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Multiplicative non-Hermitian perturbations of classical $β$-ensembles

Gökalp Alpan, Rostyslav Kozhan

Abstract

We compute the joint eigenvalue distribution for a multiplicative non-Hermitian perturbation $(I+iΓ)H$, $\operatorname{rank}\,Γ=1$ of a random matrix $H$ from the Gaussian, Laguerre, and chiral Gaussian $β$-ensembles.

Multiplicative non-Hermitian perturbations of classical $β$-ensembles

Abstract

We compute the joint eigenvalue distribution for a multiplicative non-Hermitian perturbation , of a random matrix from the Gaussian, Laguerre, and chiral Gaussian -ensembles.
Paper Structure (16 sections, 8 theorems, 89 equations, 1 figure)

This paper contains 16 sections, 8 theorems, 89 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Gaussian ensembles
  3. Gaussian $\beta$-ensemble and the spectral measure
  4. Multiplicative perturbation of $\mathrm{G\beta E}_n$
  5. Eigenvalue configurations
  6. Jacobians
  7. Proof of Theorem \ref{['Gauss']}
  8. Laguerre $\beta$-ensembles
  9. Laguerre $\beta$-ensemble and the spectral measure
  10. Eigenvalue density for multiplicative perturbations: $m\geq n$ case.
  11. Eigenvalue density for multiplicative perturbations: $m\leq n-1$ case.
  12. Multiplicative perturbations of the classical ensembles
  13. Multiplicative perturbations of $\mathrm{GOE}_n$ and $\mathrm{GUE}_n$
  14. Multiplicative perturbations of $\mathrm{LOE}_{(m,n)}$ and $\mathrm{LUE}_{(m,n)}$
  15. Multiplicative perturbations of $\mathrm{GSE}_n$ and $\mathrm{LSE}_{(m,n)}$
  16. ...and 1 more sections

Key Result

Theorem 2.1

Let $\beta>0$, $\mathcal{J}$ belong to the Gaussian $\beta$-ensemble, $l>0$ be independent of $\mathcal{J}$ with a given absolutely continuous distribution $d\nu (l)=F(l)\,dl$. Then the eigenvalues $\{z_j\}_{j=1}^n$ of $\mathcal{J}_{l,\times}$genericJ2new are distributed on according to where $l=\tan(\sum_{j=1}^n \operatorname{Arg}_{[0,\pi)} z_j)$ and $C_{\beta,n}=g_{\beta,n}c_{\beta,n} 2^{n(\be

Figures (1)

  • Figure 1: Fig. 1. One realization of eigenvalues for $\mathcal{J}_{l,\times}$ ($n=30$, $\beta=2$, $l=1$) (via Wolfram Mathematica)

Theorems & Definitions (17)

  • Theorem 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3: KT
  • Lemma 2.4
  • proof
  • Remark
  • Lemma 2.5
  • proof
  • proof : Proof of Theorem \ref{['Gauss']}
  • ...and 7 more