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Gaussian statistics for left and right eigenvectors of complex non-Hermitian matrices

Sofiia Dubova, Kevin Yang, Horng-Tzer Yau, Jun Yin

Abstract

We consider a constant-size subset of left and right eigenvectors of an $N\times N$ i.i.d. complex non-Hermitian matrix associated with the eigenvalues with pairwise distances at least $N^{-\frac12+ε}$. We show that arbitrary constant rank projections of these eigenvectors are Gaussian and jointly independent.

Gaussian statistics for left and right eigenvectors of complex non-Hermitian matrices

Abstract

We consider a constant-size subset of left and right eigenvectors of an i.i.d. complex non-Hermitian matrix associated with the eigenvalues with pairwise distances at least . We show that arbitrary constant rank projections of these eigenvectors are Gaussian and jointly independent.
Paper Structure (19 sections, 29 theorems, 278 equations)

This paper contains 19 sections, 29 theorems, 278 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Acknowledgements
  4. Gaussian divisible matrices
  5. A1:
  6. A2:
  7. A3:
  8. Change of variables
  9. Proof of Proposition \ref{['prop:single']}
  10. Proof of Proposition \ref{['prop:maingauss']}
  11. Proof of \ref{['eq:kjcomputation1']}
  12. Proof of \ref{['eq:kjcomputation2']}
  13. Concentration
  14. Proof of Lemma \ref{['lemma:simplify']}
  15. Proof of Theorem \ref{['theorem:main']}
  16. ...and 4 more sections

Key Result

Theorem 1

Let $A$ be an $N\times N$ complex random matrix with i.i.d. entries satisfying $\mathbb{E} A_{ij} = 0$, $\mathbb{E}\left|A_{ij}\right|^2 = N^{-1}$ and $\mathbb{E}\left|A_{ij}\right|^p \le C_p N^{-p/2}$. Fix $\epsilon,\tau>0$, $m_R, m_L\in \mathbb{Z}_+$, set $m=m_R+m_L$. Consider deterministic points where $Z= (Z_1, \ldots, Z_N)$ consists of independent Gaussian random variables $Z_j \sim \mathcal{

Theorems & Definitions (50)

  • Theorem 1
  • Theorem 2
  • Corollary 3
  • proof
  • Proposition 4
  • Proposition 5
  • proof : Proof of Theorem \ref{['theorem:maingauss']}
  • Lemma 6
  • proof
  • Lemma 7
  • ...and 40 more