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A matrix model of a non-Hermitian $β$-ensemble

Francesco Mezzadri, Henry Taylor

TL;DR

This work constructs the first non-Hermitian complex $β$-ensemble by introducing a complex tridiagonal matrix $T$ whose entries are independent in a prescribed way, yielding a joint eigenvalue density that generalizes the Ginibre ensemble via a Vandermonde factor $|\Delta(\boldsymbol{λ})|^{β}$. The authors derive an explicit integral representation for the eigenvector contribution $f(\boldsymbol{λ})$, controlled by a positive function $g(\boldsymbol{λ},\mathbf{r})$ and an auxiliary vector $\mathbf{r}$, and establish a precise Jacobian linking matrix elements to spectral data. The main theorem expresses the eigenvalue j.p.d.f. as $P_β(\boldsymbol{λ}) = Z_β^{-1} e^{- rac{1}{2}\sum_j|λ_j|^2} |Δ(λ)|^{β} f(λ)$, with $f(λ)$ defined by an integral over $\mathbf{r}$ of $e^{-g(λ, r)}$ weighted by $|r_j|^{β/2-2}$. The model preserves rotation invariance and, in large $n$, local eigenvalue statistics align with the Ginibre determinantal process, although for $β=2$ the additional eigenvector factor blocks a Ginibre reduction, highlighting a novel eigenvalue-eigenvector coupling in non-Hermitian random matrices.

Abstract

We introduce the first random matrix model of a complex $β$-ensemble. The matrices are tridiagonal and can be thought of as the non-Hermitian analogue of the Hermite $β$-ensembles discovered by Dumitriu and Edelman (J. Math. Phys., Vol. 43, 5830 (2002)). The main feature of the model is that the exponent $β$ of the Vandermonde determinant in the joint probability density function (j.p.d.f.) of the eigenvalues can take any value in $\mathbb{R}_+$. However, when $β=2$, the j.p.d.f. does not reduce to that of the Ginibre ensemble, but it contains an extra factor expressed as a multidimensional integral over the space of the eigenvectors.

A matrix model of a non-Hermitian $β$-ensemble

TL;DR

This work constructs the first non-Hermitian complex -ensemble by introducing a complex tridiagonal matrix whose entries are independent in a prescribed way, yielding a joint eigenvalue density that generalizes the Ginibre ensemble via a Vandermonde factor . The authors derive an explicit integral representation for the eigenvector contribution , controlled by a positive function and an auxiliary vector , and establish a precise Jacobian linking matrix elements to spectral data. The main theorem expresses the eigenvalue j.p.d.f. as , with defined by an integral over of weighted by . The model preserves rotation invariance and, in large , local eigenvalue statistics align with the Ginibre determinantal process, although for the additional eigenvector factor blocks a Ginibre reduction, highlighting a novel eigenvalue-eigenvector coupling in non-Hermitian random matrices.

Abstract

We introduce the first random matrix model of a complex -ensemble. The matrices are tridiagonal and can be thought of as the non-Hermitian analogue of the Hermite -ensembles discovered by Dumitriu and Edelman (J. Math. Phys., Vol. 43, 5830 (2002)). The main feature of the model is that the exponent of the Vandermonde determinant in the joint probability density function (j.p.d.f.) of the eigenvalues can take any value in . However, when , the j.p.d.f. does not reduce to that of the Ginibre ensemble, but it contains an extra factor expressed as a multidimensional integral over the space of the eigenvectors.
Paper Structure (7 sections, 10 theorems, 122 equations, 1 figure)

This paper contains 7 sections, 10 theorems, 122 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Spectral decomposition of tridiagonal matrices
  3. The Jacobian
  4. Proof of Theorem \ref{['main_theorem']}
  5. The multiplicity of the eigenvalues of a tridiagonal matrix
  6. The spectral parameters for matrices with off-diagonal elements equal to zero
  7. An alternative proof of Lemma \ref{['lem:jac1']}

Key Result

Theorem 1.1

The j.p.d.f. of the eigenvalues of the random matrix eq:tridiagmatrix is where $\beta\in \mathbb{R}_+$, $Z_\beta$ is a normalization constant, $\mathbf{r} =(r_1,\dotsc,r_{n}) \in \mathbb{C}^{n}$, $r_n = -r_1 - \dotsb - r_{n-1}$ and $d^2r_j=\tfrac{i}{2}dr_j \wedge d\bar{r}_j$. The exponent $g(\boldsymbol{\lambda},\mathbf{r})$ is a positive function defined by where the $a_j$s and $b_k$s are rati

Figures (1)

  • Figure 1: Density-of-states of the complex eigenvalue spectrum of $(\ref{['eq:tridiagmatrix']})$ for 250 matrices of varying $\beta$ and dimension $5000\times 5000$, whose elements are distributed according to $(\ref{['mat_el_dis']})$. The density is normalized by $\sqrt{n \beta /4}$.

Theorems & Definitions (18)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Theorem 2.2
  • proof
  • Theorem 3.1
  • Proposition 3.2
  • proof
  • Lemma 3.3
  • proof
  • ...and 8 more