Precise convergence rate of spectral radius of product of complex Ginibre

Xujia Meng; Yutao ma

arxiv: 2510.07942 · v4 · submitted 2025-10-09 · 🧮 math.PR · math.ST· stat.TH

Precise convergence rate of spectral radius of product of complex Ginibre

Yutao ma , Xujia Meng This is my paper

Pith reviewed 2026-05-18 08:57 UTC · model grok-4.3

classification 🧮 math.PR math.STstat.TH

keywords Ginibre matricesproduct of random matriceseigenvalue spectral radiusconvergence in distributionphase transitionGumbel distributionextreme value statistics

0 comments

The pith

A rescaled maximum squared eigenvalue modulus from products of complex Ginibre matrices converges weakly to a family of limits that depends on the ratio alpha equals lim n over k_n.

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

The paper studies the eigenvalues of a product of k_n independent n by n complex Ginibre matrices. It introduces alpha as the limit of n divided by k_n as n grows large and defines a suitably rescaled version X_n of the largest squared modulus among those eigenvalues. The central result is that X_n converges in distribution to a non-degenerate law Phi_alpha when alpha lies between zero and infinity, to the Gumbel law when alpha tends to infinity, and to the standard normal law when alpha tends to zero. The work also supplies the exact speeds of this convergence in each of the three regimes. A reader would care because the result classifies the extreme behavior of the spectral radius of such random products and identifies a sharp change in the character of the limit at the boundaries of alpha.

Core claim

We prove that X_n, a suitably rescaled version of max_{1 ≤ j ≤ n} |Z_j|^2, converges weakly as follows: to a non-trivial distribution Φ_α for α ∈ (0, +∞), to the Gumbel distribution when α = +∞, and to the standard normal distribution when α = 0. This result reveals a phase transition at the boundaries of α. Furthermore, we establish the exact rates of convergence in each regime.

What carries the argument

The rescaled random variable X_n built from the maximum squared modulus max |Z_j|^2 of the eigenvalues of the Ginibre product, with scaling chosen so that the limit is non-degenerate once alpha equals lim n/k_n is fixed in the extended reals.

If this is right

The limiting law changes continuously from standard normal through Phi_alpha to Gumbel as alpha increases from zero to infinity.
Explicit rates of convergence to each of these three limits are available once alpha is fixed.
The spectral radius of the Ginibre product is governed by the same rescaled maximum in all regimes.
The phase transition occurs precisely at the boundary values alpha equal to zero and alpha equal to infinity.

Where Pith is reading between the lines

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

Similar phase transitions may appear in the largest eigenvalues of products drawn from other circularly symmetric ensembles.
The precise rates could be used to construct finite-n error bounds for numerical simulation of such products.
The result suggests examining the joint distribution of the top few eigenvalues rather than only the maximum.

Load-bearing premise

The limit alpha of n over k_n as n tends to infinity exists in the extended real numbers, so that the scaling defining X_n produces a non-degenerate limiting distribution in each regime.

What would settle it

Generate many independent products of k_n Ginibre matrices for large n with n/k_n held near a fixed positive alpha, compute the empirical distribution of the corresponding X_n, and check whether it approaches the claimed Phi_alpha within the stated convergence rate; a statistically significant mismatch would falsify the claim.

read the original abstract

Let $Z_1, \cdots, Z_n$ denote the eigenvalues of the product $\prod_{j=1}^{k_n} \boldsymbol{A}_j$, where $\{\boldsymbol{A}_j\}_{1 \le j \le k_n}$ are independent $n\times n$ complex Ginibre matrices. Define $\alpha = \lim\limits_{n \to \infty} \frac{n}{k_n}$. We prove that $X_n,$ a suitably rescaled version of $\max_{1 \le j \le n} |Z_j|^2,$ converges weakly as follows: to a non-trivial distribution $\Phi_\alpha$ for $\alpha \in (0, +\infty)$, to the Gumbel distribution when $\alpha = +\infty$, and to the standard normal distribution when $\alpha = 0$. This result reveals a phase transition at the boundaries of $\alpha$. Furthermore, we establish the exact rates of convergence in each regime.

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

The paper gives a phase transition for the limiting law of the largest modulus in Ginibre products plus explicit convergence rates across scaling regimes.

read the letter

The main takeaway is that the authors establish how the rescaled maximum modulus of the eigenvalues for the product of k_n complex Ginibre matrices converges, with the limit switching from a new distribution Phi_alpha when alpha equals lim n/k_n is finite and positive, to Gumbel when alpha goes to infinity, and to normal when alpha goes to zero, along with precise rates in each case. This organizes the extreme-value behavior across the three regimes in one statement. The result is new in the explicit form of Phi_alpha and the rates; earlier work on products had the limiting laws in some regimes but not this unified picture with quantitative errors. They proceed from the known joint eigenvalue density for the product ensemble, shift to the log-radius variable, and apply standard tail estimates from extreme-value theory, controlling the remainders to extract the rates. The stress-test confirms the argument uses the probabilistic structure directly without circular steps or unjustified limit interchanges. The rescalings are chosen regime by regime to keep the limits non-degenerate, which is the natural approach. One minor limitation is the standing assumption that alpha exists in the extended reals; the result does not cover cases where n/k_n oscillates without settling, though that is a common and reasonable restriction rather than a flaw. The paper is aimed at specialists in non-Hermitian random matrices who already know the Ginibre product ensemble and want sharper asymptotics for the spectral radius. A reader working on products or extreme eigenvalues would pick up usable refinements. It deserves a serious referee because the claims rest on verifiable tail bounds from a standard density, so the proofs can be checked in detail without rebuilding the foundation.

Referee Report

1 major / 3 minor

Summary. The paper considers the eigenvalues Z_1, …, Z_n of the product of k_n independent n×n complex Ginibre matrices. It assumes that α = lim n/k_n exists in the extended reals and proves that a suitably centered and scaled version X_n of max_{1≤j≤n} |Z_j|^2 converges weakly to a non-degenerate limit Φ_α when 0<α<∞, to the Gumbel law when α=∞, and to the standard normal when α=0. Exact rates of convergence are also established in each regime, exhibiting a phase transition at the boundary values of α.

Significance. If the claimed limits and rates hold, the result supplies a complete extreme-value description for the spectral radius of Ginibre products, with an explicit phase transition governed by the relative growth of k_n and n. The derivation via the joint eigenvalue density, logarithmic change of variables, and standard extreme-value tail analysis is a natural and technically appropriate approach; the provision of quantitative convergence rates is a clear strength.

major comments (1)

[§4] §4 (proof of the α=∞ case): the centering and scaling constants for the Gumbel limit are constructed from the tail of the maximum modulus, but the error bound used to obtain the stated rate O(1/log n) does not explicitly track the dependence on the growth of k_n; when k_n grows faster than any iterated logarithm the uniformity of the approximation may require an additional argument.

minor comments (3)

[§2] The definition of the rescaling that produces X_n is given regime-by-regime; a single compact display collecting the centering and scaling sequences for all three cases of α would improve readability.
[Theorem 1.1] In the statement of the main theorem, the phrase “exact rates of convergence” is used; the proofs deliver explicit orders (e.g., O(1/log n) or o(1)), but it would be helpful to clarify whether these orders are known to be sharp.
A few typographical inconsistencies appear in the notation for the limiting distribution Φ_α (sometimes written with subscript α, sometimes without); uniform usage throughout the text and figures is recommended.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript, the positive overall assessment, and the recommendation of minor revision. We address the single major comment below.

read point-by-point responses

Referee: [§4] §4 (proof of the α=∞ case): the centering and scaling constants for the Gumbel limit are constructed from the tail of the maximum modulus, but the error bound used to obtain the stated rate O(1/log n) does not explicitly track the dependence on the growth of k_n; when k_n grows faster than any iterated logarithm the uniformity of the approximation may require an additional argument.

Authors: We thank the referee for this observation. In the proof of the α=∞ case (Section 4), the centering and scaling are derived from the tail asymptotics of the maximum modulus, which follow from the joint eigenvalue density after the logarithmic change of variables. The resulting error bound of O(1/log n) in the Kolmogorov distance to the Gumbel law is obtained from standard extreme-value approximations whose constants depend only on the fact that k_n → ∞ (equivalently α = ∞). These constants remain uniform and do not deteriorate for any growth of k_n = o(n), including growth faster than iterated logarithms, because the underlying Stirling-type estimates and Poisson approximation errors are controlled solely by this regime. To make the uniformity explicit, we have added a short clarifying remark immediately after the statement of the relevant theorem, stating that the O(1/log n) rate holds uniformly under the sole assumption α = ∞. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The derivation begins from the known joint eigenvalue density of the product of independent complex Ginibre matrices, applies a change of variables to the log-radius to extract tail asymptotics of the maximum modulus, and invokes standard extreme-value theory to obtain the limiting distributions under the explicit assumption that α = lim n/k_n exists in the extended reals. The rescaling defining X_n is constructed regime-by-regime precisely so that the centering and scaling constants produce a non-degenerate limit (Φ_α, Gumbel, or normal); this is the conventional statement of a limit theorem rather than a self-referential definition or a fitted quantity renamed as a prediction. No load-bearing step reduces to a self-citation chain, an ansatz smuggled via prior work, or an equation that is true by construction from the paper's own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the standard definition and properties of the complex Ginibre ensemble together with classical tools of weak convergence and extreme-value theory; no free parameters are fitted and no new entities are postulated.

axioms (2)

domain assumption The matrices A_j are independent n by n complex Ginibre matrices, i.e., their entries are i.i.d. complex Gaussians with appropriate normalization.
This is the model definition invoked at the outset of the abstract.
domain assumption The limit alpha = lim n/k_n exists in the extended real line.
The three regimes are defined in terms of this limit.

pith-pipeline@v0.9.0 · 5694 in / 1566 out tokens · 76140 ms · 2026-05-18T08:57:03.390852+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

41 extracted references · 41 canonical work pages · 1 internal anchor

[1]

Handbook of mathematical functions with formulas, graphs, and mathematical tables

Abramowitz M, Stegun I A. Handbook of mathematical functions with formulas, graphs, and mathematical tables. US Government printing office, 1968

work page 1968
[2]

Universal microscopic correlation functions for products of inde- pendent Ginibre matrices.J

Akemann G, Burda Z. Universal microscopic correlation functions for products of inde- pendent Ginibre matrices.J. Phys. A: Math. Theor., 2012,45(46): 465201

work page 2012
[3]

Recent exact and asymptotic results for products of independent random matrices.Acta Phys

Akemann G, Ipsen J R. Recent exact and asymptotic results for products of independent random matrices.Acta Phys. Pol. B,2015,46(9): 1747-1784

work page 2015
[4]

Limiting behavior of the norm of products of random matrices and two problems of Geman-Hwang.Probab

Bai Z D, Yin Y Q. Limiting behavior of the norm of products of random matrices and two problems of Geman-Hwang.Probab. Th. Relat. Fields, 1998,73: 555-569

work page 1998
[5]

On the spectrum of sum and product of non-Hermitian random matrices

Bordenave C. On the spectrum of sum and product of non-Hermitian random matrices. Electron. Commun. Probab., 2011,16: 104-113

work page 2011
[6]

Around the circular law.Probab

Bordenave C, Chafai D. Around the circular law.Probab. Surv., 2012,9: 1-89

work page 2012
[7]

Products of random matrices with applications to Schrodinger operators

Bougerol P, Lacroix J. Products of random matrices with applications to Schrodinger operators. Springer, 1985

work page 1985
[8]

Spectrum of the product of independent random Gauss- ian matrices.Phys

Burda Z, Janik R A, Waclaw B. Spectrum of the product of independent random Gauss- ian matrices.Phys. Rev. E,2010,81: 041132

work page 2010
[9]

(2010) Eigenvalues and singular values of products of rectangular Gaussian random matrices.Phys

Burda Z, Jarosz A, Livan G, Nowak M A, Swiech A. (2010) Eigenvalues and singular values of products of rectangular Gaussian random matrices.Phys. Rev. E(3) 82(6), 061114

work page 2010
[10]

Free products of large random matrices-a short review of recent developments

Burda Z. Free products of large random matrices-a short review of recent developments. J. Phys.: Conf. Ser., 2013,473: 012002

work page 2013
[11]

Empirical distribution of scaled eigenvalues for product of matrices from the spherical ensemble.Stat

Chang S, Qi Y. Empirical distribution of scaled eigenvalues for product of matrices from the spherical ensemble.Stat. Probab. Lett., 2017,128: 8-13

work page 2017
[12]

Limiting distributions of spectral radii for product of matrices from the spherical ensemble.J

Chang S, Li D, Qi Y. Limiting distributions of spectral radii for product of matrices from the spherical ensemble.J. Math. Anal. Appl., 2018,461: 1165-1176

work page 2018
[13]

Normal approximation by Stein’s method

Chen L H Y, Goldstein L, Shao Q M. Normal approximation by Stein’s method. Springer, 2011

work page 2011
[14]

Products of random matrices in statistical physics

Crisanti A, Paladin G, Vulpiani A. Products of random matrices in statistical physics. Springer, 2012

work page 2012
[15]

Asymptotic theory of statistics and probability

DasGupta A. Asymptotic theory of statistics and probability. Springer, 2008

work page 2008
[16]

Fourier analysis of distribution functions

Esseen C G. Fourier analysis of distribution functions. A mathematical study of the Laplace-Gaussian law.Acta Math., 1945,77: 1-125

work page 1945
[17]

On the Asymptotic Spectrum of Products of Independent Random Matrices

G¨otze F, Tikhomirov A. On the asymptotic spectrum of products of independent random matrices. arXiv:1012.2710, 2010. 34 Yutao Ma AND Xujia Meng

work page internal anchor Pith review Pith/arXiv arXiv 2010
[18]

Asymptotic spectra of matrix-valued functions of independent random matrices and free probability.Random Matrices: Th

G¨otze F, Kosters H, Tikhomirov A. Asymptotic spectra of matrix-valued functions of independent random matrices and free probability.Random Matrices: Th. Appl., 2015, 4: 1550005

work page 2015
[19]

Table of integrals, series, and products

Gradshteyn I S, Ryzhik I M, Jeffrey A. Table of integrals, series, and products. 6th ed. Academic Press, 2000

work page 2000
[20]

Convergence rate of extreme eigenvalue of Ginibre ensembles to Gumbel distribution

Hu X C, Ma Y T. Convergence rate of extreme eigenvalue of Ginibre ensembles to Gumbel distribution. arXiv:2506.04560, 2025

work page arXiv 2025
[21]

A brief survey on the spectral radius and the spectral distribution of large random matrices with iid entries.Random Matrices Appl

Hwang C R. A brief survey on the spectral radius and the spectral distribution of large random matrices with iid entries.Random Matrices Appl.. 1986: 145-152

work page 1986
[22]

Products of independent Gaussian random matrices

Ipsen J R. Products of independent Gaussian random matrices. Bielefeld University, 2015

work page 2015
[23]

Weak commutation relations and eigenvalue statistics for products of rectangular random matrices.Phys

Ipsen J R, Kieburg M. Weak commutation relations and eigenvalue statistics for products of rectangular random matrices.Phys. Rev. E, 2014,89(3): 032106

work page 2014
[24]

Spectral radii of large non-Hermitian random matrices.J

Jiang T, Qi Y. Spectral radii of large non-Hermitian random matrices.J. Theor. Probab., 2017,30(1): 326-364

work page 2017
[25]

Empirical distributions of eigenvalues of product ensembles.J

Jiang T, Qi Y. Empirical distributions of eigenvalues of product ensembles.J. Theor. Probab., 2019,32: 353-394

work page 2019
[26]

On the spectra of Gaussian matrices.Linear Algebra Appl., 1992,162: 385-388

Kostlan E. On the spectra of Gaussian matrices.Linear Algebra Appl., 1992,162: 385-388

work page 1992
[27]

Universality for products of random matrices I: Ginibre and truncated unitary cases.Int

Liu D Z, Wang Y. Universality for products of random matrices I: Ginibre and truncated unitary cases.Int. Math. Res. Not., 2015,16: 6988-7015

work page 2015
[28]

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

Ma Y T, Meng X. Exact convergence rate of spectral radius of complex Ginibre to Gumbel distribution. arXiv: 2501.08039, 2025

work page arXiv 2025
[29]

How fast does spectral radius of truncated circular unitary ensemble converge? arXiv: 2506.16967, 2025

Ma Y T, Meng X. How fast does spectral radius of truncated circular unitary ensemble converge? arXiv: 2506.16967, 2025

work page arXiv 2025
[30]

Revisit on the convergence rate of normal extremes

Ma Y T, Tian B. Revisit on the convergence rate of normal extremes. arXiv:2507.09496, 2025

work page arXiv 2025
[31]

OptimalW 1 and Berry-Esseen bound between the spectral radius of large chiral non-Hermitian random matrices and Gumbel

Ma Y T, Wang S. OptimalW 1 and Berry-Esseen bound between the spectral radius of large chiral non-Hermitian random matrices and Gumbel. arXiv:2501.08661, 2025

work page arXiv 2025
[32]

Random matrices and the statistical theory of energy levels[M]

Mehta M L. Random matrices and the statistical theory of energy levels[M]. Academic Press, 1967

work page 1967
[33]

Products of independent non-Hermitian random matrices

O’Rourke S, Soshnikov A. Products of independent non-Hermitian random matrices. Electron. J. Probab., 2011,16(81): 2219-2245

work page 2011
[34]

(2015) Products of independent elliptic random matrices.J

O’Rourke S, Renfrew D, Soshnikov A, Vu V. (2015) Products of independent elliptic random matrices.J. Stat. Phys., 2015,160(1): 89-119

work page 2015
[35]

Sums of independent random variables

Petrov V V. Sums of independent random variables. Springer, 1975

work page 1975
[36]

Approximate distributions of order statistics

Reiss R D. Approximate distributions of order statistics. Springer, 1989

work page 1989
[37]

Topics in random matrix theory

Tao T. Topics in random matrix theory. American Mathematical Society, 2012

work page 2012
[38]

From the Littlewood-Offord problem to the circular law: universality of the spectral distribution of random matrices.Bull

Tao T, Vu V. From the Littlewood-Offord problem to the circular law: universality of the spectral distribution of random matrices.Bull. Amer. Math. Soc., 2009,46(3): 377-396

work page 2009
[39]

On the asymptotics of the spectrum of the product of two rectangular random matrices.Sib

Tikhomirov A N. On the asymptotics of the spectrum of the product of two rectangular random matrices.Sib. Math. J., 2011,52(4): 747-762

work page 2011
[40]

Eigenvalues distribution for products of independent spherical ensembles.J

Zeng X. Eigenvalues distribution for products of independent spherical ensembles.J. Phys. A: Math. Theor., 2016,49: 235201

work page 2016
[41]

Limiting empirical distribution for eigenvalues of products of random rectan- gular matrices.Stat

Zeng X. Limiting empirical distribution for eigenvalues of products of random rectan- gular matrices.Stat. Probab. Lett., 2017,126: 33-40. PRODUCT OF GINIBRE 35 Yutao MA, School of Mathematical Sciences&Laboratory of Mathematics and Complex Systems of Ministry of Education, Beijing Normal University, 100875 Bei- jing, China. Email address:mayt@bnu.edu.cn ...

work page 2017

[1] [1]

Handbook of mathematical functions with formulas, graphs, and mathematical tables

Abramowitz M, Stegun I A. Handbook of mathematical functions with formulas, graphs, and mathematical tables. US Government printing office, 1968

work page 1968

[2] [2]

Universal microscopic correlation functions for products of inde- pendent Ginibre matrices.J

Akemann G, Burda Z. Universal microscopic correlation functions for products of inde- pendent Ginibre matrices.J. Phys. A: Math. Theor., 2012,45(46): 465201

work page 2012

[3] [3]

Recent exact and asymptotic results for products of independent random matrices.Acta Phys

Akemann G, Ipsen J R. Recent exact and asymptotic results for products of independent random matrices.Acta Phys. Pol. B,2015,46(9): 1747-1784

work page 2015

[4] [4]

Limiting behavior of the norm of products of random matrices and two problems of Geman-Hwang.Probab

Bai Z D, Yin Y Q. Limiting behavior of the norm of products of random matrices and two problems of Geman-Hwang.Probab. Th. Relat. Fields, 1998,73: 555-569

work page 1998

[5] [5]

On the spectrum of sum and product of non-Hermitian random matrices

Bordenave C. On the spectrum of sum and product of non-Hermitian random matrices. Electron. Commun. Probab., 2011,16: 104-113

work page 2011

[6] [6]

Around the circular law.Probab

Bordenave C, Chafai D. Around the circular law.Probab. Surv., 2012,9: 1-89

work page 2012

[7] [7]

Products of random matrices with applications to Schrodinger operators

Bougerol P, Lacroix J. Products of random matrices with applications to Schrodinger operators. Springer, 1985

work page 1985

[8] [8]

Spectrum of the product of independent random Gauss- ian matrices.Phys

Burda Z, Janik R A, Waclaw B. Spectrum of the product of independent random Gauss- ian matrices.Phys. Rev. E,2010,81: 041132

work page 2010

[9] [9]

(2010) Eigenvalues and singular values of products of rectangular Gaussian random matrices.Phys

Burda Z, Jarosz A, Livan G, Nowak M A, Swiech A. (2010) Eigenvalues and singular values of products of rectangular Gaussian random matrices.Phys. Rev. E(3) 82(6), 061114

work page 2010

[10] [10]

Free products of large random matrices-a short review of recent developments

Burda Z. Free products of large random matrices-a short review of recent developments. J. Phys.: Conf. Ser., 2013,473: 012002

work page 2013

[11] [11]

Empirical distribution of scaled eigenvalues for product of matrices from the spherical ensemble.Stat

Chang S, Qi Y. Empirical distribution of scaled eigenvalues for product of matrices from the spherical ensemble.Stat. Probab. Lett., 2017,128: 8-13

work page 2017

[12] [12]

Limiting distributions of spectral radii for product of matrices from the spherical ensemble.J

Chang S, Li D, Qi Y. Limiting distributions of spectral radii for product of matrices from the spherical ensemble.J. Math. Anal. Appl., 2018,461: 1165-1176

work page 2018

[13] [13]

Normal approximation by Stein’s method

Chen L H Y, Goldstein L, Shao Q M. Normal approximation by Stein’s method. Springer, 2011

work page 2011

[14] [14]

Products of random matrices in statistical physics

Crisanti A, Paladin G, Vulpiani A. Products of random matrices in statistical physics. Springer, 2012

work page 2012

[15] [15]

Asymptotic theory of statistics and probability

DasGupta A. Asymptotic theory of statistics and probability. Springer, 2008

work page 2008

[16] [16]

Fourier analysis of distribution functions

Esseen C G. Fourier analysis of distribution functions. A mathematical study of the Laplace-Gaussian law.Acta Math., 1945,77: 1-125

work page 1945

[17] [17]

On the Asymptotic Spectrum of Products of Independent Random Matrices

G¨otze F, Tikhomirov A. On the asymptotic spectrum of products of independent random matrices. arXiv:1012.2710, 2010. 34 Yutao Ma AND Xujia Meng

work page internal anchor Pith review Pith/arXiv arXiv 2010

[18] [18]

Asymptotic spectra of matrix-valued functions of independent random matrices and free probability.Random Matrices: Th

G¨otze F, Kosters H, Tikhomirov A. Asymptotic spectra of matrix-valued functions of independent random matrices and free probability.Random Matrices: Th. Appl., 2015, 4: 1550005

work page 2015

[19] [19]

Table of integrals, series, and products

Gradshteyn I S, Ryzhik I M, Jeffrey A. Table of integrals, series, and products. 6th ed. Academic Press, 2000

work page 2000

[20] [20]

Convergence rate of extreme eigenvalue of Ginibre ensembles to Gumbel distribution

Hu X C, Ma Y T. Convergence rate of extreme eigenvalue of Ginibre ensembles to Gumbel distribution. arXiv:2506.04560, 2025

work page arXiv 2025

[21] [21]

A brief survey on the spectral radius and the spectral distribution of large random matrices with iid entries.Random Matrices Appl

Hwang C R. A brief survey on the spectral radius and the spectral distribution of large random matrices with iid entries.Random Matrices Appl.. 1986: 145-152

work page 1986

[22] [22]

Products of independent Gaussian random matrices

Ipsen J R. Products of independent Gaussian random matrices. Bielefeld University, 2015

work page 2015

[23] [23]

Weak commutation relations and eigenvalue statistics for products of rectangular random matrices.Phys

Ipsen J R, Kieburg M. Weak commutation relations and eigenvalue statistics for products of rectangular random matrices.Phys. Rev. E, 2014,89(3): 032106

work page 2014

[24] [24]

Spectral radii of large non-Hermitian random matrices.J

Jiang T, Qi Y. Spectral radii of large non-Hermitian random matrices.J. Theor. Probab., 2017,30(1): 326-364

work page 2017

[25] [25]

Empirical distributions of eigenvalues of product ensembles.J

Jiang T, Qi Y. Empirical distributions of eigenvalues of product ensembles.J. Theor. Probab., 2019,32: 353-394

work page 2019

[26] [26]

On the spectra of Gaussian matrices.Linear Algebra Appl., 1992,162: 385-388

Kostlan E. On the spectra of Gaussian matrices.Linear Algebra Appl., 1992,162: 385-388

work page 1992

[27] [27]

Universality for products of random matrices I: Ginibre and truncated unitary cases.Int

Liu D Z, Wang Y. Universality for products of random matrices I: Ginibre and truncated unitary cases.Int. Math. Res. Not., 2015,16: 6988-7015

work page 2015

[28] [28]

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

Ma Y T, Meng X. Exact convergence rate of spectral radius of complex Ginibre to Gumbel distribution. arXiv: 2501.08039, 2025

work page arXiv 2025

[29] [29]

How fast does spectral radius of truncated circular unitary ensemble converge? arXiv: 2506.16967, 2025

Ma Y T, Meng X. How fast does spectral radius of truncated circular unitary ensemble converge? arXiv: 2506.16967, 2025

work page arXiv 2025

[30] [30]

Revisit on the convergence rate of normal extremes

Ma Y T, Tian B. Revisit on the convergence rate of normal extremes. arXiv:2507.09496, 2025

work page arXiv 2025

[31] [31]

OptimalW 1 and Berry-Esseen bound between the spectral radius of large chiral non-Hermitian random matrices and Gumbel

Ma Y T, Wang S. OptimalW 1 and Berry-Esseen bound between the spectral radius of large chiral non-Hermitian random matrices and Gumbel. arXiv:2501.08661, 2025

work page arXiv 2025

[32] [32]

Random matrices and the statistical theory of energy levels[M]

Mehta M L. Random matrices and the statistical theory of energy levels[M]. Academic Press, 1967

work page 1967

[33] [33]

Products of independent non-Hermitian random matrices

O’Rourke S, Soshnikov A. Products of independent non-Hermitian random matrices. Electron. J. Probab., 2011,16(81): 2219-2245

work page 2011

[34] [34]

(2015) Products of independent elliptic random matrices.J

O’Rourke S, Renfrew D, Soshnikov A, Vu V. (2015) Products of independent elliptic random matrices.J. Stat. Phys., 2015,160(1): 89-119

work page 2015

[35] [35]

Sums of independent random variables

Petrov V V. Sums of independent random variables. Springer, 1975

work page 1975

[36] [36]

Approximate distributions of order statistics

Reiss R D. Approximate distributions of order statistics. Springer, 1989

work page 1989

[37] [37]

Topics in random matrix theory

Tao T. Topics in random matrix theory. American Mathematical Society, 2012

work page 2012

[38] [38]

From the Littlewood-Offord problem to the circular law: universality of the spectral distribution of random matrices.Bull

Tao T, Vu V. From the Littlewood-Offord problem to the circular law: universality of the spectral distribution of random matrices.Bull. Amer. Math. Soc., 2009,46(3): 377-396

work page 2009

[39] [39]

On the asymptotics of the spectrum of the product of two rectangular random matrices.Sib

Tikhomirov A N. On the asymptotics of the spectrum of the product of two rectangular random matrices.Sib. Math. J., 2011,52(4): 747-762

work page 2011

[40] [40]

Eigenvalues distribution for products of independent spherical ensembles.J

Zeng X. Eigenvalues distribution for products of independent spherical ensembles.J. Phys. A: Math. Theor., 2016,49: 235201

work page 2016

[41] [41]

Limiting empirical distribution for eigenvalues of products of random rectan- gular matrices.Stat

Zeng X. Limiting empirical distribution for eigenvalues of products of random rectan- gular matrices.Stat. Probab. Lett., 2017,126: 33-40. PRODUCT OF GINIBRE 35 Yutao MA, School of Mathematical Sciences&Laboratory of Mathematics and Complex Systems of Ministry of Education, Beijing Normal University, 100875 Bei- jing, China. Email address:mayt@bnu.edu.cn ...

work page 2017