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Principal minors of Gaussian orthogonal ensemble

Renjie Feng, Gang Tian, Dongyi Wei, Dong Yao

Abstract

In this paper, we study the extremal process of the maxima of all the largest eigenvalues of principal minors of the classical Gaussian orthogonal ensemble (GOE). We prove that the fluctuation of the maxima is given by the Gumbel distribution in the limit. We also derive the limiting joint distribution of the maxima and the corresponding eigenvector, which implies that these two random variables are asymptotically independent.

Principal minors of Gaussian orthogonal ensemble

Abstract

In this paper, we study the extremal process of the maxima of all the largest eigenvalues of principal minors of the classical Gaussian orthogonal ensemble (GOE). We prove that the fluctuation of the maxima is given by the Gumbel distribution in the limit. We also derive the limiting joint distribution of the maxima and the corresponding eigenvector, which implies that these two random variables are asymptotically independent.
Paper Structure (8 sections, 6 theorems, 204 equations)

This paper contains 8 sections, 6 theorems, 204 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['main']}
  3. Proof of Theorem \ref{['main']}
  4. Proofs of lemmas
  5. Proof of Lemma \ref{['lem:WS']}
  6. Proof of Lemma \ref{['lem:WS1']}
  7. Proof of Lemma \ref{['t_nlimit']}
  8. Proof of Theorem \ref{['main2']}

Key Result

Theorem 1

For GOE, we have the following convergence in distribution as $n\to+\infty$ for fixed $m$, where the random variable $Y$ has the Gumbel distribution function Here, the constant where $K_m=\mu(\mathcal{S}_m)$ is the probability of the event under the uniform distribution $\mu$ on the unit sphere $S^{m-1}$. In particular,

Theorems & Definitions (9)

  • Theorem 1
  • Theorem 2
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • proof
  • proof
  • proof