Local semicircle law under moment conditions: Stieltjes transform, rigidity and delocalization

F. Götze   ; A. A. Naumov   ; A. N. Tikhomirov

Teoriâ veroâtnostej i ee primeneniâ, Tome 62 (2017) no. 1, pp. 72-103

Résumé

We consider a random symmetric matrix ${X} = [X_{jk}]_{j,k=1}^n$ where the upper triangular entries are independent identically distributed random variables with zero mean and unit variance. We additionally suppose that ${{E}} |X_{11}|^{4 + \delta} =: \mu_{4+\delta} < \infty$ for some $\delta > 0$. Under these conditions we show that the typical distance between the Stieltjes transform of the empirical spectral distribution (ESD) of the matrix $n^{-1/2} X$ and Wigner's semicircle law is of order $(nv)^{-1}$, where $v$ is the distance in the complex plane to the real line. Furthermore, we outline applications such as the rate of convergence of the ESD to the distribution function of the semicircle law, rigidity of the eigenvalues, and eigenvector delocalization.

Détail

Mots-clés : random matrices, Stieltjes transform.
Keywords: local semicircle law

Bibliographie

[1] L. Arnold, “On the asymptotic distribution of the eigenvalues of random matrices”, J. Math. Anal. Appl., 20:2 (1967), 262–268 | DOI | MR | Zbl

[2] Z. Bai, J. Hu, G. Pan, W. Zhou, “A note on rate of convergence in probability to semicircular law”, Electron. J. Probab., 16 (2011), paper No 88, 2439–2451 | DOI | MR | Zbl

[3] Z. Bai, J. W. Silverstein, Spectral analysis of large dimensional random matrices, Springer Ser. Statist., 2nd ed., Springer, New York, 2010, xvi+551 pp. | DOI | MR | Zbl

[4] M. Banna, F. Merlevède, M. Peligrad, “On the limiting spectral distribution for a large class of symmetric random matrices with correlated entries”, Stochastic Process. Appl., 125:7 (2015), 2700–2726 | DOI | MR | Zbl

[5] C. Cacciapuoti, A. Maltsev, B. Schlein, “Bounds for the Stieltjes transform and the density of states of Wigner matrices”, Probab. Theory Related Fields, 163:1-2 (2015), 1–59 | DOI | MR | Zbl

[6] L. Erdős, “Universality of Wigner random matrices: a survey of recent results”, Russian Math. Surveys, 66:3 (2011), 507–626 | DOI | DOI | MR | Zbl

[7] L. Erd{ő}s, A. Knowles, H.-T. Yau, J. Yin, “Spectral statistics of Erd{ő}s–Rényi graphs II: Eigenvalue spacing and the extreme eigenvalues”, Comm. Math. Phys., 314:3 (2012), 587–640 | DOI | MR | Zbl

[8] L. Erd{ő}s, A. Knowles, H.-T. Yau, J. Yin, “The local semicircle law for a general class of random matrices”, Electron. J. Probab., 18 (2013), paper No 59, 58 pp. | DOI | MR | Zbl

[9] L. Erd{ő}s, A. Knowles, H.-T. Yau, J. Yin, “Spectral statistics of Erdös–Rényi graphs I: Local semicircle law”, Ann. Probab., 41:3B (2013), 2279–2375 | DOI | MR | Zbl

[10] L. Erd{ő}s, B. Schlein, H.-T. Yau, “Local semicircle law and complete delocalization for Wigner random matrices”, Comm. Math. Phys., 287:2 (2009), 641–655 | DOI | MR | Zbl

[11] L. Erd{ő}s, B. Schlein, H.-T. Yau, “Semicircle law on short scales and delocalization of eigenvectors for Wigner random matrices”, Ann. Probab., 37:3 (2009), 815–852 | DOI | MR | Zbl

[12] L. Erd{ő}s, B. Schlein, H.-T. Yau, “Wegner estimate and level repulsion for Wigner random matrices”, Int. Math. Res. Not. IMRN, 2010:3 (2010), 436–479 | DOI | MR | Zbl

[13] E. Giné, R. Latała, J. Zinn, “Exponential and moment inequalities for $U$-statistics”, High dimensional probability (Seattle, WA, 1999), v. II, Progr. Probab., 47, Birkhäuser Boston, Boston, MA, 2000, 13–38 | MR | Zbl

[14] V. L. Girko, Theory of random determinants, Math. Appl. (Soviet Ser.), 45, Kluwer Acad. Publ., Dordrecht, 1988, xxv+677 pp. | MR | Zbl | Zbl

[15] V. L. Girko, “Spectral theory of random matrices”, Russian Math. Surveys, 40:1 (1985), 77–120 | DOI | MR | Zbl

[16] F. Götze, A. A. Naumov, A. N. Tikhomirov, “Limit theorems for two classes of random matrices with dependent entries”, Theory Probab. Appl., 59:1 (2015), 23–39 | DOI | DOI | Zbl

[17] F. Götze, A. Naumov, A. Tikhomirov, Local semicircle law under moment conditions. Part I: The Stieltjes transform, arXiv: 1510.07350

[18] F. Götze, A. Naumov, A. Tikhomirov, Local semicircle law under moment conditions. Part II: Localization and delocalization, arXiv: 1511.00862

[19] F. Götze, A. Tikhomirov, “Rate of convergence to the semi-circular law”, Probab. Theory Related Fields, 127:2 (2003), 228–276 | DOI | MR | Zbl

[20] F. Götze, A. Tikhomirov, On the rate of convergence to the semi-circular law, arXiv: 1109.0611

[21] F. Götze, A. Tikhomirov, Rate of convergence of the empirical spectral distribution function to the semi-circular law, arXiv: 1407.2780

[22] F. Götze, A. Tikhomirov, “Optimal bounds for convergence of expected spectral distributions to the semi-circular law”, Probab. Theory Related Fields, 165:1-2 (2016), 163–233 | DOI | Zbl

[23] J. Gustavsson, “Gaussian fluctuations of eigenvalues in the GUE”, Ann. Inst. H. Poincaré Probab. Statist., 41:2 (2005), 151–178 | DOI | MR | Zbl

[24] J. O. Lee, J. Yin, “A necessary and sufficient condition for edge universality of Wigner matrices”, Duke Math. J., 163:1 (2014), 117–173 | DOI | MR | Zbl

[25] A. A. Naumov, “Limit theorems for two classes of random matrices with Gaussian entries”, J. Math. Sci. (N. Y.), 204:1 (2015), 140–147 | DOI | MR | Zbl

[26] L. A. Pastur, “Spectra of random self adjoint operators”, Russian Math. Surveys, 28:1 (1973), 1–67 | DOI | MR | Zbl

[27] D. Shlyakhtenko, “Random Gaussian band matrices and freeness with amalgamation”, Internat. Math. Res. Notices, 1996:20 (1996), 1013–1025 | DOI | MR | Zbl

[28] Terence Tao, Van Vu, “Random matrices: the universality phenomenon for Wigner ensembles”, Modern aspects of random matrix theory, Proc. Sympos. Appl. Math., 72, Amer. Math. Soc., Providence, RI, 2014, 121–172 ; arXiv: 1202.0068 | DOI | MR | Zbl

[29] Terence Tao, Van Vu, “Random matrices: sharp concentration of eigenvalues”, Random Matrices Theory Appl., 2:3 (2013), 1350007, 31 pp. | DOI | MR | Zbl