Logarithmic Spectral Distribution of a Non-Hermitian $\beta$-Ensemble

Francesco Mezzadri; Gernot Akemann; Henry Taylor; Patricia P\"a{\ss}ler

arxiv: 2504.12120 · v3 · pith:MXA335THnew · submitted 2025-04-16 · 🧮 math-ph · cond-mat.stat-mech· math.MP· math.PR

Logarithmic Spectral Distribution of a Non-Hermitian β-Ensemble

Gernot Akemann , Francesco Mezzadri , Patricia P\"a{\ss}ler , Henry Taylor This is my paper

Pith reviewed 2026-05-22 20:13 UTC · model grok-4.3

classification 🧮 math-ph cond-mat.stat-mechmath.MPmath.PR

keywords non-Hermitian beta ensemblespectral densityfree probabilitytridiagonal random matriceslogarithmic distributionpseudospectrumcharacteristic polynomial

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The pith

In the large-beta and large-n limit the spectral density of the non-Hermitian beta-ensemble is rotationally invariant on a compact disc and given by the logarithm of the radius plus a constant.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces a non-Hermitian beta-ensemble realized as a general tridiagonal complex random matrix whose entries are normal or chi-distributed random variables. It establishes that the joint eigenvalue distribution contains a Vandermonde factor raised to the power beta together with a residual eigenvector coupling. In the combined limit of large beta and large matrix size n the ensemble reduces to a centred tridiagonal Jacobi matrix with vanishing diagonal, after which a free-probability theorem applied to the variances of the coefficients of the characteristic polynomial yields an explicit limiting density. A sympathetic reader would care because this supplies a concrete radial law for the eigenvalues of a broad class of non-normal matrices and places the model inside the pseudospectrum framework, while also showing that the identical density arises from the tridiagonal complex-symmetric case.

Core claim

The paper establishes that for the introduced non-Hermitian beta-ensemble the low-temperature limit beta much greater than one reduces the model exactly to a centred tridiagonal Jacobi matrix with vanishing diagonal. A general theorem from free probability, based on the variance of the coefficients of the characteristic polynomial, then determines the spectral density in the additional large-n limit. This density is rotationally invariant on a compact disc and is given by the logarithm of the radius plus a constant. The same density is recovered when the construction begins from a tridiagonal complex symmetric ensemble.

What carries the argument

The characteristic polynomial of a centred tridiagonal Jacobi matrix with vanishing diagonal, whose coefficients are expressed explicitly in terms of the matrix elements, together with the free-probability theorem that relates the variance of those coefficients to the limiting spectral density.

If this is right

The limiting density is identical for the tridiagonal complex symmetric ensemble.
Extensive numerical simulations confirm the analytical density and situate the ensemble relative to the pseudospectrum.
The Vandermonde-beta factor and eigenvector coupling become irrelevant once the large-beta reduction is performed.
The result supplies an exact benchmark for the eigenvalue distribution of non-normal tridiagonal matrices in the low-temperature regime.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The logarithmic radial form may indicate an underlying connection to logarithmic potentials that appear in other non-Hermitian models with circular symmetry.
It would be natural to check whether the same density appears when the tridiagonal structure is replaced by a different sparse non-normal pattern at large beta.
The explicit reduction to a vanishing diagonal suggests that temperature-driven phase transitions in non-Hermitian ensembles could be studied by tracking the diagonal entries as beta varies.

Load-bearing premise

The low-temperature limit of the ensemble reduces exactly to a centred tridiagonal Jacobi matrix with vanishing diagonal.

What would settle it

A numerical histogram of eigenvalues computed at sufficiently large beta and n that fails to match the predicted radial dependence log of radius plus constant inside the disc would falsify the limiting law.

Figures

Figures reproduced from arXiv: 2504.12120 by Francesco Mezzadri, Gernot Akemann, Henry Taylor, Patricia P\"a{\ss}ler.

**Figure 1.** Figure 1: Spectral Density of the complex eigenvalues of Tβ (left) and Teβ (right), for 100 matrices of dimension n = 5000, both with β = 6. The spectral density is normalised by √ 2nβ. In this section, we study the tridiagonal matrix ensembles numerically and compare the equilibrium densities for Tβ (1.14) and for Teβ (1.15) derived in the previous section in the limit of [PITH_FULL_IMAGE:figures/full_fig_p033_1.png] view at source ↗

**Figure 2.** Figure 2: Histograms of the radial density of of the complex eigenvalues for 100 matrices of size n = 5000 of the ensembles Tβ (left) and Teβ (right), at β = 1/2 (yellow full curve), β = 2 (green dashed curve) and β = 1000 (blue dotted curve). In [PITH_FULL_IMAGE:figures/full_fig_p034_2.png] view at source ↗

**Figure 3.** Figure 3: Comparison between analytics and numerics for Tβ and Sβ. The histogram (left) shows the distribution of the radii for an ensemble of 100 matrices of size n = 5000 for the general ensemble Tβ (blue) and symmetric ensemble Sβ (light red) for β = 100. The region where both distributions agree is purple. The black curve gives the analytical expression for the density (1.14). We also give the Kolmogorov-Smirnov… view at source ↗

**Figure 4.** Figure 4: The plots show the ϵ-psuedospectra of Tβ (left) and Teβ (right), for n = 1000, compared to its eigenvalues (dots) for β = 2. The contour scale represents 10−ϵ . This result shows that for diagonalisable matrices, the pseudospectrum Λϵ(A) is contained within a neighbourhood controlled by the condition number κ(V ) of the eigenvector matrix V . In the case of non-normal matrices, where κ(V ) can be large (se… view at source ↗

**Figure 5.** Figure 5: Histograms of the radial density of complex eigenvalues for 50000 matrices of size n = 2 of the ensemble Tβ (left) and Teβ (right) at β = 1/2 (yellow full), β = 2 (green dashed) and β = 1000 (blue dotted). have β-universality here, i.e. we see a distinct behaviour of the radial distribution for various β at n = 2. For the non-symmetric tridiagonal ensemble Teβ we see in Fig. 5b that for small β = 1/2, 2 we… view at source ↗

read the original abstract

We introduce a non-Hermitian $\beta$-ensemble and determine its spectral density in the limit of large $\beta$ 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 Mezzadri and Taylor (2025). The joint distribution of eigenvalues contains a Vandermonde determinant to the power $\beta$ 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 $\beta\gg1$, 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. The numerical study of the local nearest-neighbour spacing distribution shows agreement between the tridiagonal ensemble and two-dimensional Poisson statistics (independently of $\beta$), whereas we observe a $\beta$-dependence for the previously introduced ensemble.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper defines a new non-Hermitian beta-ensemble and derives its logarithmic limiting density on a disk via large-beta reduction to a centered Jacobi matrix plus a free-probability theorem on coefficient variances.

read the letter

This paper defines a new non-Hermitian beta-ensemble as a tridiagonal complex matrix with normal and chi entries, weighted by a Vandermonde to the beta power plus eigenvector coupling. In the large beta and large n limit they reduce it to a centered tridiagonal Jacobi matrix with zero diagonal and then apply a free probability result to get the spectral density from the variances of the characteristic polynomial coefficients. That density is rotationally symmetric and logarithmic in the radius on a disk. They also recover the same density starting from a tridiagonal complex symmetric ensemble, which suggests the symmetric case has a special status. The numerics are reported to confirm the analytic form and place it in the context of pseudospectra. The derivation looks clean once you accept the large-beta reduction. They cite a general theorem from free probability whose hypotheses are independent of this specific model, and the reduction itself comes from the ensemble definition rather than fitting. The soft spot is exactly that reduction step. The original ensemble has residual coupling to the eigenvectors, and it is not obvious that this coupling vanishes in a way that leaves the coefficient variances untouched when beta goes to infinity. If the limit and the n to infinity limit do not commute, or if the joint distribution of the entries shifts, the theorem might not apply directly. The paper should probably give more detail on the limiting measure for the matrix elements and verify the variances explicitly. This paper is for specialists in non-Hermitian random matrix theory and applications to open quantum systems. A reader who follows beta-ensembles or free probability techniques will get a concrete new density and a method that might extend to other models. The combination of analytic derivation, reduction, and numerical check makes it worth a serious referee's time. I would send it to peer review.

Referee Report

2 major / 2 minor

Summary. The paper introduces a non-Hermitian β-ensemble via a general tridiagonal complex random matrix with normal and chi-distributed entries, extending prior work, and equipped with a Vandermonde^β weight plus residual eigenvector coupling. It claims that in the joint large-β and large-n limit the spectral density is rotationally invariant on a compact disk and equals the logarithm of the radius plus a constant. The derivation proceeds by reducing the ensemble at large β to a centered tridiagonal Jacobi matrix with vanishing diagonal, explicitly determining the coefficients of its characteristic polynomial in terms of the matrix elements, and invoking a general free-probability theorem that relates the limiting density to the variances of those coefficients. The same density is recovered for the tridiagonal complex symmetric case, and the analytic result is supported by numerical simulations that also place the ensemble in the context of the pseudospectrum.

Significance. If the reduction step and the direct applicability of the free-probability theorem are placed on a rigorous footing, the result supplies an explicit, closed-form limiting spectral density for a new family of non-Hermitian β-ensembles. The explicit construction of the characteristic-polynomial coefficients and the confirmation that the complex-symmetric tridiagonal ensemble yields the identical density are concrete strengths; the numerical simulations further anchor the claim. The work therefore extends the literature on non-Hermitian random matrices and free-probability techniques in a potentially useful direction.

major comments (2)

[Abstract / reduction step] Abstract and the paragraph describing the low-temperature reduction: the claim that the original ensemble concentrates exactly onto a centered Jacobi matrix with vanishing diagonal (thereby allowing direct application of the cited free-probability theorem on coefficient variances) is load-bearing for the central density formula. The abstract notes a “residual coupling to the eigenvectors”; an explicit limiting joint law on the matrix entries must be supplied to verify that this coupling vanishes in a manner that leaves the coefficient variances unchanged and that the β→∞ and n→∞ limits commute.
[Derivation of limiting density] The invocation of the general free-probability theorem (based on variances of characteristic-polynomial coefficients) requires a check that the limiting ensemble satisfies the theorem’s hypotheses. Without an explicit computation or bound showing that the variances converge to the required values after the reduction, the theorem cannot be applied directly.

minor comments (2)

[Abstract] The abstract states that numerical simulations confirm the density but provides neither error bars nor quantitative measures of agreement; these should be added to the figures or text.
[Model definition] Notation for the tridiagonal entries (normal versus chi-distributed) and the precise form of the eigenvector coupling should be stated once in a dedicated paragraph or table for clarity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting the need for greater rigor in the reduction step and the verification of the free-probability theorem. We address each major comment below and will revise the manuscript to supply the requested details.

read point-by-point responses

Referee: [Abstract / reduction step] Abstract and the paragraph describing the low-temperature reduction: the claim that the original ensemble concentrates exactly onto a centered Jacobi matrix with vanishing diagonal (thereby allowing direct application of the cited free-probability theorem on coefficient variances) is load-bearing for the central density formula. The abstract notes a “residual coupling to the eigenvectors”; an explicit limiting joint law on the matrix entries must be supplied to verify that this coupling vanishes in a manner that leaves the coefficient variances unchanged and that the β→∞ and n→∞ limits commute.

Authors: We agree that a precise statement of the limiting joint law is required. In the revision we will derive the explicit limiting distribution of the tridiagonal entries as β→∞, showing that the residual eigenvector coupling vanishes in the sense that it leaves the variances of the characteristic-polynomial coefficients unaffected. We will also clarify that the β→∞ limit may be taken first, after which the n→∞ limit yields the stated density, with the two limits commuting under this ordering. revision: yes
Referee: [Derivation of limiting density] The invocation of the general free-probability theorem (based on variances of characteristic-polynomial coefficients) requires a check that the limiting ensemble satisfies the theorem’s hypotheses. Without an explicit computation or bound showing that the variances converge to the required values after the reduction, the theorem cannot be applied directly.

Authors: We will add an explicit computation of the coefficient variances for the reduced centered Jacobi ensemble. The revision will include a direct verification (or bound) that these variances converge to the precise values required by the cited free-probability theorem, thereby confirming that the hypotheses are satisfied in the large-n limit. revision: yes

Circularity Check

0 steps flagged

Derivation chain is self-contained via explicit limit reduction and independent free-probability theorem

full rationale

The paper starts from its defined non-Hermitian β-ensemble (tridiagonal complex matrix with normal/chi entries and Vandermonde^β weight), derives the large-β reduction to a centred zero-diagonal Jacobi matrix directly from the model definition, explicitly computes the characteristic-polynomial coefficients in terms of the matrix elements, and invokes a general free-probability theorem on coefficient variances whose statement and hypotheses are external to the present work. Prior author citations are used only for ensemble construction, not for the limiting density formula itself. The final logarithmic density on the disc is obtained by applying the theorem to the limiting variances after the n→∞ limit; this does not reduce to a fit or to a self-referential definition. No load-bearing step equates the output to the input by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The derivation rests on the large-beta reduction to a centred tridiagonal matrix and on a general free-probability theorem relating spectral density to coefficient variances; no free parameters are fitted inside the paper and no new entities are postulated.

axioms (2)

domain assumption The low-temperature limit beta >> 1 reduces the ensemble exactly to a centred tridiagonal Jacobi matrix with vanishing diagonal.
Invoked to apply the free-probability theorem; stated in the abstract as the key simplification step.
standard math A general theorem from free probability gives the spectral density from the variance of the coefficients of the characteristic polynomial.
Cited as the tool that converts the coefficient variances into the limiting density; assumed to hold for the reduced matrix.

pith-pipeline@v0.9.0 · 5774 in / 1541 out tokens · 72882 ms · 2026-05-22T20:13:34.985976+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

In the low temperature limit β ≫ 1 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... given by the logarithm of the radius plus a constant.
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

The polynomials defined in (1.16) are either even or odd... obey the three-term-recurrence-relation Pn+1(z) = z Pn(z) − eb_n Pn−1(z)

What do these tags mean?

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supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.