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arxiv: 2606.03447 · v1 · pith:CFN3SQTSnew · submitted 2026-06-02 · 🧮 math-ph · math.MP

Interpolating non-Hermitian universality classes A and AI^dagger: eigenvalue density and transition regime

Mark J. Crumpton , Francesco Mezzadri This is my paper

Pith reviewed 2026-06-28 08:05 UTC · model grok-4.3

classification 🧮 math-ph math.MP
keywords non-Hermitian random matriceseigenvalue densityuniversality classesinterpolation parameteredge behaviorKac-Rice formalismtransitional regimecircular law
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The pith

Scaling the interpolation parameter as one minus kappa over square root N produces a new transitional edge density for eigenvalues that connects two non-Hermitian classes.

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

The paper applies the Kac-Rice formalism to derive the exact finite-N joint distribution of an eigenvalue and its normalized right eigenvector in a Gaussian ensemble interpolating between complex Ginibre matrices and complex symmetric matrices. From the marginal eigenvalue density it recovers the circular law in the bulk for any interpolation strength and the class A edge density for fixed strength. A special scaling regime for the interpolation parameter reveals a new transitional edge density that smoothly interpolates the known behaviors of the two classes. The authors conjecture this transitional regime is universal and support it with numerical checks on non-Gaussian matrices.

Core claim

In the interpolating Gaussian ensemble the marginal eigenvalue density follows the circular law in the bulk independent of the parameter. At the edge, fixed values of the parameter yield the density of class A, while the scaling sigma equals one minus kappa N to the minus one half produces a new density that interpolates between class A and class AI dagger edge behaviors. This transitional regime is conjectured to hold for non-Gaussian matrices.

What carries the argument

The Kac-Rice formalism applied to the interpolating ensemble, which supplies the exact finite-N joint distribution of eigenvalue and normalized right eigenvector from which the marginal density follows.

If this is right

  • The bulk spectrum is the circular law for every fixed value of the interpolation parameter.
  • Edge eigenvalues obey class A statistics unless the parameter is tuned to the special N-dependent scaling.
  • The transitional edge density provides a continuous connection between the two previously known edge laws.
  • Numerical evidence indicates the transitional density persists for non-Gaussian matrices.

Where Pith is reading between the lines

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

  • Comparable transitional scalings may appear when other pairs of non-Hermitian ensembles are interpolated.
  • The new density supplies a concrete model for systems whose symmetry is broken at a rate tied to matrix size.
  • Exact analytic expressions for the transitional density could be extracted from the same Kac-Rice starting point.

Load-bearing premise

The Kac-Rice formalism supplies the exact joint distribution of eigenvalue and normalized right eigenvector for the Gaussian interpolating ensemble at finite N.

What would settle it

Large-N numerical sampling of a non-Gaussian interpolating ensemble with the parameter fixed at one minus kappa over square root N should display the predicted transitional edge density.

Figures

Figures reproduced from arXiv: 2606.03447 by Francesco Mezzadri, Mark J. Crumpton.

Figure 1
Figure 1. Figure 1: FIG. 1. Normalised distribution of eigenvalue moduli in the nHIE at finite [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Density of eigenvalues at the edge of the nHIE in the strong (left) and weak (right) asymmetry regimes, with fixed [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
read the original abstract

We employ the recently developed Kac-Rice formalism for non-Hermitian random matrices to derive the joint distribution of an eigenvalue and its associated normalised right eigenvector in a Gaussian ensemble that interpolates between complex Ginibre (Class A) and complex symmetric matrices (Class AI$^\dagger$). This distribution is valid at finite matrix size, $N$, for any value of the interpolation parameter $\sigma \in [0,1]$, with $0$ and $1$ corresponding to classes A and AI$^\dagger$ respectively. The marginal distribution for the density of the eigenvalues is derived at finite $N$ and then considered asymptotically as $N \to \infty$. When considering bulk eigenvalues, we recover the standard circular law for all $\sigma$. Furthermore, for edge eigenvalues we find that for fixed $\sigma$, the eigenvalues follow the edge density associated with matrices in Class A. However, a transitional regime is discovered for the interpolation parameter being scaled as $\sigma = 1 - \kappa N^{-1/2}$, where new edge behaviour is observed for the density of eigenvalues - smoothly interpolating two previously known results. This transitional regime and the associated density of eigenvalues is conjectured to be universal for non-Gaussian matrices and we provide numerical evidence in support of this.

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.

Referee Report

2 major / 2 minor

Summary. The manuscript employs the Kac-Rice formalism to derive the exact finite-N joint distribution of an eigenvalue and its normalized right eigenvector for a Gaussian ensemble interpolating between class A (Ginibre) and class AI† (complex symmetric) matrices. The marginal eigenvalue density is obtained at finite N and analyzed in the large-N limit, recovering the circular law in the bulk for all interpolation parameters σ and the class A edge density for fixed σ. A transitional regime is identified when σ = 1 - κ N^{-1/2}, yielding a new interpolating edge density conjectured to be universal, with numerical support provided.

Significance. If the transitional regime and associated density are placed on a rigorous footing, the result would identify a new universality class for the edge eigenvalues of non-Hermitian matrices under this scaling of the interpolation parameter, providing a smooth interpolation between the known class A and AI† edge laws. The exact finite-N joint distribution via Kac-Rice and the recovery of the circular law and fixed-σ edge limits are clear strengths. The numerical evidence offered for universality is a positive feature, but the conjectural status of the transitional density limits the immediate significance.

major comments (2)
  1. [Transitional regime analysis] Transitional regime (section following the definition of σ = 1 − κ N^{-1/2}): the new edge density is presented as a conjecture supported by numerics rather than derived from the finite-N marginal via explicit asymptotic analysis under the stated scaling. This is load-bearing for the central claim, since the bulk and fixed-σ results recover known laws while the transitional regime constitutes the novel contribution.
  2. [Kac-Rice joint distribution] Finite-N marginal density (Kac-Rice section): although the joint distribution is asserted to be exact, the explicit integration over the normalized right-eigenvector component to obtain the marginal eigenvalue density is not shown in sufficient detail for the transitional scaling; without these steps it is unclear whether the subsequent N → ∞ limit is fully controlled by the scaling alone.
minor comments (2)
  1. [Numerical figures] Figure captions for the numerical histograms in the transitional regime should overlay the conjectured analytic density (if available) or state the precise range of κ and N values used.
  2. [Introduction/ensemble definition] Notation: the interpolation parameter σ is introduced in the abstract and ensemble definition; a single consolidated definition early in the text would improve readability.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for their careful reading and constructive comments. We address each major comment below.

read point-by-point responses

  1. Referee: [Transitional regime analysis] Transitional regime (section following the definition of σ = 1 − κ N^{-1/2}): the new edge density is presented as a conjecture supported by numerics rather than derived from the finite-N marginal via explicit asymptotic analysis under the stated scaling. This is load-bearing for the central claim, since the bulk and fixed-σ results recover known laws while the transitional regime constitutes the novel contribution.

    Authors: We agree that the transitional edge density is presented as a conjecture rather than a fully derived result. The exact finite-N joint distribution and marginal eigenvalue density are obtained via the Kac-Rice formalism for any σ. The bulk circular law and fixed-σ edge law follow from rigorous asymptotic analysis of this expression. However, the explicit large-N asymptotics under the scaling σ = 1 − κ N^{-1/2} involve substantial technical challenges in controlling the integrals, which we have not completed. The conjectured density is supported by numerical evidence and provides a smooth interpolation between known classes. We will revise the manuscript to more explicitly state the conjectural status and the origin of the scaling, but we do not claim a complete derivation. revision: partial

  2. Referee: [Kac-Rice joint distribution] Finite-N marginal density (Kac-Rice section): although the joint distribution is asserted to be exact, the explicit integration over the normalized right-eigenvector component to obtain the marginal eigenvalue density is not shown in sufficient detail for the transitional scaling; without these steps it is unclear whether the subsequent N → ∞ limit is fully controlled by the scaling alone.

    Authors: The joint distribution of eigenvalue and normalized right eigenvector is derived exactly via Kac-Rice for finite N and any σ ∈ [0,1]. The marginal eigenvalue density is obtained by integrating out the eigenvector component. The manuscript outlines these steps, but we acknowledge that additional explicit details on the integration, particularly in preparation for the transitional scaling, would improve clarity. We will add these steps (possibly in an appendix) in the revised manuscript to show how the finite-N marginal is controlled before taking N → ∞. revision: yes

standing simulated objections not resolved
  • Explicit asymptotic derivation of the transitional edge density under the scaling σ = 1 − κ N^{-1/2}

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained from ensemble and Kac-Rice

full rationale

The paper defines the interpolating Gaussian ensemble explicitly and applies the Kac-Rice formalism to obtain the exact finite-N joint distribution of eigenvalue and normalized right eigenvector. The marginal eigenvalue density is stated to follow directly by integration, after which the bulk circular law and edge behaviors (including the transitional regime under σ = 1 − κ N^{-1/2}) are extracted by asymptotic analysis. No parameter is fitted to a data subset and then relabeled a prediction, no self-citation supplies a load-bearing uniqueness theorem or ansatz, and the universality conjecture is supported by separate numerical evidence rather than derived by construction. The central claims therefore reduce to the input ensemble definition plus standard asymptotic limits, with no reduction to self-reference.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the applicability of the Kac-Rice formalism to the interpolating Gaussian ensemble and the validity of the large-N asymptotic analysis under the stated scaling; no new entities are postulated.

axioms (1)
  • domain assumption Kac-Rice formalism applies to the joint distribution of eigenvalue and eigenvector in this ensemble
    The paper uses this to derive the distribution at finite N for any σ.

pith-pipeline@v0.9.1-grok · 5763 in / 1344 out tokens · 29300 ms · 2026-06-28T08:05:58.585266+00:00 · methodology

discussion (0)

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Reference graph

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