Abstract
In this paper we consider \(N \times N\) real generalized Wigner matrices whose entries are only assumed to have finite \((2 + \varepsilon )\)th moment for some fixed, but arbitrarily small, \(\varepsilon > 0\). We show that the Stieltjes transforms \(m_N (z)\) of these matrices satisfy a weak local semicircle law on the nearly smallest possible scale, when \(\eta = \mathfrak {I}(z)\) is almost of order \(N^{-1}\). As a consequence, we establish bulk universality for local spectral statistics of these matrices at fixed energy levels, both in terms of eigenvalue gap distributions and correlation functions, meaning that these statistics converge to those of the Gaussian orthogonal ensemble in the large N limit.
This is a preview of subscription content, access via your institution.
References
Ajanki, O.H., Erdős, L., Krüger, T.: Local spectral statistics of Gaussian matrices with correlated entries. J. Stat. Phys. 163, 280–302 (2016)
Anderson, G., Guionnet, A., Zeitouni, O.: An Introduction to Random Matrices, Studies in Advanced Mathematics 118. Cambridge University Press, Cambridge (2009)
Auffinger, A., Arous, G.Ben, Péché, S.: Poisson convergence for the largest eigenvalues of heavy-tailed random matrices. Ann. Inst. Henri Poincaré Probab. Stat. 45, 589–610 (2009)
Bauerschmidt, R., Huang, J., Yau, H.-T.: Local Kesten–McKay law for random regular graphs. Preprint arXiv:1609.09052
Bauerschmidt, R., Knowles, A., Yau, H.-T.: Local semicircle law for random regular graphs. Commun. Pure Appl. Math. 70, 1898–1960 (2017)
Bauerschmidt, R., Huang, J., Knowles, A., Yau, H.-T.: Bulk eigenvalue statistics for regular random graphs. Ann. Probab. 45, 3626–3663 (2017)
Arous, G.Ben, Guionnet, A.: The spectrum of heavy tailed random matrices. Commun. Math. Phys. 278, 715–751 (2008)
Benaych-Georges, F., Guionnet, A., Male, C.: Central limit theorem for linear statistics of heavy tailed random matrices. Commun. Math. Phys. 329, 641–686 (2014)
Biane, P.: On the free convolution with a semi-circular distribution. Indiana Univ. Math. J. 46, 705–718 (1997)
Bordenave, C., Guionnet, A.: Delocalization at small energy for heavy-tailed random matrices. Commun. Math. Phys. 354, 115–159 (2017)
Bordenave, C., Guionnet, A.: Localization and delocalization of eigenvectors for heavy-tailed random matrices. Probab. Theory Relat. Fields 157, 885–953 (2013)
Bouchaud, J., Cizeau, P.: Theory of Lévy matrices. Phys. Rev. E 3, 1810–1822 (1994)
Bourgade, P., Yau, H.-T.: The eigenvector moment flow and local quantum unique ergodicity. Commun. Math. Phys. 350, 231–278 (2017)
Bourgade, P., Erdős, L., Yau, H.-T., Yin, J.: Fixed energy universality for generalized Wigner matrices. Commun. Pure Appl. Math. 69, 1815–1881 (2016)
Cacciapuoti, C., Maltsev, A., Schlein, B.: Bounds for the Stieltjes transform and the density of states of Wigner matrices. Probab. Theory Relat. Fields 163, 1–59 (2015)
Che, Z.: Universality of random matrices with correlated entries. Electron. J. Probab. 22(30), 1–38 (2017)
Erdős, L.: Universality of Wigner random matrices: a survey of recent results. Russ. Math. Surv. 66, 527–626 (2011)
Erdős, L., Knowles, A., Yau, H.-T., Yin, J.: Spectral statistics of Erdős–Rényi graphs I: local semicircle law. Ann. Probab. 41, 2279–2375 (2013)
Erdős, L., Knowles, A., Yau, H.-T., Yin, J.: Spectral statistics of Erdős–Rényi graphs II: eigenvalue spacing and the extreme eigenvalues. Commun. Math. Phys. 314, 587–640 (2012)
Erdős, L., Knowles, A., Yau, H.-T., Yin, J.: The local semicircle law for a general class of random matrices. Electron. J. Probab. 18, 49 (2013)
Erdős, L., Péché, S., Ramírez, J.A., Schlein, B., Yau, H.-T.: Bulk universality for Wigner matrices. Commun. Pure Appl. Math. 63, 895–925 (2010)
Erdős, L., Ramírez, J.A., Schlein, B., Tao, T., Vu, V., Yau, H.-T.: Bulk universality for Wigner Hermitian matrices with subexponential decay. Math. Res. Lett. 63, 667–674 (2010)
Erdős, L., Schlein, B., Yau, H.-T.: Universality of random matrices and local relaxation flow. Invent. Math. 185, 75–119 (2011)
Erdős, L., Schlein, B., Yau, H.-T.: Wegner estimate and level repulsion for Wigner random matrices. Int. Math. Res. Not. 436–479, 2010 (2010)
Erdős, L., Schlein, B., Yau, H.-T., Yin, J.: The local relaxation flow approach to universality of the local statistics for random matrices. Ann. Henri Poincaré Probab. Stat. 48, 1–46 (2012)
Erdős, L., Schnelli, K.: Universality for random matrix flows with time-dependent density. Ann. Henri Poincaré Probab. Stat. 53, 1606–1656 (2017)
Erdős, L., Yau, H.-T.: A comment on the Wigner–Dyson–Mehta bulk universality conjecture for Wigner matrices. Electron. J. Probab. 17, 28 (2012)
Erdős, L., Yau, H.-T.: Local semicircle law and complete delocalization for Wigner random matrices. Commun. Math. Phys. 287, 641–655 (2009)
Erdős, L., Yau, H.-T.: Dynamical Approach to Random Matrix Theory. American Mathematical Society, Providence (2017)
Erdős, L., Yau, H.-T., Yin, J.: Bulk universality for generalized Wigner matrices. Probab. Theory Relat. Fields 154, 341–407 (2012)
Götze, F., Naumov, A., Tikhomirov, A.: Local semicircle law under moment conditions. Part I: the Stieltjes transform. Preprint arXiv:1510.0735
Götze, F., Naumov, A., Timushev, D., Tikhomirov, A.: On the local semicircular law for Wigner ensembles. Preprint arXiv:1602.03073
Götze, F., Tikhomirov, A.: Optimal bounds for convergence of expected spectral distributions to the semi-circular law. Probab. Theory Relat. Fields 165, 163–233 (2003)
Huang, J., Landon, B.: Spectral statistics of sparse Erdős–Rényi graph Laplacians. Ann. Henri Poincaré Probab. Stat. (to appear). arXiv:1510.06390
Huang, J., Landon, B., Yau, H.-T.: Bulk universality of sparse matrices. J. Math. Phys. 56, 123301 (2015)
Johansson, K.: Universality for certain Hermitian Wigner matrices under weak moment conditions. Ann. Henri Poincaré Probab. Stat. 48, 47–79 (2012)
Johansson, K.: Universality of the local spacing distribution in certain ensembles of Hermitian Wigner matrices. Commun. Math. Phys. 215, 683–705 (2001)
Landon, B., Yau, H.-T.: Convergence of local statistics of Dyson Brownian motion. Commun. Math. Phys. 355, 949–1000 (2017)
Landon, B., Sosoe, P., Yau, H.-T.: Fixed energy universality of Dyson Brownian motion. Preprint arXiv:1609.09011
Lee, J. O., Schnelli, K.: Local law and Tracy–Widom limit for sparse random matrices. Probab. Theory Relat. Fields (to appear). https://arxiv.org/abs/1605.08767
Lee, J.O., Yin, J.: A necessary and sufficient condition for edge universality of Wigner matrices. Duke Math. J. 163, 117–173 (2014)
Mehta, M.L.: Random Matrices. Academic Press, New York (1991)
Mehta, M.L., Gaudin, M.: On the density of eigenvalues of a random matrix. Nucl. Phys. B 18, 420–427 (1960)
Tao, T., Vu, V.: Random covariance matrices: universality of local statistics. Ann. Probab. 40, 1285–1315 (2012)
Tao, T., Vu, V.: Random matrices: localization of the eigenvalues and the necessity of four moments. Acta Math. Vietnam. 36, 431–449 (2011)
Tao, T., Vu, V.: Random matrices: the four moment theorem for Wigner ensembles. Preprint arXiv:1112.1976
Tao, T., Vu, V.: Random matrices: universality of local eigenvalue statistics. Acta Math. 206, 127–204 (2011)
Tao, T., Vu, V.: The Wigner–Dyson–Mehta bulk universality conjecture for Wigner matrices. Electron. J. Probab. 16, 2104–2121 (2011)
Tarquini, E., Biroli, G., Tarzia, M.: Level statistics and localization transitions of Lévy matrices. Phys. Rev. Lett. 116, 010601 (2016)
Wigner, E.P.: Characteristic vectors of bordered matrices with infinite dimensions. Ann. Math. 62, 548–564 (1955)
Acknowledgements
The author heartily thanks Horng-Tzer Yau for suggesting this question and for numerous valuable discussions. The author is also very grateful to Jiaoyang Huang and Benjamin Landon for many fruitful conversations and helpful explanations, and to Ji Oon Lee for several insightful comments.
Author information
Authors and Affiliations
Corresponding author
Additional information
This work was funded by the NSF Graduate Research Fellowship under Grant Number DGE1144152 and partially by the Eric Cooper and Naomi Siegel Graduate Student Fellowship I.
Appendices
Large deviation estimates
In this section we state the large deviation results that were useful to us in Sects. 4, 5, and 6. The following proposition appears as the first part of Lemma 3.8 of [18].
Proposition 24
([18, Lemma 3.8]) Let \(N \in {\mathbb {Z}}_{> 1}\) and \(q = q_N > 1\). Let \(X_1, X_2, \ldots , X_N\); \(Y_1, Y_2, \ldots , Y_N\) be centered random variables such that there exists a \(C > 0\) satisfying
for all \(2 \le p \le (\log N)^{\log \log N}\). Denote \(s_i = {\mathbb {E}} [|X_i|^2]\) for each \(1 \le i \le N\).
Let \(\{ R_i \}_{1 \le i \le N}\) and \(\{ R_{ij} \}_{1 \le i, j \le N}\) be sequences of real numbers. Then, there exists a constant \(\nu = \nu (C) > 0\), only dependent on the constant C in (138), such that the four estimates
all hold for any \(2 \le \xi \le \log \log N\).
In our setting, the random variables \(X_i\) are not centered but instead “almost centered.” The following corollary adapts the previous proposition to apply in this slightly more setting.
Corollary 25
Let \(N \in {\mathbb {Z}}_{> 1}\) and \(q = q_N \in (1, \sqrt{N})\). Let \(X_1, X_2, \ldots , X_N; Y_1, Y_2, \ldots , Y_N\) be random variables such that there exist \(\delta \in (0, 1)\) and \(C, C' > 1\) satisfying
for each \(2 \le p \le (\log N)^{\log \log N}\). Denote \(s_i = {\mathbb {E}} [|X_i|^2]\) for each \(1 \le i \le N\).
Let \(\{ R_i \}_{1 \le i \le N}\) and \(\{ R_{ij} \}_{1 \le i, j \le N}\) be sequences of real numbers. There exists a constant \(\nu = \nu ( C, C' ) > 0\) (dependent on only C and \(C'\)) such that
for all \(2 \le \xi \le \log \log N\).
Proof
The proof will follow from centering the \(X_j\) and then applying Proposition 24. To that end, let \(m_j = {\mathbb {E}} [X_j]\), for each \(j \in [1, N]\); by (143), we have that \(|m_j| \le C' N^{-1 - \delta }\).
Denote \(\widetilde{X}_i = X_i - m_j\). Then, the \(\widetilde{X}_i\) are centered and satisfy
for any \(2 \le p \le (\log N)^{\log \log N}\); in the first estimate, we used (143), and in the last estimate we used the fact that \(q, N > 1\).
Thus, Proposition 24 applies to the centered random variables \(\widetilde{X}_i\); the estimates (144), (145), (146), and (147) will follow from (139), (140), (141), and (142) respectively.
As an example, we only establish (146); the proofs of the other estimates are very similar and thus omitted. To that end, we apply (139) and (141) to obtain that
for each \(i \in [1, N]\), and
where we set \(\widetilde{\nu } = \widetilde{\nu } (C, C') = \widetilde{\nu } \big ( 2 (C + C') \big )\).
Furthermore, observe that
where we have used the fact that \(|m_i| < C' N^{-1 - \delta }\) for each \(i \in [1, N]\). Thus, applying (148) for all \(i \in [1, N]\), (149), (150), and a union estimate yields
where
In the second estimate above, we used the facts that \(q, C', N \ge 1\). Now select \(\nu = \nu (C, C') = \widetilde{\nu } / 2\), so that \((2 N + 1) \exp \big ( - \widetilde{\nu } (\log N)^{\xi } \big ) \le \exp \big ( - \nu (\log N)^{\xi } \big )\). Then, (146) follows from (149). \(\square \)
Continuity of \(G_{ij} (z)\)
In this section we establish continuity estimates on the entries of the resolvent, which allowed us to proceed with the multiscale argument in Sects. 5 and 6.
Lemma 26
Fix \(E \in {\mathbb {R}}\); \(\eta , \eta ' \in {\mathbb {R}}_{> 0}\); \(N \in {\mathbb {Z}}_{> 0}\); and an \(N \times N\) matrix \({\mathbf{H}}\). Denote \(z = E + {\mathbf{i}} \eta \), \(z' = E + {\mathbf{i}} (\eta + \eta ')\), \(G (z) = ({\mathbf{H}} - z)^{-1} = \{ G_{jk} \}\), and \(G' (z) = ( {\mathbf{H}} - z')^{-1} = \{ G_{jk}' \}\).
Fixing \(j, k \in [1, N]\), we have that
Proof
Applying the resolvent identity (13) with \(A = H - z' {{\mathrm{Id}}}\) and \(B = H - z {{\mathrm{Id}}}\), and comparing (j, k)-entries, we find that
Thus,
where the third identity was deduced from Ward’s identity (17); this implies the lemma.
Corollary 27
Adopt the notation of Lemma 26. Then,
Proof
Let \(a = |G_{jj} (e + \mathbf i \eta + \mathbf i \eta ')|\) and \(b = |G_{jj} (e + \mathbf i \eta )|\), and assume that \(a \ge b\); the case \(b > a\) is entirely analogous. Lemma 26 applied with \(j = k\) yields \(a - b < (a + b) \eta ' / 2 \eta \). Therefore,
from which we deduce the corollary. \(\square \)
Rights and permissions
About this article
Cite this article
Aggarwal, A. Bulk universality for generalized Wigner matrices with few moments. Probab. Theory Relat. Fields 173, 375–432 (2019). https://doi.org/10.1007/s00440-018-0836-y
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00440-018-0836-y
Keywords
- Random matrix
- Local semicircle law
- Universality
Mathematics Subject Classification
- 15B52
- 60B20