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Exact convergence rate of spectral radius of complex Ginibre to Gumbel distribution

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Pith's one-line read This paper proves that the spectral radius of the complex Ginibre ensemble reaches the Gumbel distribution at sharp logarithmic rates: the Wasserstein distance satisfies $\lim_{n\to\infty}\frac{\log n}{\log\log n}W_1=2$, and the…

desk verdict Plausible and likely correct rates, but the written proof has a scaling inconsistency in the core definitions and an invalid absolute-value drop in Section 3.4. read the letter →

arxiv 2501.08039 v1 pith:7V4IW4FE submitted 2025-01-14 math.PR

Yutao Ma , Xujia Meng This is my paper · ORCID
classification math.PR MSC 60F1060B2060G55
keywords GinibreensemblespectralradiusGumbeldistributionWassersteindistanceBerry-Esseenboundconvergencerateextremeeigenvalues
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 proves that the largest eigenvalue modulus of an $n\times n$ complex Ginibre random matrix approaches its Gumbel limiting distribution at a sharp, explicitly known speed. The Wasserstein distance between the distribution of the scaled spectral radius and the Gumbel law satisfies $\lim_{n\to\infty}\frac{\log n}{\log\log n}W_1(F_n,\Lambda)=2$, and the Kolmogorov (uniform) distance satisfies $\lim_{n\to\infty}\frac{\log n}{\log\log n}\sup_x|F_n(x)-e^{-e^{-x}}|=\frac2e$. The paper supplies exact rate constants for this edge statistic, turning the earlier weak-convergence theorem into a quantitative statement. The rates are slow, of order $\frac{\log\log n}{\log n}$, so the Gumbel approximation becomes accurate only after $n$ is exponentially large in the desired precision.

What carries the argument

The argument rests on Lemma 2.1, which identifies $R_n^2$ in distribution with the maximum of $n$ independent random variables $Y_j$ having density proportional to $y^{j-1}e^{-y}$ on $y>0$, i.e. Gamma variables. This reduces the two-dimensional eigenvalue problem to a one-dimensional extreme-value problem. The proof then uses Lemma 2.2, a uniform Gaussian tail expansion for $P(Y_{n-k}>a_n+b_nx)$ in terms of $u_n(k,x)=\frac{k}{\sqrt n}+\sqrt{\gamma_n}+\frac{x}{\sqrt{\gamma_n}}$, with relative error $O(u_n^{-2})$, valid for $1\ll u_n\ll n^{1/10}$. Lemma 2.3 converts the resulting sums and integrals over $k$ and $x$ into leading-order exponentials; the cutoffs $\ell_1(n)=\frac12\log\log n$ and $\ell_2(n)=\log(\sqrt{2\pi}\,\log n)$ isolate the interval that contributes the constants $2$ and $2/e$.

What would settle it

Compute, for increasing $n$ and for many choices of $k$ and $x$, the exact tail probability $P(Y_{n-k}>a_n+b_nx)$ for the Gamma variables in Lemma 2.1 and compare it with $e^{-u_n^2/2}/(\sqrt{2\pi}u_n)$ over the range $1\ll u_n\ll n^{1/10}$; a relative error larger than $O(u_n^{-2})$ at any point would falsify the uniformity on which the rates rest. Alternatively, estimate $W_1$ and $\sup_x|F_n(x)-e^{-e^{-x}}|$ by simulation for $n$ up to $10^6$; the quantities $(\log n/\log\log n)W_1$ and $(\log n/\log\log n)\sup_x|F_n-\Lambda|$ should approach $2$ and $2/e$, and any clear drift away from these constants would disprove the theorem.

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Extended reading notes

Core claim

Let $G_n$ be an $n\times n$ matrix with i.i.d. standard complex Gaussian entries, and let $R_n=\max_{1\le k\le n}|\lambda_k|$ be its spectral radius. With $\gamma_n=\log n-2\log(\sqrt{2\pi}\,\log n)$ and $X_n=\sqrt{4\gamma_n}(R_n-\sqrt n-\tfrac12\sqrt{\gamma_n})$, the paper proves two sharp limits. If $F_n$ is the distribution function of $X_n$ and $\Lambda(x)=e^{-e^{-x}}$ is the Gumbel distribution, then $\lim_{n\to\infty}\frac{\log n}{\log\log n}W_1(F_n,\Lambda)=2$, and $\lim_{n\to\infty}\frac{\log n}{\log\log n}\sup_{x\in\mathbb R}|F_n(x)-e^{-e^{-x}}|=\frac2e$. In words, both natural distances between the edge statistic and its Gumbel limit are of order $\frac{\log\log n}{\log n}$, with the Wasserstein constant exactly $2$ and the Kolmogorov constant exactly $2/e$.

Load-bearing premise

The load-bearing premise is the uniform Gaussian tail approximation of Lemma 2.2: the relative error in $P(Y_{n-k}>a_n+b_nx)$ is $O(u_n(k,x)^{-2})$ for every $k$ and $x$ in the summation range, a uniformity imported from classical Berry-Esseen bounds; if that error is larger anywhere, the constants $2$ and $2/e$ are not forced.

Editorial extensions

If this is right

  • For large $n$, $W_1(F_n,\Lambda)=(2+o(1))\frac{\log\log n}{\log n}$ and $\sup_x|F_n(x)-e^{-e^{-x}}|=(\frac2e+o(1))\frac{\log\log n}{\log n}$.
  • The worst-case approximation error is localized to $x\in(-\frac12\log\log n,\,\log(\sqrt{2\pi}\,\log n))$; outside this window both the tail and the Gumbel term are exponentially negligible.
  • Since $W_1(L(W_n),F_n)\ll \gamma_n^{-1}$, the simpler centered variable $W_n$ can be substituted for $X_n$ in rate computations without changing the leading constant.
  • The rate is a property of the maximum of independent Gamma variables, so after the reduction of Lemma 2.1 the proof bypasses the correlations among eigenvalues entirely.

Reading between the lines

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

  • Beyond the paper, the same constants should transfer to any ensemble whose squared spectral radius reduces to independent Gamma-type variables and satisfies the same uniform tail bound; the paper only claims the complex Ginibre case.
  • A practical check is to simulate the Gamma maxima of Lemma 2.1 for $n=10^3,\dots,10^6$; the ratios $(\log n/\log\log n)W_1$ and $(\log n/\log\log n)\sup|F_n-\Lambda|$ should creep toward $2$ and $2/e$, but so slowly that small-$n$ deviations are expected.
  • The method suggests an extension to the joint distribution of the largest few moduli, with constants depending on the order statistic; this is a natural next step, not contained in the paper.
  • If a future universality result for general iid matrices supplied the same uniform tail estimate, the proof strategy would transfer directly; the constants obtained here would then serve as the benchmark.
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Formalized claims in Lean

  1. Claim #1: Let $G_n$ be an $n\times n$ matrix with i.i.d. standard complex Gaussian entries, and let $R_n=\max_{1\le k\le n}|\lambda_k|$ be its spectral radius. With $\gamma_n=\log n-2\log(\sqrt{2\pi}\,\log n)$ and $X_n=\sqrt{4\gamma_n}(R_n-\sqrt n-\tfrac12\sqrt{\gamma_n})$, the paper proves two sharp limits. If $F_n$ is the distribution function of $X_n$ and $\Lambda(x)=e^{-e^{-x}}$ is the Gumbel distributi

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 3 minor

Summary. The paper studies the spectral radius R_n of an n by n complex Ginibre matrix. With X_n = sqrt(4 gamma_n)(R_n - sqrt(n) - (1/2) sqrt(gamma_n)) and gamma_n = log n - 2 log(sqrt(2 pi) log n), it claims the sharp Wasserstein rate lim_{n->infty} (log n / log log n) W_1(F_n, Lambda) = 2 and the Kolmogorov rate lim_{n->infty} (log n / log log n) sup_x |F_n(x) - exp(-exp(-x))| = 2/e, where F_n is the distribution function of X_n and Lambda is the standard Gumbel distribution. The proof uses Kostlan's representation of R_n^2 as the maximum of independent Gamma variables, a uniform Gaussian-tail expansion for Gamma tails, and detailed integral and sum estimates to extract the first-order constants. The paper also introduces an intermediate variable W_n and reduces the problem to estimates on W_1(L(W_n), Lambda) and W_1(L(W_n), F_n).

Significance. If correct, the two limits are sharp convergence-rate results for the extremal statistics of the complex Ginibre ensemble, going beyond Rider's distributional limit. The method is transparent: it reduces the problem to classical Gamma tail bounds and explicitly derives the constants 2 and 2/e from asymptotic expansions rather than fitting them. The paper states explicit error terms and does not rely on unproved heuristics. However, the current written proof contains several internal definitional inconsistencies and at least two invalid logical steps, so the claimed theorems are not yet established by the manuscript as written.

major comments (4)
  1. [Section 2, definitions of W_n, a_n, b_n and Lemma 2.2] The auxiliary variable W_n is defined by W_n = (sqrt(gamma_n)/sqrt(n))(R_n^2 - a_n) with a_n := n + sqrt(n) gamma_n. Then P(W_n >= x) = P(R_n^2 >= a_n + (sqrt(n)/sqrt(gamma_n))x), so the correct coefficient b_n is sqrt(n)/sqrt(gamma_n), not sqrt(n)/gamma_n as stated. Moreover, the quantity u_n(k,x) = k/sqrt(n) + sqrt(gamma_n) + x/sqrt(gamma_n) used throughout Lemma 2.2 and Section 3 corresponds to a_n = n + sqrt(n) sqrt(gamma_n) and b_n = sqrt(n)/sqrt(gamma_n), not to the stated a_n and b_n. This is not a purely notational issue: the expansions of P(Y_{n-k} > a_n + b_n x) in Lemma 2.2 are for a different centering than the W_n actually defined, so the proof of the main estimates does not apply to the quantity the paper calls W_n. The definitions should be corrected and carried through consistently.
  2. [Section 3.4, Eq. (3.2)] The proof of the transfer estimate W_1(L(W_n), F_n) << gamma_n^{-1} is invalid. After substituting t = x/sqrt(4 gamma_n) + sqrt(n) + sqrt(gamma_n)/4 in the first integral and t = a_n + b_n x in the second, the two integrals are combined as if the integration variables were the same, but they are different coordinates, so the displayed equality is not justified. A correct expression for the L1-Wasserstein distance between the two increasing functions of Y_(n) is sqrt(gamma_n/n) E[(sqrt(Y_(n)) - sqrt(n))^2]; integration by parts yields the paper's integral from 0 to n plus a positive tail term sqrt(gamma_n) integral_n^infty (1/sqrt(n) - 1/sqrt(t)) P(Y_(n) > t) dt, which is omitted. Since P(Y_(n) > n) -> 1, this tail is not negligible in general, and the subsequent bound sqrt(gamma_n) n P(Y_(n) <= n) does not establish (3.2). The estimate may be repairable by a direct analysis of the omitted tail, but as written Theorem 1 lacks a valid transfer estimate.
  3. [Section 4, Eq. (4.2)] The assertion that sup_x |F_n(x) - bar(F_n)(x)| << gamma_n^{-1} log log n follows from W_1(L(W_n), F_n) << gamma_n^{-1} is false: a small L1-Wasserstein distance between two probability distributions does not control the uniform sup-norm difference of their distribution functions without additional assumptions. No extra regularity or direct argument is supplied. Since (4.1) and (4.2) together imply Theorem 2, this is a load-bearing gap in the proof; a separate uniform estimate for |F_n - bar(F_n)| is needed.
  4. [Introduction and Section 3.4] The centering of X_n is inconsistent across the manuscript. The Abstract and Sections 2 and 3 define X_n = sqrt(4 gamma_n)(R_n - sqrt(n) - (1/2) sqrt(gamma_n)), whereas the Introduction and the computation in Section 3.4 use the shift sqrt(gamma_n)/4. These two variables are not asymptotically equivalent on the Gumbel scale: with the 1/4 shift, sqrt(4 gamma_n)(R_n - sqrt(n) - sqrt(gamma_n)/4) diverges in probability to +infty along the Rider scaling. Since Section 3.4's transfer estimate is computed with the 1/4 shift, the proof of Theorems 1 and 2 is not for the X_n stated in those theorems.
minor comments (3)
  1. [Section 1, notation] The definition 't_n = O(z_n) if lim t_n/z_n = c != 0' is nonstandard and conflicts with the usual meaning of O; elsewhere in the paper O is used with its standard interpretation, which is likely to confuse readers.
  2. [Lemma 2.2] The precise asymptotic in the second part of Lemma 2.2 should state explicitly that the error is uniform in k and x in the indicated ranges, since this uniformity is used implicitly in the summations in Section 3.2 and in the proof of (3.9).
  3. [Throughout] The symbol tilde(O) is defined as 'limit exists' in the introduction but is later used to indicate a bounded relative error with possible logarithmic factors; the paper should adopt a single, clearly defined convention for tilde(O).

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the rate derivation is self-contained and uses only external prior results (Kostlan representation, Rider Gumbel limit, Petrov/CGS tail bounds) with no fitted parameters or self-citations.

full rationale

The paper's central claim is an asymptotic rate for W1 and Kolmogorov distance; it is derived by direct estimates. The Gumbel limit and the scaling gamma_n are taken from Rider [32], an external prior result, not from the authors' own work. The radial representation R_n^2 = Y_(n) is imported from Kostlan [26], and the Gaussian tail expansion in Lemma 2.2 is imported from Petrov [31] and Chen-Goldstein-Shao [14]; these are published benchmarks rather than outputs of this paper, and no parameter is fitted to force the stated constants 2 and 2/e. The proof proceeds by bounding the integral of |P(Y_(n) <= an + bn x) - exp(-e^{-x})| using Lemma 2.3 summation formulas and separate estimates of the three ranges; the final constants emerge from the choice gamma_n = log n - 2 log(sqrt(2 pi) log n) and elementary integrals, not from any equation that is equivalent to the theorem by definition. There are no self-citations by the authors and no uniqueness theorem is invoked. The possible defect flagged by a reviewer in Section 3.4 concerns the validity of a substitution in the transfer estimate W1(L(Wn), Fn) <= gamma_n^{-1}; if real, that is a mathematical error in the proof, not a circularity. The correctness risk does not affect the circularity score. Therefore no significant circularity is present.

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

No fitted constants or invented entities are introduced. The only hand-chosen quantities are asymptotic cutoffs such as ell_1, ell_2, j_n, m_i(n), p_n and t_n; their values do not enter the final constants, and any choice satisfying the stated asymptotic conditions yields the same limits, so they are not free parameters.

assumptions (5)
  • standard math Kostlan representation: R_n^2 has the same distribution as the maximum of independent Gamma(k,1) variables Y_k for k=1..n.
    Invoked as Lemma 2.1 via Corollary 1.2 of [26]; it is the gateway that converts the matrix problem into a one-dimensional extreme value problem.
  • standard math Mills ratio tail expansion 1 - Phi(t) = exp(-t^2/2) / (sqrt(2 pi) t) (1 + O(t^{-2})).
    Used throughout Lemma 2.2 and Section 3 to evaluate Gamma tail probabilities.
  • domain assumption Non-uniform Berry-Esseen bounds for sums of i.i.d. exponentials (Petrov [31], Chen-Goldstein-Shao [14]) apply with the stated uniform errors O(n^{-1/2} u_n^3) and O(n^{-1/2} u_n^{-3}).
    The uniformity of these error terms over 0 <= k << n and x in the chosen intervals is load-bearing; the paper does not re-derive it.
  • standard math The integral expansion (2.5): integral_z^infty t^{-m} exp(-c t^2) dt = exp(-c z^2) / (2 c z^{m+1}) (1 - (m+1)/(2c) z^{-2} + O(z^{-4})).
    Used in Lemma 2.3 to sum the Gamma tail bounds over k; this is a standard integration-by-parts expansion.
  • domain assumption Rider's limit theorem [32] supplies the correct centering and scaling for the Gumbel limit of the spectral radius.
    The paper targets this reference normalization and uses it to identify the Gumbel law; it is external prior work, not self-cited.

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Pith. "Pith review of Exact convergence rate of spectral radius of complex Ginibre to Gumbel distribution." pith.science (2026). https://pith.science/paper/7V4IW4FE

@misc{pith2026250108039,
  author       = {Pith},
  title        = {Pith review of: Exact convergence rate of spectral radius of complex Ginibre to Gumbel distribution},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7V4IW4FE}},
  note         = {Machine review of arXiv:2501.08039}
}
abstract

Consider the complex Ginibre ensemble, whose eigenvalues are $(\lambda_i)_{1\le i\le n}$ and the spectral radius $R_n=\max_{1\le i\le n}|\lambda_i|.$ Set $X_n=\sqrt{4 \gamma_{n}}(R_{n}-\sqrt{n}-\frac12\sqrt{\gamma_{n}})$ and $F_n$ be its distribution function, where $\gamma_{n}=\log n-2\log(\sqrt{2\pi}\log n).$ It was proved in \cite{Rider 2003} that $F_n$ converges weakly to the Gumbel distribution $\Lambda.$ We prove in further in this paper that $$\lim_{n\to\infty} \frac{\log n}{\log\log n}\, W_1\left(F_n, \Lambda\right)=2$$ and the Berry-Esseen bound $$\lim\limits_{n\to \infty} \frac{\log n}{\log\log n}\sup_{x\in \mathbb{R}}|F_{n}(x)-e^{-e^{-x}}|=\frac{2}{e}.$$

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Forward citations

Cited by 2 Pith papers

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    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.

  2. Convergence rate of extreme eigenvalue of Ginibre ensembles to Gumbel distribution

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    The Kolmogorov distance from the rightmost Ginibre eigenvalue to Gumbel is exactly 25 log log n/(4e log n) and the W1 distance is exactly 25 log log n/(4 log n), with analogous rates for the spectral radius.

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Works this paper leans on

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

  1. [1]

    Akemann and M.J

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

    work page 2014

  2. [2]

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

    work page 1997

  3. [3]

    Z. D. Bai and Y. Q. Yin. Limiting behavior of the norm of products o f random matrices and two problems of Geman-Hwang. Probab. Th. Relat. Fields , 73(1986), 555-569

    work page 1986

  4. [4]

    Biely and S

    C. Biely and S. Thurner. Random Matrix Ensembles of Time-Lagged Correlation Ma- trices: Derivation of Eigenvalue Spectra and Analysis of Financial Tim e-Series. Quant. Financ., 8(7)(2008), 705-722

    work page 2008

  5. [5]

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

    work page 2010

  6. [6]

    Bordenave and D

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

    work page 2012

  7. [7]

    Bordenave, P

    C. Bordenave, P. Caputo, D. Chafa¨ ı and K. Tikhomirov. On the spectral radius of a random matrix: An upper bound without fourth moment. Ann. Probab., 46(2018), 2268-2286

    work page 2018

  8. [8]

    Bordenave, D

    C. Bordenave, D. Chafa¨ ı and D. Garcia-Zelada. Convergence of the spectral radius of a random matrix through its characteristic polynomial. Probab. Th. Relat. Fields , 182(2022), 1163-1181

    work page 2022

Show all 39 references
  1. [9]

    Bourgade, G

    P. Bourgade, G. Cipolloni and J. Y. Huang. Fluctuations for non- Hermitian dynamics. arXiv:2409.02902v1

  2. [10]

    Byun and P

    S.-S. Byun and P. J. Forrester. Progress on the Study of the Ginibre Ensembles . 1st ed., Springer Singapore, 2025

    2025

  3. [11]

    Byun and P

    S.-S. Byun and P. J. Forrester. Progress on the Study of the Ginibre Ensembles II: GinOE and GinSE . arXiv.2301.05022

    arXiv

  4. [12]

    D. Chafai. Around the circular law: an update. (Webblog) https://djalil.chafai.net/blog/2018/11/04/around-the-circular-law-an-update

    2018

  5. [13]

    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

  6. [14]

    L. H. Y. Chen, L. Goldstein and Q.-M. Shao. Normal Approximation by Stein’s Method . 1st ed, Springer Berlin Heidelberg, 2011

    2011

  7. [15]

    Cipolloni, L

    G. Cipolloni, L. Erd¨ os and D. Schr¨oder. Central Limit Theorem for Linear Eigenvalue Statistics of non-Hermitian Random Matrices. Commun. Pure Appl. Math. , 76(5)(2023), 946-1034

    2023

  8. [16]

    Cipolloni, L

    G. Cipolloni, L. Erd¨ os and D. Schr¨ oder. Mesoscopic Central Limit Theorem for non- Hermitian Random Matrices. Probab. Th. Relat. Fields , 188(5)(2024), 1131-1182

    2024

  9. [17]

    Cipolloni, L

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

  10. [18]

    Cipolloni, L

    G. Cipolloni, L. Erd¨os, D. Schr¨oder and Y. Xu. Directional extremal statistics for Ginibre eigenvalues. J. Math. Phys. , 63(10)(2022), 103303. 18 YUTAO MA AND XUJIA MENG

    2022

  11. [19]

    A. Edelman. The probability that a random real Gaussian matrix h as k real eigenvalues, related distributions, and the circular law. J. Multi. Anal. , 60(1997), 203-232

    1997

  12. [20]

    Forrester

    P. Forrester. Fluctuation formula for complex random matrice s. J. Physics A: Math. Gen., 32(1999), 159-163

    1999

  13. [21]

    S. Geman. The spectral radius of large random matrices. Ann. Probab. , 14(1986), 1318- 1328

    1986

  14. [22]

    J. Ginibre. Statistical Ensembles of Complex, Quaternion, and R eal Matrices. J. Math. Physics, 6 (1965), 440-449

    1965

  15. [23]

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

    1984

  16. [24]

    I. S. Gradstein, I. M. Ryzhik and A. Jeffrey. Table of Integrals , Series, and Products. 6th ed., Academic Press, San Diego, 2000

    2000

  17. [25]

    N. Karoui. A rate of convergence result for the largest eigenv alue of complex white Wishart matrices. Ann. Probab., 34(2006)(6): 2077-2117 (2006)

    2006

  18. [26]

    E. Kostlan. On the spectra of Gaussian matrices. Lin. Alg. Appl. , 162(1992), 385-388

    1992

  19. [27]

    M. L. Mehta. Random Matrices and the Statistical Theory of Energy Levels . Academic Press, New York-London, 1967

    1967

  20. [28]

    E. S. Meckes and M. W. Meckes. A rate of convergence for the circular law for the complex Ginibre ensemble, AFST: Math. , 24(1)(2015), 93-117

    2015

  21. [29]

    M. Lovric. International Encyclopedia of Statistical Science . 1st ed., Springer-Verlag Berlin Heidelberg, 2011

    2011

  22. [30]

    V. M. Panaretos and Y. Zemel. Statistical Aspects of Wassers tein Distances. Annu. Rev. Stat. Appl. , 6(2019), 405-431

    2019

  23. [31]

    V. V. Petrov. Sums of independent random variables. Springer -Verlag Berlin Heidelberg New York, 1975

    1975

  24. [32]

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

    2003

  25. [33]

    Rider and J

    B. Rider and J. W. Silverstein. Gaussian Fluctuations for non-He rmitian random matrix ensembles. Ann. Probab., 34(6)(2006), 2118-2143

    2006

  26. [34]

    Rider and C

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

    2014

  27. [35]

    Rider and B

    B.C. Rider and B. Virag. The noise in the circular law and the Gaussia n free field. Int. Math. Res. Not. , 2007(2007), rnm006

    2007

  28. [36]

    Schnelli and Y

    K. Schnelli and Y. Xu. Quantitative Tracy-Widom laws for the larg est eigenvalue of generalized Wigner matrices. Electron. J. Probab., 28(2023), 1-38

    2023

  29. [37]

    Tao and V

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

    2008

  30. [38]

    Tao and V

    T. Tao and V. Vu. Random matrices: Universality of local spectr al statistics of non- Hermitian matrices. Ann. Probab., 43(2)(2015), 782-874

    2015

  31. [39]

    C. Villani. Optimal transport: old and new . Springer-Verlag Berlin Heidelberg, 2000. Yutao MA, School of Mathematical Sciences & Laboratory of Mathematics and Complex Systems of Ministry of Education, Beijing Normal Un iversity, 100875 Bei- jing, China. Email address : mayt@b...

    2000

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