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RANDOM MATRICES WITH SLOW CORRELATION DECAY

Part of: Probability theory on algebraic and topological structures Special matrices

Published online by Cambridge University Press:  26 March 2019

LÁSZLÓ ERDŐS Open the ORCID record for LÁSZLÓ ERDŐS [Opens in a new window] ,
TORBEN KRÜGER  and
DOMINIK SCHRÖDER Open the ORCID record for DOMINIK SCHRÖDER [Opens in a new window]
LÁSZLÓ ERDŐS
Affiliation:
IST Austria, 3400 Klosterneuburg, Austria; lerdos@ist.ac.at, dschroed@ist.ac.at
TORBEN KRÜGER
Affiliation:
University of Bonn, 53115 Bonn, Germany; torben.krueger@uni-bonn.de
DOMINIK SCHRÖDER
Affiliation:
IST Austria, 3400 Klosterneuburg, Austria; lerdos@ist.ac.at, dschroed@ist.ac.at

Abstract

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We consider large random matrices with a general slowly decaying correlation among its entries. We prove universality of the local eigenvalue statistics and optimal local laws for the resolvent away from the spectral edges, generalizing the recent result of Ajanki et al. [‘Stability of the matrix Dyson equation and random matrices with correlations’, Probab. Theory Related Fields173(1–2) (2019), 293–373] to allow slow correlation decay and arbitrary expectation. The main novel tool is a systematic diagrammatic control of a multivariate cumulant expansion.

MSC classification

Primary: 60B20: Random matrices (probabilistic aspects; for algebraic aspects see ) 15B52: Random matrices
Type
Research Article
Information
Forum of Mathematics, Sigma , Volume 7 , 2019 , e8
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s) 2019

References

Ajanki, O. H., Erdős, L. and Krüger, T., ‘Local spectral statistics of Gaussian matrices with correlated entries’, J. Stat. Phys. 163(2) (2016), 280–302.Google Scholar
Ajanki, O. H., Erdős, L. and Krüger, T., ‘Universality for general Wigner-type matrices’, Probab. Theory Related Fields 169(3-4) (2017), 667–727.Google Scholar
Ajanki, O. H., Erdős, L. and Krüger, T., ‘Stability of the matrix Dyson equation and random matrices with correlations’, Probab. Theory Related Fields 173(1–2) (2019), 293–373.Google Scholar
Alt, J., ‘The local semicircle law for random matrices with a fourfold symmetry’, J. Math. Phys. 56(10)103301, 20 pp (2015).Google Scholar
Alt, J., Erdős, L., Krüger, T. and Nemish, Y., ‘Location of the spectrum of Kronecker random matrices’. Preprint, 2017, arXiv:1706.08343.Google Scholar
Anderson, G. W. and Zeitouni, O., ‘A law of large numbers for finite-range dependent random matrices’, Commun. Pure Appl. Math. 61(8) (2008), 1118–1154.Google Scholar
Bai, Z. D. and Yin, Y. Q., ‘Convergence to the semicircle law’, Ann. Probab. 16(2) (1988), 863–875.Google Scholar
Banna, M., Merlevède, F. and Peligrad, M., ‘On the limiting spectral distribution for a large class of symmetric random matrices with correlated entries’, Stochastic Process. Appl. 125(7) (2015), 2700–2726.Google Scholar
Boutet de Monvel, A., Khorunzhy, A. and Vasilchuk, V., ‘Limiting eigenvalue distribution of random matrices with correlated entries’, Markov Process. Related Fields 2(4) (1996), 607–636.Google Scholar
Bradley, R. C., ‘Basic properties of strong mixing conditions. A survey and some open questions’, Probab. Surv. 2 (2005), 107–144. Update of, and a supplement to, the 1986 original.Google Scholar
Che, Z., ‘Universality of random matrices with correlated entries’, Electron. J. Probab. 22(30) (2017), 38 pp.Google Scholar
Duneau, M., Iagolnitzer, D. and Souillard, B., ‘Decrease properties of truncated correlation functions and analyticity properties for classical lattices and continuous systems’, Comm. Math. Phys. 31 (1973), 191–208.Google Scholar
Erdős, L., Knowles, A., Yau, H.-T. and Yin, J., ‘The local semicircle law for a general class of random matrices’, Electron. J. Probab. 18(59) (2013), 58 pp.Google Scholar
Erdős, L., Péché, S., Ramírez, J. A., Schlein, B. and Yau, H.-T., ‘Bulk universality for Wigner matrices’, Commun. Pure Appl. Math. 63(7) (2010), 895–925.Google Scholar
Erdős, L., Schlein, B. and Yau, H.-T., ‘Universality of random matrices and local relaxation flow’, Invent. Math. 185(1) (2011), 75–119.Google Scholar
Erdős, L., Schlein, B., Yau, H.-T. and Yin, J., ‘The local relaxation flow approach to universality of the local statistics for random matrices’, Ann. Inst. Henri Poincaré Probab. Stat. 48(1) (2012), 1–46.Google Scholar
Erdős, L. and Schnelli, K., ‘Universality for random matrix flows with time-dependent density’, Ann. Inst. Henri Poincaré Probab. Stat. 53(4) (2017), 1606–1656.Google Scholar
Erdős, L. and Schröder, D., ‘Fluctuations of rectangular Young diagrams of interlacing Wigner eigenvalues’, Int. Math. Res. Not. IMRN 2018(10) (2016), 3255–3298.Google Scholar
Erdős, L. and Yau, H.-T., A Dynamical Approach to Random Matrix Theory, Courant Lecture Notes in Mathematics, 28 (Courant Institute of Mathematical Sciences, American Mathematical Society, New York, Providence, RI, 2017), pp. ix+226.Google Scholar
Erdős, L. and Yau, H.-T., ‘Universality of local spectral statistics of random matrices’, Bull. Amer. Math. Soc. (N.S.) 49(3) (2012), 377–414.Google Scholar
Erdős, L., Yau, H.-T. and Yin, J., ‘Bulk universality for generalized Wigner matrices’, Probab. Theory Related Fields 154(1–2) (2012), 341–407.Google Scholar
Erdős, L., Yau, H.-T. and Yin, J., ‘Universality for generalized Wigner matrices with Bernoulli distribution’, J. Comb. 2(1) (2011), 15–81.Google Scholar
Giraitis, L. and Surgailis, D., ‘Multivariate Appell polynomials and the central limit theorem’, inDependence in Probability and Statistics (Oberwolfach, 1985), Progr. Probab. Statist., 11 (Birkhäuser Boston, Boston, MA, 1986), 21–71.Google Scholar
Girko, V. L., ‘Asymptotics of the distribution of the spectrum of random matrices’, Uspekhi Mat. Nauk 44(4(268)) (1989), 7–34.Google Scholar
Girko, V. L., Theory of Stochastic Canonical Equations, Vol. I, Mathematics and its Applications, 535 (Kluwer Academic Publishers, Dordrecht, 2001), pp. xxiv+497.Google Scholar
Hachem, W., Loubaton, P. and Najim, J., ‘The empirical eigenvalue distribution of a Gram matrix: from independence to stationarity’, Markov Process. Related Fields 11(4) (2005), 629–648.Google Scholar
He, Y. and Knowles, A., ‘Mesoscopic eigenvalue statistics of Wigner matrices’, Ann. Appl. Probab. 27(3) (2017), 1510–1550.Google Scholar
He, Y., Knowles, A. and Rosenthal, R., ‘Isotropic self-consistent equations for mean-field random matrices’, Probab. Theory Related Fields 171(1–2) (2018), 203–249.Google Scholar
Helton, J. W., Rashidi Far, R. and Speicher, R., ‘Operator-valued semicircular elements: solving a quadratic matrix equation with positivity constraints’, Int. Math. Res. Not. IMRN 2007(22) (2007), 15 pp.Google Scholar
Khorunzhy, A. M., Khoruzhenko, B. A. and Pastur, L. A., ‘Asymptotic properties of large random matrices with independent entries’, J. Math. Phys. 37(10) (1996), 5033–5060.Google Scholar
Landon, B., Sosoe, P. and Yau, H.-T., ‘Fixed energy universality for Dyson Brownian motion’. Preprint, 2016, arXiv:1609.09011.Google Scholar
Landon, B. and Yau, H.-T., ‘Convergence of local statistics of Dyson Brownian motion’, Comm. Math. Phys. 355(3) (2017), 949–1000.Google Scholar
Lee, J. O. and Schnelli, K., ‘Edge universality for deformed Wigner matrices’, Rev. Math. Phys. 27(8) (2015), 1550018, 94 pp.Google Scholar
Lee, J. O., Schnelli, K., Stetler, B. and Yau, H.-T., ‘Bulk universality for deformed Wigner matrices’, Ann. Probab. 44(3) (2016), 2349–2425.Google Scholar
McCullagh, P., ‘Tensor notation and cumulants of polynomials’, Biometrika 71(3) (1984), 461–476.Google Scholar
Mehta, M. L., Random Matrices and the Statistical Theory of Energy Levels (Academic Press, New York-London, 1967), pp. x+259.Google Scholar
O’Rourke, S. and Vu, V., ‘Universality of local eigenvalue statistics in random matrices with external source’, Random Matrices Theory Appl. 3(2) (2014), 1450005, 37 pp.Google Scholar
Pastur, L. A., ‘Spectra of random selfadjoint operators’, Uspehi Mat. Nauk 28(1(169)) (1973), 3–64.Google Scholar
Rashidi Far, R., Oraby, T., Bryc, W. and Speicher, R., ‘On slow-fading MIMO systems with nonseparable correlation’, IEEE Trans. Inform. Theory 54(2) (2008), 544–553.Google Scholar
Schenker, J. H. and Schulz-Baldes, H., ‘Semicircle law and freeness for random matrices with symmetries or correlations’, Math. Res. Lett. 12(4) (2005), 531–542.Google Scholar
Speed, T. P., ‘Cumulants and partition lattices’, Austral. J. Statist. 25(2) (1983), 378–388.Google Scholar
Tao, T. and Vu, V., ‘Random matrices: universality of local eigenvalue statistics’, Acta Math. 206(1) (2011), 127–204.Google Scholar