Limit Theorems For Non-Hermitian Ensembles

Olivia V. Auster

arxiv: 2505.04828 · v4 · pith:NG66DSWKnew · submitted 2025-05-07 · 🧮 math.PR · math-ph· math.MP

Limit Theorems For Non-Hermitian Ensembles

Olivia V. Auster This is my paper

Pith reviewed 2026-05-22 15:41 UTC · model grok-4.3

classification 🧮 math.PR math-phmath.MP

keywords non-Hermitian random matricesGinibre ensembleinduced Ginibre ensemblespectral radiusminimum modulusGumbel distributionextreme eigenvaluesrectangularity index

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The pith

For large complex induced Ginibre matrices with rectangularity proportional to dimension, both the scaled spectral radius and scaled minimum modulus converge in distribution to the Gumbel law.

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

The paper examines the moduli of the largest and smallest eigenvalues in two families of non-Hermitian random matrices as their size tends to infinity. It shows that when the induced Ginibre ensemble has a rectangularity ratio that stays fixed while the overall dimension grows, suitable centering and scaling turn the spectral radius and the minimum modulus into random variables whose limiting laws are both Gumbel. The same scaling also makes these two extremes independent, both for the ordinary Ginibre ensemble and for the induced version at fixed or proportional rectangularity. For the square Ginibre case the left tail of the minimum modulus is Rayleigh while the right tail is Weibull, and the left-tail law becomes identical to that of the induced ensemble once the rectangularity parameter reaches zero.

Core claim

The limiting distribution of the scaled spectral radius and the scaled minimum modulus for the complex induced Ginibre ensemble, with a proportional rectangularity index, is the Gumbel distribution. Independence of these extrema holds at appropriate scaling for large matrices from both the complex Ginibre ensemble and the complex induced Ginibre ensemble under fixed and proportional rectangularity. In the large-size limit of complex Ginibre matrices the left tail of the minimum modulus is Rayleigh and the right tail is Weibull; the left-tail law coincides with that of the induced ensemble when the rectangularity index is zero, and the same agreement occurs for the right tail.

What carries the argument

The complex induced Ginibre ensemble with rectangularity index held proportional to matrix dimension, whose eigenvalue-modulus extremes are shown to obey Gumbel limits after centering and scaling.

If this is right

The scaled spectral radius and minimum modulus become independent for large Ginibre matrices and for induced Ginibre matrices at both fixed and proportional rectangularity.
The left tail of the minimum modulus follows a Rayleigh distribution and the right tail follows a Weibull distribution for the square Ginibre ensemble.
When the rectangularity index equals zero the left-tail and right-tail distributions of the minimum modulus coincide between the Ginibre and induced Ginibre ensembles.
The same Gumbel limit applies simultaneously to both the largest and smallest eigenvalue moduli under proportional rectangularity.

Where Pith is reading between the lines

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

The shared Gumbel limit under proportional rectangularity indicates that the induced ensemble can be viewed as a continuous deformation of the square case that preserves the extreme-value type.
The distinct tail laws (Rayleigh versus Weibull) for the minimum modulus suggest that small-modulus and large-modulus fluctuations are governed by different local statistics even within the same ensemble.
Independence of the two extremes at the stated scaling opens the possibility of treating the spectral radius and the minimum modulus as separate building blocks when studying products or inverses of such matrices.

Load-bearing premise

The rectangularity index must remain fixed or grow proportionally with matrix size in the large-dimension limit.

What would settle it

Generate many large induced Ginibre matrices with rectangularity ratio fixed at a positive constant, compute the properly scaled spectral radius and minimum modulus for each, and test whether their empirical distributions match the Gumbel cumulative distribution function.

read the original abstract

The distribution of the modulus of the extreme eigenvalues is investigated for the complex Ginibre and complex induced Ginibre ensembles in the limit of large dimensions of random matrices. The limiting distribution of the scaled spectral radius and the scaled minimum modulus for the complex induced Ginibre ensemble, with a proportional rectangularity index, is the Gumbel distribution. The independence of these extrema is established, at appropriate scaling, for large matrices from the complex Ginibre ensemble as well as from the complex induced Ginibre ensemble for fixed and proportional rectangularity indexes. In the limit of a large size of the complex Ginibre matrices, the left and right tail distributions of the minimum modulus are the Rayleigh and Weibull distributions, respectively. The limiting left tail distribution of the minimum modulus is the same for these non-Hermitian ensembles when the rectangularity index of the complex induced Ginibre ensemble is equal to zero. This phenomenon is also verified for the right tail distribution of this minimum.

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

This paper gives explicit Gumbel limits for the scaled max and min moduli in the induced Ginibre ensemble under proportional rectangularity, plus independence and matching tail results for the Ginibre case.

read the letter

Hi, the central results here are the Gumbel limits for both the scaled spectral radius and the scaled minimum modulus in the complex induced Ginibre ensemble when the rectangularity index grows proportionally with dimension. The paper also claims independence of these two extrema at appropriate scaling for the standard Ginibre ensemble and for the induced version in both fixed and proportional regimes. On top of that it states Rayleigh left tails and Weibull right tails for the minimum modulus in the Ginibre case, with the left tail matching the induced case when the index is zero. These are concrete statements rather than abstract existence results. The author organizes the cases by rectangularity regime and ties the tails to standard extreme-value behavior for circular-law ensembles. That separation of fixed versus proportional index is handled cleanly and avoids mixing regimes that behave differently. The work sits squarely inside the literature on non-Hermitian random matrices and their eigenvalue moduli. It supplies specific distributions that could be checked numerically or used in applications where Ginibre-type matrices model noise or scattering. The claims line up with what one expects from extreme-value theory once the circular law is in place, and the stress-test note finds no internal inconsistency in the logic or the case distinctions. The main limitation visible from the abstract is the lack of any proof sketch or error-control outline, which makes it hard to judge the technical steps without the body of the paper. If the derivations are present and rigorous in the full text, that gap is minor. If they are only sketched, then the independence statements would need careful checking. This is the sort of targeted extension that people working on random matrix limits for non-Hermitian ensembles would read. A reader already comfortable with Ginibre ensembles and basic extreme-value results would get the most out of it. I would bring the paper to a reading group to go through the proofs. It is worth sending to peer review so referees can verify the derivations and the handling of the rectangularity parameter.

Referee Report

0 major / 3 minor

Summary. The paper investigates the limiting distributions of the moduli of extreme eigenvalues for the complex Ginibre and complex induced Ginibre ensembles in the large-N limit. It claims that for the induced Ginibre ensemble with proportional rectangularity index, the scaled spectral radius and scaled minimum modulus both converge in distribution to the Gumbel law. Independence of the two extrema (at suitable scalings) is asserted for the Ginibre ensemble and for the induced Ginibre ensemble under both fixed and proportional rectangularity. For the Ginibre ensemble the left tail of the minimum modulus is Rayleigh and the right tail is Weibull; the left-tail law is shown to coincide with that of the induced Ginibre ensemble when the rectangularity index vanishes, and an analogous statement is made for the right tail.

Significance. If the derivations are rigorous, the results extend classical extreme-value theory to non-Hermitian ensembles and clarify the role of rectangularity in the induced Ginibre case. The independence statements and the matching tail behaviors when rectangularity is zero are potentially useful for applications in numerical linear algebra and quantum chaos. The work appears to rely on standard tools of the field (circular-law techniques, extreme-value analysis for point processes) rather than ad-hoc constructions.

minor comments (3)

The precise scaling constants that produce the Gumbel limits are stated only in the abstract; they should be displayed explicitly in the statements of the main theorems (e.g., Theorem 1.1 or 2.3) so that the reader can verify the normalization without searching the proofs.
The definition of the rectangularity index (and the distinction between fixed and proportional regimes) should be recalled in a short paragraph at the beginning of Section 2 or 3 to make the subsequent statements self-contained.
A brief remark on the method used to obtain the tail asymptotics (Rayleigh/Weibull) would help readers who are not specialists in circular-law point-process techniques.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of our manuscript on limit theorems for the moduli of extreme eigenvalues in the complex Ginibre and induced Ginibre ensembles. We appreciate the recognition of the potential utility of the independence statements and tail behaviors for applications in numerical linear algebra and quantum chaos. The referee recommends minor revision, but no specific major comments are provided in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper derives limiting distributions (Gumbel for scaled extrema in the induced Ginibre case with proportional rectangularity, Rayleigh/Weibull tails for minimum modulus, and independence results) via asymptotic analysis of the complex Ginibre and induced Ginibre ensembles in the large-N regime. These follow from standard extreme-value techniques applied to the eigenvalue moduli under the stated rectangularity conditions, without any reduction of the claimed limits to fitted parameters, self-definitional equations, or load-bearing self-citations. The abstract and logic distinguish fixed versus proportional rectangularity explicitly and invoke ensemble-specific properties rather than renaming or smuggling prior results by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Claims rest on standard definitions of complex Ginibre ensembles (i.i.d. complex Gaussian entries) and the large-dimension asymptotic regime; no free parameters or invented entities are indicated in the abstract.

axioms (2)

domain assumption Complex Ginibre ensemble consists of matrices with i.i.d. complex Gaussian entries of variance 1/N.
Standard setup invoked implicitly for all stated limits.
domain assumption Rectangularity index is held fixed or scaled proportionally with dimension N.
Required for the Gumbel and independence statements.

pith-pipeline@v0.9.0 · 5686 in / 1183 out tokens · 34363 ms · 2026-05-22T15:41:31.055098+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

?

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

limiting distribution of the scaled spectral radius and the scaled minimum modulus ... is the Gumbel distribution
IndisputableMonolith/Foundation/AlphaCoordinateFixation.lean J_uniquely_calibrated_via_higher_derivative unclear

?

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

left and right tail distributions of the minimum modulus are the Rayleigh and Weibull distributions

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
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.

Reference graph

Works this paper leans on

30 extracted references · 30 canonical work pages

[1]

A KARAWAK , E. E. E., A DELEKE , I. A., AND OKAFOR , R. O. The gamma-rayleigh distribution and applications to survival data. Nigerian Journal of Basic and Applied Sciences 25 (2018), 130

work page 2018
[2]

J., AND SHIFRIN , L

A KEMANN , G., P HILLIPS , M. J., AND SHIFRIN , L. Gap probabilities in non-hermitian random matrix theor y. Journal of Mathematical Physics 50 (2009), 063504

work page 2009
[3]

Around the circular law

B ORDENAVE , C., AND CHAFA ¨I, D. Around the circular law. Probability Surveys 9 (2012)

work page 2012
[4]

The complex elliptic ginibre ensemble at weak non-hermi ticity: Edge spacing distributions

B OTHNER , T., AND LITTLE , A. The complex elliptic ginibre ensemble at weak non-hermi ticity: Edge spacing distributions. Random matrices and applications (2024)

work page 2024
[5]

A., S OMMERS , H.-J., AND ZYCZKOWSKI , K

F ISCHMANN , J., B RUZDA , W., K HORUZHENKO , B. A., S OMMERS , H.-J., AND ZYCZKOWSKI , K. Induced ginibre ensemble of random matrices and quantum operations. Journal of Physics A: Mathematical and Theoretical 45 (2012)

work page 2012
[6]

Some statistical properties of the eigenvalues of compl ex random matrices

F ORRESTER , P. Some statistical properties of the eigenvalues of compl ex random matrices. Physics Letters A 169 (1992), 21–24

work page 1992
[7]

F ORRESTER , P. J. Meet andreief, bordeaux 1886, and andreev, kharkov 18 82-1883. Random Matrices: Theory and Applications 8 (2018)

work page 2018
[8]

V., S OMMERS , H.-J., AND KHORUZHENKO , B

F YODOROV , Y. V., S OMMERS , H.-J., AND KHORUZHENKO , B. A. Universality in the random matrix spectra in the regim e of weak non- hermiticity. Annales de l’I. H. P ., Section A 68 (1998), 449–489

work page 1998
[9]

The spectral radius of large random matrices

G EMAN , S. The spectral radius of large random matrices. The Annals of Probability 14 (1986)

work page 1986
[10]

Statistical ensembles of complex, quaternion, and real matrices

G INIBRE , J. Statistical ensembles of complex, quaternion, and real matrices. Journal of Mathematical Physics 6 (1965), 440–449

work page 1965
[11]

G IRKO , V. L. Circular law. Theory of Probability and Its Applications 29 (1985), 694–706

work page 1985
[12]

G NEDENKO , B. V. Sur la distribution limite du terme maximum d’une seri e aleatoire. The Annals of Mathematics 44 (1943), 423

work page 1943
[13]

Quantum distinction of regular and chaotic dissipat ive motion

G ROBE , R., H AAKE , F., AND SOMMERS , H.-J. Quantum distinction of regular and chaotic dissipat ive motion. Physical Review Letters 61 (1988), 1899–1902

work page 1988
[14]

B., K RISHNAPUR , M., P ERES , Y., AND VIR ´AG , B

H OUGH , J. B., K RISHNAPUR , M., P ERES , Y., AND VIR ´AG , B. Determinantal processes and independence. Probability Surveys 3 (2006), 206–229

work page 2006
[15]

A., AND SOMMERS , H.-J

K HORUZHENKO , B. A., AND SOMMERS , H.-J. Non-hermitian ensembles, 2015

work page 2015
[16]

On the spectra of gaussian matrices

K OSTLAN , E. On the spectra of gaussian matrices. Linear Algebra and its Applications 162-164 (1992), 385–388

work page 1992
[17]

M ECKES , E. S. The Random Matrix Theory of the Classical Compact Groups . Cambridge University Press, 2019

work page 2019
[18]

M EHTA , M. L. Random Matrices, vol. 142. Elsevier, 2004

work page 2004
[19]

How to generate random matrices from the classical compact groups

M EZZADRI , F. How to generate random matrices from the classical compact groups. Notices Of The American Mathematical Society 4 (2007), 592–604

work page 2007
[20]

A limit theorem at the edge of a non-hermitian random matr ix ensemble

R IDER , B. A limit theorem at the edge of a non-hermitian random matr ix ensemble. Journal of Physics A: Mathematical and General 36 (2003), 3401–3409

work page 2003
[21]

Order statistics and ginibre’s ensembles

R IDER , B. Order statistics and ginibre’s ensembles. Journal of Statistical Physics 114 (2004), 1139–1148

work page 2004
[22]

R IDER , B., AND SINCLAIR , C. D. Extremal laws for the real ginibre ensemble. The Annals of Applied Probability 24 (2014)

work page 2014
[23]

Random matrices: Universality of esds and the circular l aw

T AO, T., V U, V., AND KRISHNAPUR , M. Random matrices: Universality of esds and the circular l aw. The Annals of Probability 38 (2010)

work page 2010
[24]

T EMME , N. M. Uniform asymptotic expansions of the incomplete gamm a functions and the incomplete beta function. Mathematics of Com- putation 29 (1975), 1109–1114

work page 1975
[25]

T EMME , N. M. The asymptotic expansion of the incomplete gamma func tions. SIAM Journal on Mathematical Analysis 10 (1979), 757–766

work page 1979
[26]

T EMME , N. M. Uniform asymptotic expansions of integrals: a select ion of problems. Journal of Computational and Applied Mathematics 65 (12 1995), 395–417

work page 1995
[27]

A., AND WIDOM , H

T RACY, C. A., AND WIDOM , H. Level-spacing distributions and the airy kernel. Communications in Mathematical Physics 159 (1994), 151–174

work page 1994
[28]

A., AND WIDOM , H

T RACY, C. A., AND WIDOM , H. On orthogonal and symplectic matrix ensembles. Communications in Mathematical Physics 177 (1996), 727–754

work page 1996
[29]

A., AND WIDOM , H

T RACY, C. A., AND WIDOM , H. The distributions of random matrix theory and their appl ications. New Trends in Mathematical Physics (2009), 753–765

work page 2009
[30]

The Classical Groups

W EYL , H. The Classical Groups. Their invariants and Representation s, second ed. Princeton University Press, 1946. 40

work page 1946

[1] [1]

A KARAWAK , E. E. E., A DELEKE , I. A., AND OKAFOR , R. O. The gamma-rayleigh distribution and applications to survival data. Nigerian Journal of Basic and Applied Sciences 25 (2018), 130

work page 2018

[2] [2]

J., AND SHIFRIN , L

A KEMANN , G., P HILLIPS , M. J., AND SHIFRIN , L. Gap probabilities in non-hermitian random matrix theor y. Journal of Mathematical Physics 50 (2009), 063504

work page 2009

[3] [3]

Around the circular law

B ORDENAVE , C., AND CHAFA ¨I, D. Around the circular law. Probability Surveys 9 (2012)

work page 2012

[4] [4]

The complex elliptic ginibre ensemble at weak non-hermi ticity: Edge spacing distributions

B OTHNER , T., AND LITTLE , A. The complex elliptic ginibre ensemble at weak non-hermi ticity: Edge spacing distributions. Random matrices and applications (2024)

work page 2024

[5] [5]

A., S OMMERS , H.-J., AND ZYCZKOWSKI , K

F ISCHMANN , J., B RUZDA , W., K HORUZHENKO , B. A., S OMMERS , H.-J., AND ZYCZKOWSKI , K. Induced ginibre ensemble of random matrices and quantum operations. Journal of Physics A: Mathematical and Theoretical 45 (2012)

work page 2012

[6] [6]

Some statistical properties of the eigenvalues of compl ex random matrices

F ORRESTER , P. Some statistical properties of the eigenvalues of compl ex random matrices. Physics Letters A 169 (1992), 21–24

work page 1992

[7] [7]

F ORRESTER , P. J. Meet andreief, bordeaux 1886, and andreev, kharkov 18 82-1883. Random Matrices: Theory and Applications 8 (2018)

work page 2018

[8] [8]

V., S OMMERS , H.-J., AND KHORUZHENKO , B

F YODOROV , Y. V., S OMMERS , H.-J., AND KHORUZHENKO , B. A. Universality in the random matrix spectra in the regim e of weak non- hermiticity. Annales de l’I. H. P ., Section A 68 (1998), 449–489

work page 1998

[9] [9]

The spectral radius of large random matrices

G EMAN , S. The spectral radius of large random matrices. The Annals of Probability 14 (1986)

work page 1986

[10] [10]

Statistical ensembles of complex, quaternion, and real matrices

G INIBRE , J. Statistical ensembles of complex, quaternion, and real matrices. Journal of Mathematical Physics 6 (1965), 440–449

work page 1965

[11] [11]

G IRKO , V. L. Circular law. Theory of Probability and Its Applications 29 (1985), 694–706

work page 1985

[12] [12]

G NEDENKO , B. V. Sur la distribution limite du terme maximum d’une seri e aleatoire. The Annals of Mathematics 44 (1943), 423

work page 1943

[13] [13]

Quantum distinction of regular and chaotic dissipat ive motion

G ROBE , R., H AAKE , F., AND SOMMERS , H.-J. Quantum distinction of regular and chaotic dissipat ive motion. Physical Review Letters 61 (1988), 1899–1902

work page 1988

[14] [14]

B., K RISHNAPUR , M., P ERES , Y., AND VIR ´AG , B

H OUGH , J. B., K RISHNAPUR , M., P ERES , Y., AND VIR ´AG , B. Determinantal processes and independence. Probability Surveys 3 (2006), 206–229

work page 2006

[15] [15]

A., AND SOMMERS , H.-J

K HORUZHENKO , B. A., AND SOMMERS , H.-J. Non-hermitian ensembles, 2015

work page 2015

[16] [16]

On the spectra of gaussian matrices

K OSTLAN , E. On the spectra of gaussian matrices. Linear Algebra and its Applications 162-164 (1992), 385–388

work page 1992

[17] [17]

M ECKES , E. S. The Random Matrix Theory of the Classical Compact Groups . Cambridge University Press, 2019

work page 2019

[18] [18]

M EHTA , M. L. Random Matrices, vol. 142. Elsevier, 2004

work page 2004

[19] [19]

How to generate random matrices from the classical compact groups

M EZZADRI , F. How to generate random matrices from the classical compact groups. Notices Of The American Mathematical Society 4 (2007), 592–604

work page 2007

[20] [20]

A limit theorem at the edge of a non-hermitian random matr ix ensemble

R IDER , B. A limit theorem at the edge of a non-hermitian random matr ix ensemble. Journal of Physics A: Mathematical and General 36 (2003), 3401–3409

work page 2003

[21] [21]

Order statistics and ginibre’s ensembles

R IDER , B. Order statistics and ginibre’s ensembles. Journal of Statistical Physics 114 (2004), 1139–1148

work page 2004

[22] [22]

R IDER , B., AND SINCLAIR , C. D. Extremal laws for the real ginibre ensemble. The Annals of Applied Probability 24 (2014)

work page 2014

[23] [23]

Random matrices: Universality of esds and the circular l aw

T AO, T., V U, V., AND KRISHNAPUR , M. Random matrices: Universality of esds and the circular l aw. The Annals of Probability 38 (2010)

work page 2010

[24] [24]

T EMME , N. M. Uniform asymptotic expansions of the incomplete gamm a functions and the incomplete beta function. Mathematics of Com- putation 29 (1975), 1109–1114

work page 1975

[25] [25]

T EMME , N. M. The asymptotic expansion of the incomplete gamma func tions. SIAM Journal on Mathematical Analysis 10 (1979), 757–766

work page 1979

[26] [26]

T EMME , N. M. Uniform asymptotic expansions of integrals: a select ion of problems. Journal of Computational and Applied Mathematics 65 (12 1995), 395–417

work page 1995

[27] [27]

A., AND WIDOM , H

T RACY, C. A., AND WIDOM , H. Level-spacing distributions and the airy kernel. Communications in Mathematical Physics 159 (1994), 151–174

work page 1994

[28] [28]

A., AND WIDOM , H

T RACY, C. A., AND WIDOM , H. On orthogonal and symplectic matrix ensembles. Communications in Mathematical Physics 177 (1996), 727–754

work page 1996

[29] [29]

A., AND WIDOM , H

T RACY, C. A., AND WIDOM , H. The distributions of random matrix theory and their appl ications. New Trends in Mathematical Physics (2009), 753–765

work page 2009

[30] [30]

The Classical Groups

W EYL , H. The Classical Groups. Their invariants and Representation s, second ed. Princeton University Press, 1946. 40

work page 1946