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Logarithmic Spectral Distribution of a non-Hermitian $β$-Ensemble

Gernot Akemann, Francesco Mezzadri, Patricia Päßler, Henry Taylor

TL;DR

This work introduces a non-Hermitian $β$-ensemble realized as a general complex tridiagonal matrix and derives its global spectral density in the joint limit $n,β\to∞$; the leading law is a rotationally invariant, logarithmic density on a disc of radius $r_0=\sqrt{2/e}$, with $ρ(z)=\frac{e}{2π}(\ln 2-1-\ln|z|^2)$ for $|z|\le r_0$. A central analytic tool is a centred Jacobi polynomial associated with the truncated tridiagonal, enabling a low-temperature expansion $β\gg1$ that decouples eigenvalues from eigenvectors and reduces the problem to analysing the coefficients $\kappa_ℓ^{(n)}$ of the characteristic polynomial. Using free-probability insights (Barbarino–Noferini), the zeros of the random characteristic polynomials converge to the equilibrium density, yielding the logarithmic radial law for the general and symmetric ensembles, while the non-symmetric case collapses to a Dirac delta at the origin. Numerical simulations corroborate the analytic density and reveal pseudospectral features and measures of non-normality, highlighting universal radial behavior for the stable ensembles and strong non-normality in the non-symmetric case. Overall, the paper connects tridiagonal complex random matrices, characteristic polynomials, and free-probability methods to establish a novel logarithmic spectral distribution in non-Hermitian $β$-ensembles with clear implications for pseudospectra and stability analysis.

Abstract

We introduce a non-Hermitian $β$-ensemble and determine its spectral density in the limit of large $β$ and large matrix size $n$. The ensemble is given by a general tridiagonal complex random matrix of normal and chi-distributed random variables, extending previous work of two of the authors. The joint distribution of eigenvalues contains a Vandermonde determinant to the power $β$ and a residual coupling to the eigenvectors. A tool in the computation of the limiting spectral density is a single characteristic polynomial for centred tridiagonal Jacobi matrices, for which we explicitly determine the coefficients in terms of its matrix elements. In the low temperature limit $β\gg1$ our ensemble reduces to such a centred matrix with vanishing diagonal. A general theorem from free probability based on the variance of the coefficients of the characteristic polynomial allows us to obtain the spectral density when additionally taking the large-$n$ limit. It is rotationally invariant on a compact disc, given by the logarithm of the radius plus a constant. The same density is obtained when starting form a tridiagonal complex symmetric ensemble, which thus plays a special role. Extensive numerical simulations confirm our analytical results and put this and the previously studied ensemble in the context of the pseudospectrum.

Logarithmic Spectral Distribution of a non-Hermitian $β$-Ensemble

TL;DR

This work introduces a non-Hermitian -ensemble realized as a general complex tridiagonal matrix and derives its global spectral density in the joint limit ; the leading law is a rotationally invariant, logarithmic density on a disc of radius , with for . A central analytic tool is a centred Jacobi polynomial associated with the truncated tridiagonal, enabling a low-temperature expansion that decouples eigenvalues from eigenvectors and reduces the problem to analysing the coefficients of the characteristic polynomial. Using free-probability insights (Barbarino–Noferini), the zeros of the random characteristic polynomials converge to the equilibrium density, yielding the logarithmic radial law for the general and symmetric ensembles, while the non-symmetric case collapses to a Dirac delta at the origin. Numerical simulations corroborate the analytic density and reveal pseudospectral features and measures of non-normality, highlighting universal radial behavior for the stable ensembles and strong non-normality in the non-symmetric case. Overall, the paper connects tridiagonal complex random matrices, characteristic polynomials, and free-probability methods to establish a novel logarithmic spectral distribution in non-Hermitian -ensembles with clear implications for pseudospectra and stability analysis.

Abstract

We introduce a non-Hermitian -ensemble and determine its spectral density in the limit of large and large matrix size . The ensemble is given by a general tridiagonal complex random matrix of normal and chi-distributed random variables, extending previous work of two of the authors. The joint distribution of eigenvalues contains a Vandermonde determinant to the power and a residual coupling to the eigenvectors. A tool in the computation of the limiting spectral density is a single characteristic polynomial for centred tridiagonal Jacobi matrices, for which we explicitly determine the coefficients in terms of its matrix elements. In the low temperature limit our ensemble reduces to such a centred matrix with vanishing diagonal. A general theorem from free probability based on the variance of the coefficients of the characteristic polynomial allows us to obtain the spectral density when additionally taking the large- limit. It is rotationally invariant on a compact disc, given by the logarithm of the radius plus a constant. The same density is obtained when starting form a tridiagonal complex symmetric ensemble, which thus plays a special role. Extensive numerical simulations confirm our analytical results and put this and the previously studied ensemble in the context of the pseudospectrum.

Paper Structure

This paper contains 21 sections, 26 theorems, 118 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction and Main Results
  2. General Complex Tridiagonal Ensemble
  3. Joint Distribution and Limiting Spectral Density
  4. Complex $\beta$-Ensemble
  5. Spectral Decomposition of Tridiagonal Matrices
  6. Jacobian
  7. Proof of Theorem \ref{['jpdf_theorem']}
  8. Low Temperature Expansion $\beta\gg1$
  9. General Ensemble
  10. Non-Symmetric Ensemble
  11. Characteristic Polynomials of Centred Jacobi Matrices
  12. Equilibrium Density via Free Probability for $\beta \gg 1$ and $n\rightarrow \infty$
  13. General Ensemble
  14. Proof of Theorem \ref{['thm-density beta']} eq. \ref{['eq-limiting density T,S']}
  15. Non-Symmetric Ensemble
  16. ...and 6 more sections

Key Result

Theorem 1.1

The jpdf of the eigenvalues of the random matrix $T$eq-gen_trid is given by where $\beta\in \mathbb{R}_+$, $\Lambda=\operatorname{diag}(\lambda_1,\ldots,\lambda_n)$, and $Z_\beta^T$ is the normalization constant The function $f_T(\Lambda)$ that contains a coupling between eigenvalues and eigenvectors reads where $\mathbf{r} =(r_1,\dotsc,r_{n}) \in \mathbb{C}^{n}$, $1 = r_1 +\ldots+ r_{n}$ denot

Figures (5)

  • Figure 1: Spectral Density of the complex eigenvalues of $T_\beta$ (left) and $\widetilde{T}_\beta$ (right), for $100$ matrices of dimension $n=5000$, both with $\beta=6$. The spectral density is normalised by $\sqrt{2n\beta}$.
  • Figure 2: Histograms of the radial density of of the complex eigenvalues for $100$ matrices of size $n=5000$ of the ensembles $T_\beta$ (left) and $\widetilde{T}_\beta$ (right), at $\beta=1/2$ (yellow full curve), $\beta=2$ (green dashed curve) and $\beta=1000$ (blue dotted curve).
  • Figure 3: Comparison between analytics and numerics for $T_\beta$ and $S_\beta$. The histogram (left) shows the distribution of the radii for an ensemble of $100$ matrices of size $n=5000$ for the general ensemble $T_\beta$ (blue) and symmetric ensemble $S_\beta$ (light red) for $\beta=100$. The region where both distributions agree is purple. The black curve gives the analytical expression for the density \ref{['eq-limiting density T,S']}. We also give the Kolmogorov-Smirnov distances (right) in units $10^{-2}$ between the numerics and the analytical result \ref{['eq-limiting density T,S']} for both ensembles for various matrix sizes $n$, ensemble sizes $m$ and $\beta$ values.
  • Figure 4: The plots show the $\epsilon$-psuedospectra of $T_{\beta}$ (left) and $\widetilde{T}_{\beta}$ (right), for $n=1000$, compared to its eigenvalues (dots) for $\beta=2$. The contour scale represents $10^{-\epsilon}$.
  • Figure 5: Histograms of the radial density of complex eigenvalues for $50000$ matrices of size $n=2$ of the ensemble $T_\beta$ (left) and $\widetilde{T}_\beta$ (right) at $\beta=1/2$ (yellow full), $\beta=2$ (green dashed) and $\beta=1000$ (blue dotted).

Theorems & Definitions (34)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 1.3
  • Remark 2.1
  • Lemma 2.2
  • Theorem 2.3
  • Lemma 2.4
  • Theorem 2.5
  • Lemma 3.1
  • Lemma 3.2
  • ...and 24 more