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Central limit theorem for the complex eigenvalues of Gaussian random matrices

Advay Goel, Patrick Lopatto, Xiaoyu Xie

Abstract

We establish a central limit theorem for the eigenvalue counting function of a matrix of real Gaussian random variables.

Central limit theorem for the complex eigenvalues of Gaussian random matrices

Abstract

We establish a central limit theorem for the eigenvalue counting function of a matrix of real Gaussian random variables.
Paper Structure (11 sections, 12 theorems, 67 equations)

This paper contains 11 sections, 12 theorems, 67 equations.

Table of Contents

  1. Introduction
  2. Main Result
  3. Background
  4. Outline
  5. Acknowledgments
  6. Preliminary Results
  7. Proof of Theorem \ref{['t:main']}
  8. Variance Calculation
  9. Higher Cumulants
  10. Conclusion
  11. Technical Estimates

Key Result

Theorem 1.3

Let $A$ be an admissible Lipschitz domain. Then we have the weak convergence

Theorems & Definitions (24)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3
  • Lemma 2.1
  • Remark 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Lemma 3.1
  • proof
  • ...and 14 more