  • Remark 4.2
  • Corollary 4.3
  • Proposition 4.4
  • proof
  • Proposition 4.5
  • proof
  • ...and 42 more