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Local semicircle law under fourth moment condition. (English) Zbl 1444.60013

Summary: We consider a random symmetric matrix \(\mathbf{X} = [X_{jk}]_{j, k = 1}^n\) with upper triangular entries being independent random variables with mean zero and unit variance. Assuming that \(\max_{jk} \mathbb{E} |X_{jk}|^{4 + \delta} < \infty, \delta > 0\), it was proved in [the first author et al., Bernoulli 24, No. 3, 2358–2400 (2018; Zbl 1429.60010)] that with high probability the typical distance between the Stieltjes transforms \(m_n(z), z = u + i v\), of the empirical spectral distribution (ESD) and the Stieltjes transforms \(m_{\text{sc}}(z)\) of the semicircle law is of order \((nv)^{-1} \log n\). The aim of this paper is to remove \(\delta > 0\) and show that this result still holds if we assume that \(\max_{jk} \mathbb{E} |X_{jk}|^4 < \infty\). We also discuss applications to the rate of convergence of the ESD to the semicircle law in the Kolmogorov distance, rates of localization of the eigenvalues around the classical positions and rates of delocalization of eigenvectors.

MSC:

60B20 Random matrices (probabilistic aspects)
60F05 Central limit and other weak theorems

Citations:

Zbl 1429.60010
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References:

[1] Aggarwal, A., Bulk universality for generalized Wigner matrices with few moments, Probab. Theory Relat. Fields, 173, 1-2, 375-432 (2019) · Zbl 1442.15055
[2] Anderson, G.; Guionnet, A.; Zeitouni, O., An Introduction to Random Matrices, Volume 118 of Cambridge Studies in Advanced Mathematics (2010), Cambridge: Cambridge University Press, Cambridge · Zbl 1184.15023
[3] Bai, Z.; Silverstein, J., Spectral Analysis of Large Dimensional Random Matrices (2010), New York: Springer, New York · Zbl 1301.60002
[4] 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, 1-59 (2015) · Zbl 1330.60014
[5] 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, 3, 587-640 (2012) · Zbl 1251.05162
[6] Erdős, L.; Knowles, A.; Yau, H-T; Yin, J., The local semicircle law for a general class of random matrices, Electron. J. Probab., 18, 59, 58 (2013) · Zbl 1373.15053
[7] Erdős, L.; Knowles, A.; Yau, H-T; Yin, J., Spectral statistics of Erdős-Rényi graphs I: Local semicircle law, Ann. Probab., 41, 3, 2279-2375 (2013) · Zbl 1272.05111
[8] Erdős, L.; Schlein, B.; Yau, H-T, Local semicircle law and complete delocalization for Wigner random matrices, Commun. Math. Phys., 287, 2, 641-655 (2009) · Zbl 1186.60005
[9] Erdős, L.; Schlein, B.; Yau, H-T, Semicircle law on short scales and delocalization of eigenvectors for Wigner random matrices, Ann. Probab., 37, 3, 815-852 (2009) · Zbl 1175.15028
[10] Erdős, L.; Schlein, B.; Yau, H-T, Wegner estimate and level repulsion for Wigner random matrices, Int. Math. Res. Not. IMRN, 3, 436-479 (2010) · Zbl 1204.15043
[11] Erdős, L.; Yau, H-T, A Dynamical Approach to Random Matrix Theory. Courant Lecture Notes (2017), Providence: AMS, Providence · Zbl 1379.15003
[12] Erdős, L.; Yau, H-T; Yin, J., Rigidity of eigenvalues of generalized Wigner matrices, Adv. Math., 229, 3, 1435-1515 (2012) · Zbl 1238.15017
[13] Götze, F., Naumov, A., Tikhomirov, A.: Local semicircle law under moment condtions. Part I: the Stieltjes transfrom. arXiv:1510.07350 (2015)
[14] Götze, F., Naumov, A., Tikhomirov, A.: Local semicircle law under moment condtions. Part II: localization and delocalization. arXiv:1511.00862 (2015)
[15] Götze, F.; Naumov, A.; Tikhomirov, A., Local semicircle law under moment conditions: the Stieltjes transform, rigidity, and delocalization, Theory Probab. Appl., 62, 1, 58-83 (2018) · Zbl 1426.15056
[16] Götze, F., Naumov, A., Tikhomirov, A.: On local laws for non-Hermitian random matrices and their products. arXiv:1708.06950 (2017) · Zbl 1392.60012
[17] Götze, F.; Naumov, A.; Tikhomirov, A.; Timushev, A., On the local semicircle law for Wigner ensembles, Bernoulli, 24, 3, 2358-2400 (2018) · Zbl 1429.60010
[18] Götze, F.; Tikhomirov, A., Optimal bounds for convergence of expected spectral distributions to the semi-circular law, Probab. Theory Relat. Fields, 165, 1-2, 163-233 (2016) · Zbl 1338.60014
[19] Gustavsson, J., Gaussian fluctuations of eigenvalues in the GUE, Ann. Inst. H. Poincaré Probab. Statist., 41, 2, 151-178 (2005) · Zbl 1073.60020
[20] Lee, J.; Yin, J., A necessary and sufficient condition for edge universality of Wigner matrices, Duke Math. J., 163, 1, 117-173 (2014) · Zbl 1296.60007
[21] Rosenthal, H., On the subspaces of \(L^p(p>2)\) spanned by sequences of independent random variables, Israel J. Math., 8, 273-303 (1970) · Zbl 0213.19303
[22] Rudelson, M.; Vershynin, R., Hanson-Wright inequality and sub-Gaussian concentration, Electron. Commun. Probab., 18, 82, 1-9 (2013) · Zbl 1329.60056
[23] Tao, T., Topics in Random Matrix Theory, Volume 132 of Graduate Studies in Mathematics (2012), Providence: American Mathematical Society, Providence · Zbl 1256.15020
[24] Tao, T., Vu, V.: Random matrices: the universality phenomenon for Wigner ensembles. arXiv:1202.0068 · Zbl 1310.15077
[25] Tao, T.; Vu, V., Random matrices: universality of local eigenvalue statistics, Acta Math., 206, 1, 127-204 (2011) · Zbl 1217.15043
[26] Wigner, E., Characteristic vectors of bordered matrices with infinite dimensions, Ann. Math. (2), 62, 548-564 (1955) · Zbl 0067.08403
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