Pith. sign in

REVIEW 3 major objections 3 minor 2 cited by

Convergence rate of extreme eigenvalue of Ginibre ensembles to Gumbel distribution

T0 review · 3 major / 3 minor · reviewed 2026-08-07 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read This paper proves that Ginibre rightmost eigenvalues approach Gumbel with sharp rates $\frac{25\log\log n}{4e\log n}$ and $\frac{25\log\log n}{4\log n}$, universal for complex i.i.d. entries.

desk verdict Sharp Ginibre edge rates are real; the paper's flaws are presentational, not load-bearing. read the letter →

arxiv 2506.04560 v3 pith:SAFFGCG4 submitted 2025-06-05 math.PR

Xinchen Hu , Yutao Ma This is my paper · ORCID
classification math.PR MSC 60G7060B2060B10
keywords GumbeldistributionrightmosteigenvaluespectralradiusWassersteindistanceBerry-EsseenboundGinibreensembleextremeuniversality
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper pins down exactly how fast the largest real-part eigenvalue of a Ginibre matrix approaches the Gumbel law. For the rescaled rightmost eigenvalue $Z_n$, the Kolmogorov distance to the Gumbel distribution $e^{-\frac{\beta}{2}e^{-x}}$ is asymptotically $\frac{25\log\log n}{4e\log n}$, and the $W_1$ Wasserstein distance is $\frac{25\log\log n}{4\log n}$. The spectral radius converges at the faster order $(\log\log n)/\log n$ with constants $2/e$ and $2$. These exact rates replace the previous $O((\log\log n)^2/\log n)$ bound and carry over to complex i.i.d. matrices under moment conditions. The prefactors emerge from a secondary polynomial correction in the trace of the edge correlation kernel.

What carries the argument

The central object is the correlation kernel $\widetilde K_n(z,w)$ of the complex Ginibre eigenvalues, with the Pfaffian analogue $K_n^{C,C}(z,w)$ for the real case. The machinery is the identity $\mathbb{P}(Z_n\le t)=\det(1-W_n^{(t)})$, where $W_n^{(t)}$ is the kernel compressed to the half-plane $A(t)$. Since $\|W_n^{(t)}\|_2\lesssim e^{-\sqrt{\log n}/32}$, the determinant is well approximated by $\exp(-\operatorname{Tr} W_n^{(t)})$. The proof's heart is Lemma 2.3, which computes $\operatorname{Tr} W_n^{(t)}=e^{-t}(1+(c_n-t^2-5t)/\log n)(1+O((\log n)^{-1}))$ uniformly in the edge regime; the term $25\log\log n$ inside $c_n=(25\log\log n+5\log(2\pi^4)-35)/4$ is the source of the sharp constant. The analogous trace asymptotics for the spectral radius and for the real Pfaffian kernel are proved in Lemmas 2.4 through 2.6.

What would settle it

Simulate many real and complex Ginibre matrices at a large finite $n$, compute the rescaled rightmost eigenvalue $Z_n$, and estimate the Kolmogorov distance $\sup_x|\mathbb{P}(Z_n\le x)-e^{-\frac{\beta}{2}e^{-x}}|$ and the $W_1$ distance. If the theorem is correct, these quantities multiplied by $\log n/\log\log n$ should approach $25/(4e)$ and $25/4$ (respectively $2/e$ and $2$ for $Y_n$) as $n$ grows. A ratio that drifts or settles on a different constant would falsify the claimed asymptotics.

Watch

Extended reading notes

Core claim

The central discovery is that the speed of Gumbel convergence at the Ginibre edge has a closed-form asymptotic with explicit constants. Theorem 1 shows that for real and complex Ginibre matrices, with $Z_n=\sqrt{4n\gamma_n}(\max_i\Re\sigma_i-1-\sqrt{\gamma_n/(4n)})$ and $\gamma_n=\tfrac12(\log n-5\log\log n-\log(2\pi^4))$, the Kolmogorov distance satisfies $\sup_x|\mathbb{P}(Z_n\le x)-e^{-\frac{\beta}{2}e^{-x}}|=\frac{25\log\log n}{4e\log n}(1+o(1))$ and the $W_1$ distance satisfies $W_1(\mathcal{L}(Z_n),\Lambda_\beta)=\frac{25\log\log n}{4\log n}(1+o(1))$. Theorem 2 gives the analogous statements for the spectral radius $Y_n$ with constants $2/e$ and $2$. Theorem 3 extends the rightmost-eigenvalue rate to complex i.i.d. matrices satisfying moment conditions, and the accompanying remark carries the same rate to the spectral radius. Theorem 4 shows that under the alternative scaling equations $64\tilde\gamma_n^5\pi^4 e^{2\tilde\gamma_n}=n$ and $2\pi(\tilde\gamma'_n)^2 e^{\tilde\gamma'_n}=n$, the rates improve to $\kappa_1/(4\log n)$ and $\kappa_2/\log n$ with $\kappa_1\approx 15.4$ and $\kappa_2\approx 1.48$.

Load-bearing premise

The argument relies on a quoted asymptotic expansion for the incomplete gamma function being uniform at the edge scale where $|z|^2-1$ is of order $\sqrt{\log n/n}$; if that uniformity fails, the sharp constants in the theorems lose their justification.

Editorial extensions

If this is right

  • The previously known error bound $O((\log\log n)^2/\log n)$ for the rightmost eigenvalue of Ginibre matrices is upgraded to an exact asymptotic with explicit constant, for both real and complex ensembles.
  • The spectral-radius analogue of the exact-rate result now holds for rightmost eigenvalues and for the real Ginibre ensemble, not only for the complex spectral radius.
  • For complex i.i.d. matrices satisfying the stated moment conditions, the same sharp rates hold for both the rightmost eigenvalue and the spectral radius, so the rate is universal.
  • Under the alternative scalings defined by $64\tilde\gamma_n^5\pi^4 e^{2\tilde\gamma_n}=n$ and $2\pi(\tilde\gamma'_n)^2 e^{\tilde\gamma'_n}=n$, the convergence rate improves to $1/\log n$ with explicit constants $\kappa_1/4$ and $\kappa_2$.
  • The $W_1$ result is an integral statement, not just a pointwise one, so it controls the full tail of the distance between the rescaled edge distribution and Gumbel.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Because the proof relies only on local asymptotics of the correlation kernel near the spectral edge, the same sharp constants should control other directional extremes, such as the largest imaginary part or the extreme in a fixed angular sector, by rotation symmetry of the Ginibre kernel.
  • The new scaling theorem suggests that choosing the centering and scaling constants optimally removes the $\log\log n$ factor entirely, leaving a rate of order $1/\log n$; further optimization of the scaling constants may push the rate even lower, though the paper stops at the stated constants.
  • The Pfaffian results for the real Ginibre ensemble are a first step toward a sharp-rate universality statement for real i.i.d. matrices, where the correlation structure is more delicate than in the complex case.
  • The explicit dependence of the constant on the polynomial correction $4t^2+20t+35$ indicates that the sharp prefactor is governed by the second-order edge profile of the kernel, so any model sharing that profile should reproduce the same constants.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 3 minor

Summary. The paper studies the extreme eigenvalues of real and complex Ginibre matrices, focusing on the rightmost eigenvalue (largest real part) and on the spectral radius. The main results, Theorems 1 and 2, claim sharp first-order asymptotics for both the Kolmogorov distance and the W1 Wasserstein distance to the appropriate Gumbel distribution, with explicit constants: 25/(4e) and 25/4 for the rightmost eigenvalue, and 2/e and 2 for the spectral radius. Theorem 3 extends the complex Ginibre rates to complex i.i.d. matrices under Assumption 1.1, and Theorem 4 discusses the effect of changing the scaling constants. The proofs use the determinantal/Pfaffian structure of Ginibre ensembles, with detailed asymptotic analysis of the correlation kernel and of the trace of the associated integral operator.

Significance. If the main claims are correct, the paper sharpens the previously known O((log log n)^2/log n) bound from Cipolloni–Erdős–Schröder–Xu [15] to the exact order and sharp constant, and similarly gives the first sharp W1 rate for these edge statistics. The extension to the real Ginibre ensemble and to spectral radius is a natural and valuable step. The paper contains substantial technically detailed kernel asymptotics, particularly Lemmas 2.2, 2.3, and 2.5, and the overall strategy is coherent. The main reservations concern the rigor of the uniform control in Proposition 1 and a numerical/algebraic inconsistency in Lemma 2.4, as well as the sketchy treatment of the universality theorem.

major comments (3)
  1. [Section 3, Proposition 1, Eq. (3.1)] The uniform statement in Proposition 1 is not justified as written. In the proof, the factor |c_n - t^2 - 5t| is replaced by c_n(1+o(1)), but on the stated interval t in [-1/4 log log n, (log n)^{1/4}] the term t^2 can be of order (log n)^{1/2}, which is not o(log log n); for instance at t=(log n)^{1/8} we have t^2/(log log n) -> infinity. Thus the displayed pointwise asymptotic fails on parts of the interval. The final supremum or integral may still be dominated by the region t=O(1), but this requires a separate argument; as it stands, the proof of the Berry-Esseen bound in Section 3.1 and the computation of I2 in Section 3.2 are incomplete.
  2. [Lemma 2.4, Eq. (2.19) and (2.23)-(2.26)] The expansion of a_t in (2.22) gives a_t - 1 = sqrt(gamma'_n/n) + t/sqrt(n gamma'_n) + O(gamma'_n/n), and hence n mu^2(a_t) = gamma'_n/2 + t + t^2/(2 gamma'_n) + o(gamma'_n^{-1}), not the displayed gamma'_n/2 + t + t^2/gamma'_n. Tracing this through (2.24)-(2.26) changes the numerator in (2.19) from d_n - t^2 - 4t to d_n - 2t - t^2/2. The stated lemma therefore appears to be incorrect as it stands. Since Proposition 1(3.2) and Theorem 2 rely on this lemma, the proof needs revision. The leading 2 log log n/log n constant is likely unaffected because the polynomial correction is O(1) near the maximizing t=O(1), but the lemma must be corrected.
  3. [Section 4, Theorem 3, Eq. (4.2)] The decisive comparison sup_{x in [-ell_n, ell_n]} |F_n(x)-F_n^Gin(x)| <~ n^{-epsilon} with ell_n=(log n)^{1/4} is asserted by 'scrutinizing the proof of [17, Theorem 4.1]', but the cited theorem is not stated in that form. The introduction only quotes pointwise convergence results (1.2)-(1.3), and a uniform comparison on a growing interval with a polynomial error is a stronger statement. Without a precise statement of the version used, and a verification that it applies to the scaling here, the universality of the convergence rate - a central advertised conclusion - is not established. Please provide the comparison as a lemma with full hypotheses and proof, or at least state the exact theorem from [17] that implies (4.2).
minor comments (3)
  1. [Introduction, definition of O(.)] The definition of O(.) in the introduction is nonstandard and conflicts with the usual meaning: requiring lim t_n/z_n = c != 0 excludes terms that are actually o(z_n) and makes statements such as 0=O(z_n) false. Please use standard O notation, with a separate symbol such as '=' or '~' for asymptotic equivalence.
  2. [Section 3.2] The integration limits in the W1 decomposition appear reversed: the text writes \int_{-\ell_1(n)}^{-\infty} and later \int_{-\ell_1(n)}^{-\ell_2(n)}. The intended intervals are clearly (-\infty, -\ell_1(n)] and [-\ell_2(n), -\ell_1(n)]; please correct the typography.
  3. [Theorem 4] The numerical values kappa_1 ≈ 15.4 and kappa_2 ≈ 1.48 are stated without derivation. If these constants are included, please add a short justification or a computational remark so the reader can verify them.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the rate constants are derived from explicit trace asymptotics and external estimates, not fitted or imported by self-citation.

full rationale

The paper's central claims (Theorems 1 and 2) are proved by computing Tr(W_n^{(t)}) and ||W_n^{(t)}||_2 directly from the explicit Ginibre correlation kernel, with asymptotic expansions quoted from external sources [15, 31, 24]. The only self-citations, [26] and [27], appear as background and motivation; they are not used as load-bearing inputs in any proof. In fact, the paper independently re-derives the spectral-radius rate for complex Ginibre that [26] had obtained, rather than importing it. The constants 25/4, 25/(4e), 2, and 2/e arise algebraically from the expansions in Lemmas 2.3 and 2.4 and from the explicit supremum/integral of e^{-e^{-t}-t}; no parameter is fitted to the announced rate. Theorem 3's universality relies on the external comparison theorem [17, Theorem 4.1], whose authors do not overlap with the present authors, and the proof only uses it to transfer the Ginibre rate to i.i.d. matrices. The uniformity concern around [31, Lemma 3.2] raised in the reader's take is a potential rigor issue about the validity of an external asymptotic estimate in the edge regime, not a circularity within this paper. No definitional equivalence, fitted-input-as-prediction, or self-citation chain that forces the result was found.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No parameters are fitted. The paper depends on external correlation kernel and incomplete-Gamma asymptotics from [15,31], and on the Green function comparison theorem from [17]; all are cited, not re-derived. The scaling sequences gamma_n and gamma'_n are fixed, not tuned to data.

assumptions (5)
  • domain assumption The incomplete-Gamma ratio expansion from [31, Lemma 3.2] is valid uniformly in the edge regime (2.7).
    Used as equation (2.9) in Lemma 2.2 to derive the trace asymptotics; errors of order 1/(n||z|^2-1|) multiply a n^{1/4} factor and feed the constants in Theorem 1.
  • standard math The determinant-to-exponential bound (2.4) from [24] applies to the integral operator W_n^{(t)}.
    Bridges det(1-W) to exp(-Tr W), used in equations (2.4) through (2.6).
  • domain assumption The Green function comparison theorem of [17, Theorem 4.1] gives sup_{x in [-l,l]} |F_n - F_n^Gin| bounded by n^{-epsilon}.
    Load-bearing for Theorem 3; the paper sketches the extraction without reproducing the proof.
  • domain assumption For real Ginibre, the largest real eigenvalue is O(1/sqrt n) away from the edge and can be ignored relative to complex eigenvalues.
    Justifies using the Pfaffian kernel K^{C,C} restricted to C without the real axis in Section 2.2; cited from [29].
  • domain assumption The norm bound ||W_n^{(t)}||_2 bounded by e^{-sqrt(log n)/32} from [15] holds uniformly on |t| <= sqrt(log n)/10.
    Used to replace the determinant by exp(-Tr(W)) with negligible error in equations (2.5) and (2.6).

how reviews work

0 comments
Cite this review

Pith. "Pith review of Convergence rate of extreme eigenvalue of Ginibre ensembles to Gumbel distribution." pith.science (2026). https://pith.science/paper/SAFFGCG4

@misc{pith2026250604560,
  author       = {Pith},
  title        = {Pith review of: Convergence rate of extreme eigenvalue of Ginibre ensembles to Gumbel distribution},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SAFFGCG4}},
  note         = {Machine review of arXiv:2506.04560}
}
abstract

Let $X$ be a real $(\beta=1)$ or complex $(\beta=2)$ Ginibre ensemble. Let $\{\sigma_i\}_{1\le i\le n}$ be the eigenvalues of $X,$ and $Z_n$ be some rescaled version of $\max_i \Re \sigma_i.$ It was proved that $Z_n$ converges weakly to the Gumbel distribution $\Lambda_{\beta}$ with distribution function $e^{-\frac{\beta}{2}e^{-x}}.$ We further prove that $$\sup_{x\in \mathbb{R}}|\mathbb{P}(Z_n \leq x)-e^{-\frac{\beta}{2}e^{-x}}|=\frac{25\log \log n}{4e \log n}(1+o(1))$$ and $$ W_1\left(\mathcal{L}(Z_n), \Lambda_{\beta}\right)=\frac{25\log \log n}{4\log n}(1+o(1))$$ for sufficiently large $n$, where $\mathcal{L}(Z_n)$ is the distribution of $Z_n$ and $W_1$ is the Wasserstein distance. Similar results hold for $\max_{i} |\sigma_i|.$ Furthermore, the convergence rates of the complex Ginibre ensemble are universal for complex iid random matrices under certain moment conditions on entries.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Moderate-to-large deviation asymptotics for real eigenvalues of the elliptic Ginibre matrices

    math.PR 2025-11 accept novelty 7.0 of 10

    For elliptic real Ginibre matrices, probabilities of rare counts of real eigenvalues have explicit exponential rate functions in the strong- and weak-asymmetry regimes, new even for the real Ginibre ensemble.

  2. Revisit on the convergence rate of normal extremes

    math.PR 2025-07 conditional novelty 7.0 of 10

    Gaussian maxima to Gumbel convergence rates are computed exactly for the Kolmogorov, W1, total variation, KL and Fisher metrics, with explicit constants depending on powers of log log n and log n.

Reference graph

Works this paper leans on

36 extracted references · 33 canonical work pages · cited by 2 Pith papers

  1. [26]

    Y. T. Ma and X. Meng. Exact convergence rate of spectral radius of complex Ginibre to Gumbel distribution. arXiv:2501.08039

    arXiv

  2. [15]

    Cipolloni, L

    G. Cipolloni, L. Erd¨ os, D. Schr¨ oder and Y. Xu. Directional extremal statistics for Gini- bre eigenvalues.J. Math. Phys.,63(10)(2022), 103303

    work page 2022

  3. [17]

    Cipolloni, L

    G. Cipolloni, L. Erd¨ os and Y. Xu. Universality of extremal eigenvalues of large random matrices. arXiv:2312.08325

    arXiv

  4. [1]

    Interpolation between Airy and Poisson statistics for unitary chiral non-Hermitian random matrix ensembles

    G, Akemann and M, Bender(2010). Interpolation between Airy and Poisson statistics for unitary chiral non-Hermitian random matrix ensembles. J. Math. Phys.51(10): 103524

    work page 2010

  5. [2]

    Universality Conjecture for all Airy, Sine and Bessel Kernels in the Complex Plane

    G. Akemann and M. J. Phillips. Universality Conjecture for all Airy, Sine and Bessel Kernels in the Complex Plane. arXiv:1204.2740

    work page Pith review arXiv

  6. [3]

    Akemann and M.J

    G. Akemann and M.J. Phillips. The Interpolating Airy Kernels for theβ= 1 andβ= 4 Elliptic Ginibre Ensembles.J. Stat. Phys.,155(2014), 421-465

    work page 2014

  7. [4]

    J. Alt, L. Erd¨ os and T. Kr¨ uger. Spectral radius of random matrices with independent entries.Probab. Math. Phys.,2(2)(2021), 221-280

    work page 2021

  8. [5]

    Z. D. Bai. Circular law.Ann. Probab.,25(1)(1997), 494-529

    work page 1997

Show all 36 references
  1. [6]

    Z. D. Bai. Convergence Rate of Expected Spectral Distributions of Large Random Ma- trices. Part I. Wigner Matrices.Ann. Probab.,21(2)(1993), 625-648

    1993

  2. [7]

    Z. D. Bai. Convergence Rate of Expected Spectral Distributions of Large Random Ma- trices. Part II. Sample Covariance Matrices.Ann. Probab.,21(2)(1993), 649-672

    1993

  3. [8]

    Z. D. Bai and J. Hu. A revisit of the circular law. arXiv:2408.13490

    arXiv

  4. [9]

    Z. D. Bai and Y. Q. Yin. Convergence to the semicircle law.Ann. Probab.,16(1988), 863-875

    1988

  5. [10]

    M. Bender. Edge scaling limits for a family of non-Hermitian random matrix ensembles. Probab. Th. Relat. Fields,147(2010), 241-271

    2010

  6. [11]

    Bordenave and D

    C. Bordenave and D. Chafa¨ı. Around the circular law.Probab. Surv.,9(2012), 1-89

    2012

  7. [12]

    Borodin and C

    A. Borodin and C. D. Sinclair. The Ginibre ensemble of real random matrices and its scaling limits.Commun. Math. Phys.,291(2009), 177-224

    2009

  8. [13]

    Borodin, M

    A. Borodin, M. Poplavskyi, C. D. Sinclair, R. Tribe, and O. Zaboronski, Erratum to: The Ginibre ensemble of real random matrices and its scaling limits [MR2530159],Comm. Math. Phys.,346(2016), 1051-1055

    2016

  9. [14]

    S. S. Byun and P. J. Forrester.Progress on the study of the Ginibre ensembles.1st ed., Springer Singapore, 2024

    2024

  10. [16]

    Cipolloni, L

    G. Cipolloni, L. Erd¨ os, D. Schr¨ oder and Y. Xu. On the rightmost eigenvalue of non- Hermitian random matrices.Ann. Probab.,51(6)(2022), 2192-2242

    2022

  11. [18]

    Chafai and S

    D. Chafai and S. P´ ech´ e. A note on the second order universality at the edge of Coulomb gases on the plane.J. Stat. Phys.,156(2014), 368-383

    2014

  12. [19]

    A. Edelman. The Probability that a random real Gaussian matrix haskreal eigenvalues, related distributions, and the circular law.J. Multivar. Anal.,60(2)(1997), 203-232

    1997

  13. [20]

    Forrester and T

    P. Forrester and T. Nagao. Eigenvalue statistics of the real Ginibre ensemble.Phys. Rev. Lett.,99(2007), 050603

    2007

  14. [21]

    Garrod, M

    B. Garrod, M. Poplavskyi, R. Tribe and O. Zaboronski. Examples of interacting particle systems onZas Pfaffian point processes: annihilating and coalescing random walks. Ann. Henri. Poincar´e,19(2018), 3635–3662

    2018

  15. [22]

    J. Ginibre. Statistical Ensembles of Complex, Quaternion, and Real Matrices.J. Math. Phys.,6(1965), 440-449

    1965

  16. [23]

    V. L. Girko. The circular law.Teor. Veroyatnost. i Primenen.,29(1984), 669-679

    1984

  17. [24]

    Gohberg, S

    I. Gohberg, S. Goldberg, and N. Krupnik.Traces and determinants of linear operators. Operator Theory: Advances and Applications.116Birkhauser, Basel, 2000

    2000

  18. [25]

    On the spectra of Gaussian matrices.Linear Algebra Appl.162/164(1992), 385-388

    E., Kostlan. On the spectra of Gaussian matrices.Linear Algebra Appl.162/164(1992), 385-388

    1992

  19. [27]

    Y. T. Ma and S. Wang. OptimalW 1 and Berry-Esseen bound between the spectral radius of large Chiral non-Hermitian random matrices and Gumbel. arXiv:2501.08661. UNIVERSALITY OF CONVERGENCE RATE 27

    arXiv

  20. [28]

    Meckes and M

    E. Meckes and M. Meckes. A rate of convergence for the circular law for the complex Ginibre ensemble.Ann. Fac. Sci. Toulouse Math.,24(2015), 93-117

    2015

  21. [29]

    Poplavskyi, R

    M. Poplavskyi, R. Tribe and O. Zaboronski. On the distribution of the largest real eigenvalue for the real Ginibre ensemble.Ann. Appl. Probab.,27(3)(2017), 1395-1413

    2017

  22. [30]

    B. Rider. A limit theorem at the edge of a non-Hermitian random matrix ensemble.J. Physics A.,36(2003), 3401-3409

    2003

  23. [31]

    Rider and C

    B. Rider and C. D. Sinclair. Extremal laws for the real Ginibre ensemble.Ann. Appl. Probab.,24(4) (2014), 1621-1651

    2014

  24. [32]

    Silverstein and Z

    J.W. Silverstein and Z. D. Bai. On the Empirical Distribution of Eigenvalues of a Class of Large Dimensional Random Matrices.J. Multivariate. Anal.,54(6)(1995), 175-192

    1995

  25. [33]

    H.-J. Sommers. Symplectic structure of the real Ginibre ensemble.J. Phys. A40(2007), F671-F676, MR2371225

    2007

  26. [34]

    Tao and V

    T. Tao and V. Vu. Random matrices: The circular law.Commun. Contemp. Math., 10(2008), 261-307

    2008

  27. [35]

    Tao and V

    T. Tao and V. Vu. Random matrices: Universality of local spectral statistics of non- Hermitian matrices.Ann. Probab.,38(5)(2009), 2023-2065

    2009

  28. [36]

    Temme, N. M. (1996). Asymptotic Expansions for the Incomplete Gamma Functions. SIAM J. Math. Anal.,27(1), 20-36. Xinchen Hu, School of Mathematical Sciences&Laboratory of Mathematics and Complex Systems of Ministry of Education, Beijing Normal University, 100875 Beijing, China...

    1996

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.