Some estimates for generalized Wigner matrix linear spectral statistics

Benjamin Landon

REVIEW 2 major objections 4 minor 34 references

Some estimates for generalized Wigner matrix linear spectral statistics

T0 review · 2 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read For generalized Wigner matrices, the eigenvalue-statistic law matches a Gaussian to order 1/N, with explicit non-Gaussian corrections.

desk verdict Main theorem as stated is false: the cubic coefficient B(f) has the wrong sign and is off by a factor of 2; the paper's real value is the loop/line machinery, which looks sound. read the letter →

arxiv 2412.14070 v1 pith:TJIJ66VE submitted 2024-12-18 math.PR

Benjamin Landon This is my paper · ORCID

classification math.PR MSC 15B5260B20

keywords generalizedWignermatriceslinearspectralstatisticscharacteristicfunctionexpansionlocalsemicirclelawcumulantStein'smethodmesoscopicCLTlog-characteristicpolynomial

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

The paper proves a sharp expansion for the law of linear spectral statistics of generalized Wigner matrices. For an admissible test function $f$, the characteristic function is shown to equal $\exp(-\lambda^2 V_\beta(f)/2 + i\lambda^3 B(f)/3 + i\lambda E_\beta(f))$ with an error of order $N^{-1}$ (up to factors $(1+|\lambda|)$ and $\|f''\|_{1,w}$), provided the Gaussian variance $V_\beta(f)$ is bounded below, and with a slightly weaker error otherwise. Previously, such $O(N^{-1})$ accuracy was known only for Wigner matrices, whose constant variance profile allows a direct simplification; only a polynomial error rate was available for general variance profiles. The corrections $B(f)$ and $E_\beta(f)$ are explicit: they depend on the low-order cumulants of the matrix entries and the Chebyshev coefficients of $f$, and are of order $N^{-1/2}$. In effect, the paper supplies the missing optimal-rate input needed to extend optimal CLT and extremal-statistics results from Wigner to generalized Wigner ensembles.

What carries the argument

The argument is carried by an iterated cumulant expansion applied to monomials of resolvent entries, organized graphically as 'loops' and 'lines'. A loop is a product $G_{i_1i_2}(z_1)\cdots G_{i_k i_1}(z_k)$ over distinct indices; a line is the analogue with distinct endpoints $G_{i_1i_2}(z_1)\cdots G_{i_k i_{k+1}}(z_k)$. The key estimates (Propositions 3.2 and 4.2) show that the expectation of a loop with a distinguished spectral parameter $z$ gains a factor $\Psi_1(z)=(N|\operatorname{Im}z|)^{-1/2}$ over the naive entry-wise local-law size, and that the expectation of a line gains $N^{-1/2}$, with the gain coming from cancellation in the cumulant expansion. The resolvent entries are packaged with the variance matrix $S=A+N^{-1}ee^{\mathsf T}$, where $A$ has spectral norm at most $1-c$; inverting the resulting self-consistent equation for vectors such as $v_j=E[e_a G_{j2}(z)G_{2j}(w)]$ uses bounds on $(1-m_{sc}(z)m_{sc}(w)(A-B))^{-1}$, the identity $(m_{sc}(z)-m_{sc}(w))/(z-w)=m_{sc}(z)m_{sc}(w)/(1-m_{sc}(z)m_{sc}(w))$, and Sherman–Morrison. The expansion is then fed into the Helffer–Sjostrand representation and Stein's method to derive the differential equation for the characteristic function.

What would settle it

Choose a generalized Wigner matrix with a doubly stochastic two-block variance profile $S$ that is not constant, set $f(x)=x^3$ (or another smooth compactly supported admissible function), and compute $E[e^{i\lambda(\mathrm{LSS}(f))}]$ for $\lambda=1$ at increasing $N$; compare the difference with the right-hand side of Theorem 1.6. If the discrepancy after removing the explicit $N^{-1/2}$ corrections does not decay like $N^{-1}$, the expansion is wrong.

Watch

Extended reading notes

Core claim

The central claim is Theorem 1.6: for a real symmetric or complex Hermitian generalized Wigner matrix $H$ with doubly stochastic variance matrix $S_{ij}=\mathbb E[|H_{ij}|^2]$, and an admissible test function $f$, the characteristic function of the centered linear spectral statistic $\mathrm{LSS}(f)=\mathrm{tr}\,f(H)-N\int f(x)\rho_{sc}(x)\,dx$ obeys $$E[$e^{{i\lambda\mathrm{LSS}}$(f)}]=\exp\left(-\frac{\$lambda^{{2}}$V_{\$\beta$}(f)}2+\frac{i\$lambda^{{3}}$B(f)}3+i\$\lambda$ E_{\$\beta$}(f)\right)+O\left(\|f''\|_{1,w}$N^{{-1}}$(1+|\$\lambda$|)+$N^{{-1}}$|\$\lambda$|^{2}\right)$$ when $V_\beta(f)\geq c>0$ and $|\lambda|\leq N^{-\varepsilon}\sqrt{N\|f''\|_1^{-1}}$; a related estimate holds without the lower-bound condition. The paper's own way of stating the result is that this is an expansion around the Gaussian limit with error $O(N^{-1})$ and an explicit sub-leading non-Gaussian correction $N^{-1/2}P_f(\lambda)$. The functionals $V_\beta$, $E_\beta$ and $B$ are deterministic and explicit in the cumulants $\kappa_k(H_{ij})$ and the Chebyshev coefficients $t_j(f)$ of $f$; $B(f)$ is a cubic term controlled by the third cumulant of the diagonal entries. This is the first such optimal-rate expansion for full linear spectral statistics of generalized Wigner matrices.

Load-bearing premise

The entire expansion rests on the quoted local semicircle law for generalized Wigner matrices holding with sharp entry-wise, isotropic, and fluctuation-averaging rates; if any of those rates degrades for a particular variance profile, the $O(N^{-1})$ error estimate would not follow.

Editorial extensions

If this is right

If Theorem 1.6 is correct, the CLT for full linear spectral statistics of generalized Wigner matrices holds with the same $O(N^{-1})$ characteristic-function accuracy as for Wigner matrices, closing the gap left by polynomial-error estimates.
The explicit expressions for $V_\beta(f)$, $E_\beta(f)$ and $B(f)$ give a computationally usable formula for the leading and sub-leading moments of $\mathrm{LSS}(f)$, including the non-Gaussian cubic correction.
The same expansion applies to mesoscopic test functions that vary on scales $N^{\alpha-1}$, yielding CLT accuracy there rather than only at global scale.
Theorem 9.2 extends the recent Wigner-matrix result [11] on the maximum of the log-characteristic polynomial, and the associated optimal rigidity of extreme eigenvalues, to generalized Wigner matrices.

Reading between the lines

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

Extension (not in the paper): the same loop/line machinery should apply to any ensemble whose local law, fluctuation averaging, and variance matrix with a spectral gap hold at the stated rates—most directly to band matrices and Wigner-type profiles—so the optimal characteristic-function expansion is likely a benchmark for the whole family.
Extension (not in the paper): because $B(f)$ is proportional to the third cumulant of diagonal entries, one could use the formula to engineer variance profiles that suppress or amplify skewness in the limiting fluctuations; the paper does not discuss this design consequence.
Extension (not in the paper): numerically testing the two-point resolvent estimate (Corollary 6.7) on a two-block variance profile would isolate the new technical step; a clean $N^{-1}$ decay there would corroborate the full expansion.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The paper is worth a serious look, but the headline result is not correct as written. Take an admissible f that equals x on [-3,3] and is smoothly cut off to zero. With overwhelming probability all eigenvalues lie in [-3,3], so LSS(f)=tr H exactly. The exact characteristic function factorizes over the diagonal entries, giving a log-cubic term of -iλ³ŝ3/6. Theorem 1.6, with B(f)=ŝ3 t1(f)^3/8 and t1(f)=2, predicts +iλ³ŝ3/3. That is a factor of 2 and a sign error, and it is not absorbed by the error term at, say, λ=N^{-1/8}.

That said, the paper has real substance. The loop estimate (Proposition 3.2) and line estimate (Proposition 4.2) are genuinely new and are the key to closing the expansion at order N^{-1} for non-constant variance profiles. The iterative cumulant-expansion argument is intricate and, as far as I can verify, coherent. The internal derivation actually points to the correct answer: combining (8.19) with Lemma 8.4 gives B_a = -ŝ3 t1^3/16, which reproduces the exact solvable case. So the mistake is localized to the statement of the coefficient in Definition 1.4 and the corresponding line in Lemma 8.7, probably a sign/factor slip in translating a_{1,a}= -t1/2 into the final formula.

The other soft spot is Section 9: the application to the logarithm of the characteristic polynomial goes beyond the compactly supported admissibility condition, and the paper relies on sketched fixes to gaps in [11]. That part is rougher, but it is secondary to the main technical contribution.

Who gets value from this: anyone working on mesoscopic linear spectral statistics, generalized Wigner matrices, or extremal eigenvalue statistics. The loop/line estimates will likely be cited even after the B(f) issue is fixed. This deserves a serious referee, but the paper should not be accepted in its current form. The author needs to correct the cubic coefficient, re-check the deterministic coefficient computations, and tighten the application section. With that done, the core result should be publishable.

Referee Report

2 major / 4 minor

Summary. The paper studies the characteristic function of linear spectral statistics (LSS) for generalized Wigner matrices. It develops a graphical cumulant-expansion framework, proves estimates for loop and line hypergraphs, and derives an expansion of the characteristic function with error O(N^{-1}) around the Gaussian limit, with explicit sub-leading terms of size O(N^{-1/2}) controlled by functionals V_beta(f), E_beta(f), and B(f). The main theorem (Theorem 1.6) states this expansion in two regimes depending on whether V_beta(f) is bounded below. As an application, the paper extends results of Bourgade-Lopatto-Zeitouni on the maximum of the log-characteristic polynomial to generalized Wigner matrices.

Significance. The paper addresses an important open step in random matrix theory: an O(N^{-1}) characteristic-function expansion for LSS of generalized Wigner matrices, previously available only for Wigner matrices. The graphical loop/line estimates and the Stein-method implementation are serious technical advances, and the proposed applications to log-characteristic polynomial maxima are natural. However, the explicit cubic coefficient B(f) in the central theorem is incorrect as stated: it is off by a factor of 2 and has the wrong sign relative to both the paper's own internal computation and an exactly solvable example. This makes Theorem 1.6 false as written, though the error appears localized and correctable. The paper's framework remains potentially valuable once the coefficient is fixed and the consequences re-derived.

major comments (2)

[Definition 1.4 (1.18); Proposition 8.3 (8.19); Lemma 8.4 (8.26); Lemma 8.7 (8.46)] The cubic coefficient B(f) is internally inconsistent and incorrect as stated. Substituting a_{1,a} = -t_1(f)/2 from Lemma 8.4 into B_a(f) = tilde{s}_3/(2 N^{1/2}) (a_{1,a})^3 of (8.19) gives B_a = -tilde{s}_3 t_1(f)^3/(16 N^{1/2}) = -hat{s}_3 t_1(f)^3/16, whereas (8.46) and Definition 1.4 assert +hat{s}_3 t_1(f)^3/8. These formulas differ by a factor of 2 and a sign. This is not a minor typo: in the exactly solvable case f(x)=x on [-3,3] with a real symmetric generalized Wigner matrix having S_ij=1/N and diagonal third cumulants, LSS(f)=tr H with overwhelming probability and the exact log-characteristic function has cubic term -i lambda^3 hat{s}_3/6, so Theorem 1.6 would require B(f) = -hat{s}_3/2. The stated B(f) is +hat{s}_3. At lambda = N^{-1/8} the discrepancy in the exponent is O(N^{-7/8}), which exceeds the claimed O(N^{-1}) error in Theorem 1.6. The derivation in (8.19) appears to yield the correct coefficient, so the error likely lies in the transcription to (8.46) and (1.18); the author should correct these formulas, fix the sign conventions, and re-verify Lemma 8.9 and the bounds in Section 9 that use B(f).
[Section 9, Proposition 9.3] The application of Theorem 1.6 in Proposition 9.3 uses the functions f_z(x) = Re log(z-x) and Im log(z-x), which are not compactly supported and are not admissible under Definition 1.2. The proof says it uses the coefficients V_z, E_z, B_z of Theorem 1.6 for these functions and then applies (9.8), but no truncation or approximation argument is given. Since the spectrum lies in a compact interval with overwhelming probability, a truncation to a fixed interval should be possible, but the needed uniformity in eta >= exp(-K (log log N)^2) is not demonstrated. As written, the passage from Theorem 1.6 to (9.8) is unjustified.

minor comments (4)

[Section 1, Eq. (1.18); Section 8, Eq. (8.46)] After correcting the cubic coefficient, the author should ensure that all displayed formulas for B(f) match the deterministic computation in Section 8.1 and that the sign convention for the third cumulant is stated consistently.
[Section 8, Eq. (8.14)] The relationship between tilde{s}_3 in (8.14) and hat{s}_3 in (1.16) should be made explicit (tilde{s}_3 = N^{1/2} hat{s}_3 up to the conventions for s^{(3)}_{aa}), as the current notation invites confusion in the verification of the coefficient.
[Section 9.1] The proof of Theorem 9.2 relies on the arguments of [11], including the derivation of (6.7) there. The author notes that (6.7) is not fully proven in [11] and asserts that rigidity estimates suffice, but no proof is given in the present paper. This gap should be closed or the dependence made explicit.
[Throughout] There are many small typos and missing differentials (e.g., in (2.20) and (8.18)); a careful proofreading pass is needed.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the characteristic-function expansion is derived from cumulant expansions and explicit coefficient computations, with same-author citations only for auxiliary deterministic identities.

full rationale

The central result, Theorem 1.6, is an O(N^{-1}) expansion of E[e^{iλ LSS(f)}] obtained by repeated cumulant expansions (Lemma 2.6), resolvent estimates built on the external local semicircle law (Theorem 2.4, quoted from [2,3,9]), and Stein's method in Section 8. The coefficients V_β(f), E_β(f), and B(f) are evaluated in Sections 7 and 8 from explicit formulas involving entry cumulants and Chebyshev coefficients; they are computed, not fitted or imposed. The paper's reliance on prior work by the same author ([26], [27], [25]) is confined to deterministic semicircle-law integrals, the identity (2.19), and an application-side comparison theorem. These are parameter-free auxiliary identities that do not assume the target expansion, so they do not make the derivation circular. The one limitation passage — in Section 9.1, noting an incomplete proof in [11] for a deterministic upper bound — concerns an application to the maximum of the log-characteristic polynomial, not the main theorem. A skeptical concern about a possible factor-of-two error in B(f) would be a correctness issue, not a circularity issue, and does not change this verdict.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No free parameters are fitted; all constants in the expansion are explicit. The main external inputs are the local semicircle law and the cumulant expansion lemma. The application section introduces an additional tail assumption.

assumptions (4)

domain assumption Local semicircle law for generalized Wigner matrices, Theorem 2.4 (from [2,3,9])
Quoted, not proved; supplies entry-wise and isotropic resolvent estimates and fluctuation averaging used throughout Sections 3-8.
standard math Cumulant expansion lemma, Lemma 2.6 (from [28])
Standard cumulant expansion for centered random variables; used to move derivatives off resolvent products.
domain assumption Moment bounds (1.7) and doubly stochastic variance profile (1.6) in Definition 1.1
Defines the model class; the spectral gap property of S (Lemma 2.8) follows from double stochasticity.
domain assumption Subexponential tail condition, Assumption 9.1
Only needed for the application in Section 9, not for the main CLT theorem.

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Pith. "Pith review of Some estimates for generalized Wigner matrix linear spectral statistics." pith.science (2026). https://pith.science/paper/TJIJ66VE

@misc{pith2026241214070,
  author       = {Pith},
  title        = {Pith review of: Some estimates for generalized Wigner matrix linear spectral statistics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TJIJ66VE}},
  note         = {Machine review of arXiv:2412.14070}
}

abstract

We consider the characteristic function of linear spectral statistics of generalized Wigner matrices. We provide an expansion of the characteristic function with error $\mathcal{O} ( N^{-1})$ around its limiting Gaussian form, and identify sub-leading non-Gaussian corrections of size $\mathcal{O} (N^{-1/2})$. Prior expansions with this error rate held only for Wigner matrices; only a weaker error rate was available for more general matrix ensembles. We provide some applications.

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Works this paper leans on

34 extracted references · 29 canonical work pages

[11]

Bourgade, P

P. Bourgade, P. Lopatto, and O. Zeitouni. Optimal rigid ity and maximum of the char- acteristic polynomial of Wigner matrices. arXiv preprint arXiv:2312.13335 , 2023

arXiv 2023
[1]

Adhikari, I

K. Adhikari, I. Jana, and K. Saha. Linear eigenvalue stat istics of random matrices with a variance proﬁle. Random Matrices: Theory and Applications , 10(03):2250004, 2021

work page 2021
[2]

O. H. Ajanki, L. Erd˝ os, and T. Kr¨ uger. Universality for general Wigner-type matrices. Probability Theory and Related Fields , 169:667–727, 2017

work page 2017
[3]

J. Alt, L. Erd˝ os, T. Kr¨ uger, and D. Schr¨ oder. Correlated random matrices: band rigidity and edge universality. 2020

work page 2020
[4]

G. W. Anderson and O. Zeitouni. A CLT for a band matrix mode l. Probab. Theory Related Fields, 134(2):283–338, 2006

work page 2006
[5]

Z. Bai, X. Wang, and W. Zhou. CLT for linear spectral stati stics of Wigner matrices. Electron. J. Probab., 14:2391–2417, 2009

work page 2009
[6]

Bai and J

Z. Bai and J. Yao. On the convergence of the spectral empir ical process of Wigner matrices. Bernoulli, 11(6):1059–1092, 2005

work page 2005
[7]

Baik and J

J. Baik and J. O. Lee. Fluctuations of the free energy of th e spherical Sherrington– Kirkpatrick model. Journal of Statistical Physics , 165:185–224, 2016

work page 2016

Show all 34 references

[8]

Bao and Y

Z. Bao and Y. He. Quantitative CLT for linear eigenvalue s tatistics of Wigner matrices. The Annals of Applied Probability , 33(6B):5171–5207, 2023. 80

2023
[9]

Bloemendal, L

A. Bloemendal, L. Erd˝ os, A. Knowles, H.-T. Yau, and J. Yi n. Isotropic local laws for sample covariance and generalized Wigner matrices. 2014

2014
[10]

Bourgade

P. Bourgade. Extreme gaps between eigenvalues of Wigne r matrices. J. Eur. Math. Soc., to appear, 2021

2021
[12]

Chatterjee

S. Chatterjee. Fluctuations of eigenvalues and second order Poincar´ e inequalities.Prob- ability Theory and Related Fields , 143(1):1–40, 2009

2009
[13]

M. Diaz, J. A. Mingo, and S. T. Belinschi. On the global ﬂu ctuations of block Gaussian matrices. Probability Theory and Related Fields , 176(1):599–648, 2020

2020
[14]

Erd˝ os, A

L. Erd˝ os, A. Knowles, H.-T. Yau, and J. Yin. The local se micircle law for a general class of random matrices. 2013

2013
[15]

L. Erd˝ os. The matrix Dyson equation and its applicatio ns for random matrices. arXiv preprint arXiv:1903.10060, 2019

1903 arXiv
[16]

Erd˝ os and A

L. Erd˝ os and A. Knowles. The Altshuler–Shklovskii for mulas for random band matrices I: the unimodular case. Communications in Mathematical Physics , 333(3):1365–1416, 2015

2015
[17]

Erd˝ os and A

L. Erd˝ os and A. Knowles. The Altshuler–Shklovskii for mulas for random band matrices II: the general case. In Annales Henri Poincar´ e, volume 16, pages 709–799. Springer, 2015

2015
[18]

Erd˝ os, A

L. Erd˝ os, A. Knowles, H.-T. Yau, and J. Yin. Delocaliza tion and diﬀusion proﬁle for random band matrices. Communications in Mathematical Physics , 323:367–416, 2013

2013
[19]

Erd˝ os and H.-T

L. Erd˝ os and H.-T. Yau. A dynamical approach to random matrix theory , volume 28. American Mathematical Soc., 2017

2017
[20]

Erd˝ os, H.-T

L. Erd˝ os, H.-T. Yau, and J. Yin. Rigidity of eigenvalue s of generalized Wigner matrices. Advances in Mathematics , 229(3):1435–1515, 2012

2012
[21]

Guionnet

A. Guionnet. Large deviations upper bounds and central limit theorems for non- commutative functionals of Gaussian large random matrices . Ann. Inst. Henri Poincar´ e Probab. Stat., 38(3):341–384, 2002

2002
[22]

I. Jana, K. Saha, and A. Soshnikov. Fluctuations of line ar eigenvalue statistics of random band matrices. Theory of Probability & Its Applications , 60(3):407–443, 2016

2016
[23]

Khorunzhy, B

A. Khorunzhy, B. Khoruzhenko, and L. Pastur. On the 1/n c orrections to the green func- tions of random matrices with independent entries. Journal of Physics A: Mathematical and General, 28(1):L31, 1995

1995
[24]

A. M. Khorunzhy, B. A. Khoruzhenko, and L. A. Pastur. Asy mptotic properties of large random matrices with independent entries. Journal of Mathematical Physics , 37(10):5033–5060, 1996. 81

1996
[25]

Landon, P

B. Landon, P. Lopatto, and J. Marcinek. Comparison theo rem for some extremal eigen- value statistics. The Annals of Probability , 48(6):2894–2919, 2020

2020
[26]

Landon and P

B. Landon and P. Sosoe. Applications of mesoscopic CLTs in random matrix theory. Ann. Appl. Probab. , 30(6):2769–2795, 2020

2020
[27]

Landon and P

B. Landon and P. Sosoe. Almost-optimal bulk regularity conditions in the CLT for Wigner matrices. arXiv preprint arXiv:2204.03419 , 2022

2022 arXiv
[28]

J. O. Lee and K. Schnelli. Local law and Tracy–Widom limi t for sparse random matrices. Probab. Theory Related Fields , 171(1):543–616, 2018

2018
[29]

Li and A

L. Li and A. Soshnikov. Central limit theorem for linear statistics of eigenvalues of band random matrices. arXiv preprint arXiv:1304.6744 , 2013

2013 arXiv
[30]

Li and Y

Y. Li and Y. Xu. On ﬂuctuations of global and mesoscopic l inear statistics of generalized Wigner matrices. Bernoulli, 2(27):1057–1076, 2021

2021
[31]

Lodhia and N

A. Lodhia and N. Simm. Mesoscopic linear statistics of W igner matrices. arXiv preprint arXiv:1503.03533, 2015

2015 arXiv
[32]

Lytova and L

A. Lytova and L. Pastur. Central limit theorem for linea r eigenvalue statistics of random matrices with independent entries. Ann. Probab., 37(5):1778–1840, 2009

2009
[33]

Shcherbina

M. Shcherbina. Central limit theorem for linear eigenv alue statistics of the Wigner and sample covariance random matrices. J. Math. Phys., Analysis, Geometry , (7):176–192, 2011

2011
[34]

Shcherbina

M. Shcherbina. On ﬂuctuations of eigenvalues of random band matrices. Journal of Statistical Physics , 161:73–90, 2015. 82

2015

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [11]

Bourgade, P

P. Bourgade, P. Lopatto, and O. Zeitouni. Optimal rigid ity and maximum of the char- acteristic polynomial of Wigner matrices. arXiv preprint arXiv:2312.13335 , 2023

arXiv 2023

[2] [1]

Adhikari, I

K. Adhikari, I. Jana, and K. Saha. Linear eigenvalue stat istics of random matrices with a variance proﬁle. Random Matrices: Theory and Applications , 10(03):2250004, 2021

work page 2021

[3] [2]

O. H. Ajanki, L. Erd˝ os, and T. Kr¨ uger. Universality for general Wigner-type matrices. Probability Theory and Related Fields , 169:667–727, 2017

work page 2017

[4] [3]

J. Alt, L. Erd˝ os, T. Kr¨ uger, and D. Schr¨ oder. Correlated random matrices: band rigidity and edge universality. 2020

work page 2020

[5] [4]

G. W. Anderson and O. Zeitouni. A CLT for a band matrix mode l. Probab. Theory Related Fields, 134(2):283–338, 2006

work page 2006

[6] [5]

Z. Bai, X. Wang, and W. Zhou. CLT for linear spectral stati stics of Wigner matrices. Electron. J. Probab., 14:2391–2417, 2009

work page 2009

[7] [6]

Bai and J

Z. Bai and J. Yao. On the convergence of the spectral empir ical process of Wigner matrices. Bernoulli, 11(6):1059–1092, 2005

work page 2005

[8] [7]

Baik and J

J. Baik and J. O. Lee. Fluctuations of the free energy of th e spherical Sherrington– Kirkpatrick model. Journal of Statistical Physics , 165:185–224, 2016

work page 2016