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arxiv: 2602.16878 · v2 · pith:U5DD3PKFnew · submitted 2026-02-18 · ❄️ cond-mat.dis-nn · math-ph· math.MP· math.PR

Spectral boundaries of deterministic matrices deformed by rotationally invariant random non-Hermitian ensembles

Pierre Bousseyroux , Marc Potters This is my paper

Pith reviewed 2026-05-15 20:45 UTC · model grok-4.3

classification ❄️ cond-mat.dis-nn math-phmath.MPmath.PR
keywords random matrix theorynon-Hermitian matricesspectral boundarieseigenvalue distributionR-transformdeformed matriceslarge-N limitrotationally invariant ensembles
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The pith

The complex eigenvalues of a deterministic matrix plus a rotationally invariant random matrix lie inside boundaries given by simple equations from the R1 and R2 transforms of the random matrix.

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

This paper studies matrices of the form A plus B, where A is a fixed deterministic matrix and B is a random non-Hermitian matrix whose statistics are unchanged under rotations. In the limit where the matrix dimension grows to infinity, the region in the complex plane that contains the eigenvalues of the sum obeys boundary equations that depend only on two transforms of B. A reader would care because these equations replace what would otherwise be intractable finite-size calculations with compact rules that are easy to apply once the transforms are known. The approach covers a broad class of random ensembles and is checked against direct numerical diagonalization.

Core claim

In the large-N limit, the complex eigenvalue distribution of A + B satisfies remarkably simple boundary equations that depend on the R1 and R2 transforms of B.

What carries the argument

The R1 and R2 transforms of the rotationally invariant random matrix B, which encode its eigenvalue statistics in the complex plane and enter directly into the equations that locate the boundary of the eigenvalue support.

Load-bearing premise

The random matrix B must belong to a rotationally invariant ensemble so that its statistics are completely captured by the R1 and R2 transforms.

What would settle it

Compute the R1 and R2 transforms for a concrete rotationally invariant ensemble such as the Ginibre ensemble, derive the predicted boundary curves, and check whether large-N numerical eigenvalue scatter plots fall inside those curves to within sampling error.

Figures

Figures reproduced from arXiv: 2602.16878 by Marc Potters, Pierre Bousseyroux.

Figure 1
Figure 1. Figure 1: Empirical eigenvalues from single realizations of non-Hermitian deformations of [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Complex eigenvalues of a matrix of size N = 500 given by Wq + σG, where Wq is a complex Wishart matrix with parameter q = 1/4, σ = 0.3, and G is a complex Ginibre matrix. The theoretical spectral boundary is also shown using the equation (31). 3.3 Samoussa Proposition 10. We consider two independent realizations of complex Wishart matrices W1 and W2 with parameter q (see Definition 6), and we study the eig… view at source ↗
Figure 3
Figure 3. Figure 3: The right panel shows the eigenvalues from a single realization of the matrix [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Empirical eigenvalues of U + σE, where U is a random unitary matrix, σ = 0.8, and E is a complex elliptic matrix with parameter τ = 0.8. The boundaries are given by Proposition 11. 3.5 Two rings triangularly deformed We conclude with a more involved example. We consider the matrix Cr1,r2 defined in Definition 7. We define M := Cr1,r2 + D, (43) where D is a diagonal matrix in which one third of the coeffici… view at source ↗
Figure 5
Figure 5. Figure 5: Complex eigenvalues of the matrix M defined in Eq.(43) (N = 500) for several parameter pairs (r1, r2). The theoretical spectral boundaries predicted by Conjecture 12 are also shown. 11 [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗
read the original abstract

One of the great miracles of random matrix theory is that, in the $N \to \infty$ limit, many otherwise intractable matrix problems with horrendously complicated finite-$N$ expressions admit remarkably simple and elegant asymptotic solutions. In this paper, we illustrate this phenomenon in the context of spectral boundaries (or spectral edges) for deformed random matrices. Specifically, we consider matrices of the form $\mathbf{A} + \mathbf{B}$, where $\mathbf{A}$ is a deterministic $N\times N$ matrix (not necessarily Hermitian) and $\mathbf{B}$ is a rotationally invariant random matrix. In the large-$N$ limit, we show that the complex eigenvalue distribution of $\mathbf{A} + \mathbf{B}$ satisfies remarkably simple boundary equations that depend on the $\mathcal{R}_1$ and $\mathcal{R}_2$ transforms of $\mathbf{B}$. We illustrate our results on several explicit random matrix ensembles and support them with numerical simulations.

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

0 major / 2 minor

Summary. The manuscript claims that in the large-N limit, the complex eigenvalue distribution of a deterministic matrix A plus a rotationally invariant random non-Hermitian matrix B obeys simple boundary equations determined by the R1 and R2 transforms of B. The result is illustrated on explicit ensembles and backed by numerical simulations.

Significance. If the derivation holds, the work supplies an elegant, parameter-free method for locating spectral boundaries in non-Hermitian deformations via free-probability tools, extending subordination relations to the complex plane. This could streamline analysis in disordered systems and non-Hermitian physics where such sums arise.

minor comments (2)
  1. Abstract: the phrase 'remarkably simple boundary equations' is used without displaying the equations themselves; inserting the explicit forms (even in compact notation) would improve immediate readability.
  2. Section on numerical simulations: error bars or convergence diagnostics for the large-N asymptotics are not described; adding a brief statement on finite-N scaling would strengthen the validation.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our work, the recognition of its potential utility in non-Hermitian random matrix problems, and the recommendation for minor revision. No major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity in the derivation chain

full rationale

The paper derives the boundary equations for the eigenvalue support of A + B in the large-N limit from standard subordination and free-convolution relations involving the R1 and R2 transforms of the rotationally invariant ensemble B, together with the deterministic resolvent of A. These transforms are external inputs from free probability theory and are not fitted or redefined within the paper to produce the claimed boundaries. No load-bearing step reduces by construction to a self-citation, ansatz smuggled via citation, or renaming of a known result; the central claim remains independent of the paper's own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Only abstract available; ledger populated from stated assumptions in abstract.

axioms (2)
  • domain assumption Large-N limit exists and eigenvalue distribution converges to a deterministic shape
    Invoked for all asymptotic statements in abstract
  • domain assumption B is rotationally invariant so its law is determined by R1 and R2 transforms
    Central to the claimed simplification

pith-pipeline@v0.9.0 · 5472 in / 1136 out tokens · 18770 ms · 2026-05-15T20:45:40.667687+00:00 · methodology

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Reference graph

Works this paper leans on

30 extracted references · 30 canonical work pages · 3 internal anchors

  1. [1]

    The Brown measure of a sum of two free random variables, one of which is R-diagonal.arXiv preprint arXiv:2209.12379, 2022

    Hari Bercovici and Ping Zhong. The Brown measure of a sum of two free random variables, one of which is R-diagonal.arXiv preprint arXiv:2209.12379, 2022

    work page arXiv 2022

  2. [2]

    Computation of some examples of Brown's spectral measure in free probability

    Philippe Biane and Franz Lehner. Computation of some examples of Brown’s spectral measure in free probability.arXiv preprint math/9912242, 1999

    work page internal anchor Pith review Pith/arXiv arXiv 1999

  3. [3]

    Computation of some examples of Brown’s spectral measure in free probability.Colloquium Mathematicae, 90(2):181–211, 2001

    Philippe Biane and Franz Lehner. Computation of some examples of Brown’s spectral measure in free probability.Colloquium Mathematicae, 90(2):181–211, 2001

    work page 2001

  4. [4]

    The eigenvalues and eigenvectors of finite-rank normal perturbations of large rotationally invariant non-Hermitian matrices

    Pierre Bousseyroux and Marc Potters. The eigenvalues and eigenvectors of finite-rank normal per- turbations of large rotationally invariant non-hermitian matrices.arXiv preprint arXiv:2601.10427, 2026

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  5. [5]

    Pierre Bousseyroux and Marc Potters.R-transforms for non-hermitian matrices: A spherical integral approach.arXiv preprint arXiv:2601.09360, 2026

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  6. [6]

    Lidskii’s theorem in the type II case, Geometric methods in operator algebras (Kyoto 1983), H

    LG Brown. Lidskii’s theorem in the type II case, Geometric methods in operator algebras (Kyoto 1983), H. Araki and E. Effros (Eds.) Pitman Res. notes in Math. Ser 123, 1986

    work page 1983

  7. [7]

    J. T. Chalker and Z. Jane Wang. Diffusion in a random velocity field: spectral properties.Phys. Rev. Lett., 79:1797–1800, 1997

    work page 1997

  8. [8]

    Feinberg and A

    J. Feinberg and A. Zee. Non-Gaussian non-Hermitian random matrix theory: Phase transition and addition formalism.Nucl. Phys. B, 501:643–669, 1997

    work page 1997

  9. [9]

    Feinberg and A

    J. Feinberg and A. Zee. Non-Hermitian random matrix theory: Method of hermitization.Nucl. Phys. B, 504:579–608, 1997

    work page 1997

  10. [10]

    V. L. Girko. Circular law.Theory of Probability & Its Applications, 29(4):694–706, 1985. Original: Teor. Veroyatnost. i Primenen. 29 (1984)

    work page 1985

  11. [11]

    Elliptic law.Theory of Probability & Its Applications, 30(4):677–690, 1986

    VL Girko. Elliptic law.Theory of Probability & Its Applications, 30(4):677–690, 1986

    work page 1986

  12. [12]

    Circular law.Theory of Probability & Its Applications, 29(4):694–706, 1985

    Vyacheslav L Girko. Circular law.Theory of Probability & Its Applications, 29(4):694–706, 1985

    work page 1985

  13. [13]

    The single ring theorem.Annals of mathematics, pages 1189–1217, 2011

    Alice Guionnet, Manjunath Krishnapur, and Ofer Zeitouni. The single ring theorem.Annals of mathematics, pages 1189–1217, 2011

    work page 2011

  14. [14]

    Brown’s spectral distribution measure for R-diagonal elements in finite von Neumann algebras.Journal of Functional Analysis, 176(2):331–367, 2000

    Uffe Haagerup and Flemming Larsen. Brown’s spectral distribution measure for R-diagonal elements in finite von Neumann algebras.Journal of Functional Analysis, 176(2):331–367, 2000

    work page 2000

  15. [15]

    Brown measures of unbounded operators affiliated with a finite von Neumann algebra.Mathematica Scandinavica, pages 209–263, 2007

    Uffe Haagerup and Hanne Schultz. Brown measures of unbounded operators affiliated with a finite von Neumann algebra.Mathematica Scandinavica, pages 209–263, 2007

    work page 2007

  16. [16]

    Hall and Ching-Wei Ho

    Brian C. Hall and Ching-Wei Ho. The Brown measure of the sum of a self-adjoint element and an imaginary multiple of a semicircular element.Letters in Mathematical Physics, 112, 2022. Paper No. 19, 61 pp

    work page 2022

  17. [17]

    The Brown measure of the sum of a self-adjoint element and an elliptic element

    Ching-Wei Ho. The Brown measure of the sum of a self-adjoint element and an elliptic element. Electronic Journal of Probability, 27:1–32, 2022

    work page 2022

  18. [18]

    Outlier eigenvalues for full rank deformed single ring random matrices.arXiv preprint, 2025

    Ching-Wei Ho, Zhi Yin, and Ping Zhong. Outlier eigenvalues for full rank deformed single ring random matrices.arXiv preprint, 2025

    work page 2025

  19. [19]

    Deformed single ring theorems.Journal of Functional Analysis, 288(5):110797, 2025

    Ching-Wei Ho and Ping Zhong. Deformed single ring theorems.Journal of Functional Analysis, 288(5):110797, 2025. Issue date: 1 Mar 2025

    work page 2025

  20. [20]

    R. A. Janik, M. A. Nowak, G. Papp, J. Wambach, and I. Zahed. Aspect of non-Hermitian random matrix models.Phys. Rev. E, 55:4100–4107, 1997. 12

    work page 1997

  21. [21]

    Non-hermitian random matrix models.Nuclear Physics B, 501(3):603–642, 1997

    Romuald A Janik, Maciej A Nowak, Gabor Papp, and Ismail Zahed. Non-hermitian random matrix models.Nuclear Physics B, 501(3):603–642, 1997

    work page 1997

  22. [22]

    Distribution of eigenvalues for some sets of random matrices.Matematicheskii Sbornik, 114(4):507–536, 1967

    Vladimir Alexandrovich Marchenko and Leonid Andreevich Pastur. Distribution of eigenvalues for some sets of random matrices.Matematicheskii Sbornik, 114(4):507–536, 1967

    work page 1967

  23. [23]

    Elsevier, 2004

    Madan Lal Mehta.Random matrices, volume 142. Elsevier, 2004

    work page 2004

  24. [24]

    Bouchaud M.Potters.A First Course in Random Matrix Theory: For Physicists, Engineers and Data Scientists

    J.-P. Bouchaud M.Potters.A First Course in Random Matrix Theory: For Physicists, Engineers and Data Scientists. Cambridge University Press, 2020

    work page 2020

  25. [25]

    G. J. Rodgers. Non-Hermitian random matrices and Brown measures.J. Math. Phys., 51:093304, 2010

    work page 2010

  26. [26]

    Spectrum of large random asymmetric matrices.Physical review letters, 60(19):1895, 1988

    Hans Juergen Sommers, Andrea Crisanti, Haim Sompolinsky, and Yaakov Stein. Spectrum of large random asymmetric matrices.Physical review letters, 60(19):1895, 1988

    work page 1988

  27. [27]

    Addition of certain non-commuting random variables.Journal of functional anal- ysis, 66(3):323–346, 1986

    Dan Voiculescu. Addition of certain non-commuting random variables.Journal of functional anal- ysis, 66(3):323–346, 1986

    work page 1986

  28. [28]

    The generalised product moment distribution in samples from a normal multivariate population.Biometrika, 20(1/2):32–52, 1928

    John Wishart. The generalised product moment distribution in samples from a normal multivariate population.Biometrika, 20(1/2):32–52, 1928

    work page 1928

  29. [29]

    Brown measure of the sum of an elliptic operator and a free random variable in a finite von Neumann algebra.arXiv preprint, 2021

    Ping Zhong. Brown measure of the sum of an elliptic operator and a free random variable in a finite von Neumann algebra.arXiv preprint, 2021. v5 (Aug 2025); to appear in American Journal of Mathematics. 13 A APPENDICES A.1 Proof of the first theorem Once again, we strongly encourage the reader to consult [5], where the notions we will use here are develop...

    work page 2021

  30. [30]

    First case.We assume here that∂ αR1,B(0,g M(z′)) exists

    Taking the limitω→0 in (55) yields: GM(0, z) =G A − R1,B ioM(z),g M(z) , z− R 2,B ioM(z),g M(z) .(56) We now consider two cases. First case.We assume here that∂ αR1,B(0,g M(z′)) exists. Then using (56), we get GM(0, z) = z→z ′ GA(−∂αR1,B(0,g M(z′))io M(z), z ′ − R2,B(0,g M(z′))).(57) Looking only at the upper-left coefficient, we obtain: io(z) = z→z ′ g1(...

    work page