Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Smallest gaps between eigenvalues of real Gaussian matrices

Patrick Lopatto, Matthew Meeker

Abstract

We consider an $n\times n$ matrix of independent real Gaussian random variables and determine the asymptotic distribution of the smallest gaps between complex eigenvalues.

Smallest gaps between eigenvalues of real Gaussian matrices

Abstract

We consider an matrix of independent real Gaussian random variables and determine the asymptotic distribution of the smallest gaps between complex eigenvalues.
Paper Structure (16 sections, 16 theorems, 148 equations)

This paper contains 16 sections, 16 theorems, 148 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Main Result
  4. Proof Ideas
  5. Outline
  6. Acknowledgments
  7. Preliminary Results
  8. Correlation Functions for the GinOE
  9. Poisson Convergence
  10. Proof of Main Result
  11. Modified Point Process
  12. Proofs of Main Lemmas
  13. Proof of Lemma \ref{['l:bunching']}
  14. Proof of Lemma \ref{['l:33']}
  15. Proof of Lemma \ref{['l:integration']}
  16. ...and 1 more sections

Key Result

Theorem 1.6

Let $\Omega$ be an admissible domain. As $n\rightarrow \infty$, the processes $\chi^{(n)}_{\Omega}$ converge weakly to a Poisson point process $\chi_{\Omega}$ on $\mathbb{R}^+$ with intensity for any bounded Borel set $A \subset \mathbb{R}^+$.

Theorems & Definitions (38)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Remark 1.5
  • Theorem 1.6
  • Corollary 1.7
  • Remark 1.8
  • Definition 2.1
  • Lemma 2.2
  • ...and 28 more