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Random matrices: universality of local spectral statistics of non-Hermitian matrices. (English) Zbl 1316.15042
The complex Gaussian matrix ensemble is constituted by random matrices whose entries are independent and identically distributed with the distribution of a complex Gaussian \(N(0, 1/2)_{\mathbf{C}}\) with mean zero and variance one, i.e., the probability distribution of each entry is \(\omega(z)= \frac{1}{\pi} \exp (- |z|^{2}),\) and the real and imaginary parts of each entry independently have the distribution \(N(0, 1/2)_{\mathbf{R}}.\) The correlation functions \(\rho_{n}^{(k)}(z_{1}, \cdot\cdot\cdot, z_{k})\) of a complex Gaussian matrix are given by the Ginibre formula (see [J. Ginibre, J. Math. Phys. 6, 440–449 (1965; Zbl 0127.39304)]) \(\rho_{n}^{(k)}(z_{1}, \cdot\cdot\cdot, z_{k})= \det (K_{n}(z_{i}, z_{j}))_{i,j=1}^{k}\), where \(K_{n}(z,y)= \omega(z)^{1/2} \omega(y)^{1/2} \sum_{j=0}^{n-1} \frac{(z\bar{y})^{j}}{j!}\) is the \(n\)th kernel associated with the Gaussian distribution. For \(k=1\), one has the asymptotics \(\rho_{n}^{(1)} (n^{1/2} z) \rightarrow \frac{1}{\pi} 1_{|z|\leq 1}\) a.e. which yields the circular law for complex Gaussian matrices.
In the paper under review, the authors prove, as a first result, the universality of the asymptotic law for Gaussian ensembles among all random \(n\times n\) matrices whose entries are jointly independent, have independent real and imaginary parts, exponentially decaying, the so-called (C1) condition, and whose moments match those of the complex Gaussian ensemble to fourth order.
In the Hermitian case, four moment theorems can be used to extend various facts concerning the asymptotic spectral distribution of special matrix ensembles to other matrix ensembles which satisfy appropriate moment matching conditions. In such a sense, a partial extension of a central limit theorem (see [B. Rider, Probab. Theory Relat. Fields 130, No. 3, 337–367 (2004; Zbl 1071.82029)]) in the small radius case is deduced. In the case of real matrices, a four moment theorem, an universality result as well as the asymptotic behavior of real matrices satisfying the (C1) condition are also obtained. As a direct and quick application, the authors show that for many ensembles of independent-entry matrices \(M_{n}\) satisfying the condition (C1) and matching moments with the real or complex Gaussian matrix to fourth order (in the real case it is assumed that \(n\) is even) most of the eigenvalues are simple in the sense that with probability \(1-O(n^{-c})\), at most \(O(n^{1-c})\) of the complex eigenvalues, and \(O(n^{1/2-c})\) of the real eigenvalues, are repeated, for some fixed \(c>0\).
Since the spectrum of non-Hermitian matrices is unstable, the authors use the log-determinants \(\log|\det (M_{n}- z_{0})|\) instead of the resolvent \((M_{n}-z_{0})^{-1}\) or the closely related Stieltjes transform \(\frac{1}{n} \operatorname{trace}(M_{n}-z_{0})\) taking into account the connection between spectral statistics and the log-determinant which goes back to the Girko’s Hermitization method (see [V. L. Girko, Teor. Veroyatn. Primen. 29, No. 4, 669–679 (1984; Zbl 0565.60034)]). Notice that the log-determinant is connected to the eigenvalues of the independent and identically distributed matrix \(M_{n}\) via the identity \(\log|\det (M_{n}- z_{0})|= \sum_{i=0}^{n} \log| \lambda_{i}(M_{n})- z_{0}|.\) The main tools are a four moment theorem for these log-determinants together with a strong concentration result for the log-determinants in the Gausssian case. This is done from the analysis of the solutions of a certain nonlinear stochastic difference equation which is governed by the dynamics of the maps \(a\mapsto \frac{|z_{0}|a n^{1/2}}{(|a|^{2} + n -i)^{1/2}}\) as \(i\) increases from \(1\) to \(n-1\). A crude lower bound, lower bounds at early times, concentration at late times based on repulsive properties near the origin to propagate the initial largeness to latter values of \(i\) are obtained for real and Gaussian matrices. From here, the concentration bound on log-determinant for an independent-entry matrix \(M_{n}\) satisfying the (C1) condition and matching the real or complex Gaussian ensembles to third order means that for any fixed \(C>0\) and any \(z_{0}\in B(0, C)\), \(\log|\det (M_{n}- z_{0} n^{1/2})|\) concentrates around \(\frac{n \log n}{2} + \frac{1}{2} n (|z_{0}|^{2}-1)\) for \(|z_{0}|\leq 1\) and around \(\frac{n \log n}{2} + n \log |z_{0}|\) for \(|z_{0}|\geq 1\), uniformly in \(|z_{0}|\).

15B52 Random matrices (algebraic aspects)
60B20 Random matrices (probabilistic aspects)
15A15 Determinants, permanents, traces, other special matrix functions
39A50 Stochastic difference equations
44A15 Special integral transforms (Legendre, Hilbert, etc.)
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[1] Akemann, G. and Kanzieper, E. (2007). Integrable structure of Ginibre’s ensemble of real random matrices and a Pfaffian integration theorem. J. Stat. Phys. 129 1159-1231. · Zbl 1136.82019 · doi:10.1007/s10955-007-9381-2 · arxiv:math-ph/0703019
[2] Alon, N. and Spencer, J. H. (2008). The Probabilistic Method , 3rd ed. Wiley, Hoboken, NJ. · Zbl 1333.05001
[3] Bai, Z. D. (1997). Circular law. Ann. Probab. 25 494-529. · Zbl 0871.62018 · doi:10.1214/aop/1024404298
[4] Bai, Z. D. and Yin, Y. Q. (1986). Limiting behavior of the norm of products of random matrices and two problems of Geman-Hwang. Probab. Theory Related Fields 73 555-569. · Zbl 0586.60021 · doi:10.1007/BF00324852
[5] Bordenave, C. and Chafaï, D. (2012). Around the circular law. Probab. Surv. 9 1-89. · Zbl 1243.15022 · doi:10.1214/11-PS183 · euclid:ps/1325604980 · arxiv:1109.3343
[6] Borodin, A. and Sinclair, C. D. Correlation functions of ensembles of asymmetric real matrices. Preprint. Available at . arXiv:0706.2670v1 · Zbl 1184.82004 · arxiv.org
[7] Borodin, A. and Sinclair, C. D. (2009). The Ginibre ensemble of real random matrices and its scaling limits. Comm. Math. Phys. 291 177-224. · Zbl 1184.82004 · doi:10.1007/s00220-009-0874-5 · arxiv:0805.2986
[8] Bourgade, P., Yau, H.-T. and Yin, J. (2014). The local circular law II: The edge case. Probab. Theory Related Fields 159 619-660. · Zbl 1342.15028 · doi:10.1007/s00440-013-0516-x · arxiv:1206.3187
[9] Bourgade, P., Yau, H.-T. and Yin, J. (2014). Local circular law for random matrices. Probab. Theory Related Fields 159 545-595. · Zbl 1301.15021 · doi:10.1007/s00440-013-0514-z · arxiv:1206.1449
[10] Brown, L. G. (1986). Lidskiĭ’s theorem in the type II case. In Geometric Methods in Operator Algebras ( Kyoto , 1983). Pitman Res. Notes Math. Ser. 123 1-35. Longman Sci. Tech., Harlow.
[11] Chatterjee, S. (2006). A generalization of the Lindeberg principle. Ann. Probab. 34 2061-2076. · Zbl 1117.60034 · doi:10.1214/009117906000000575 · arxiv:math/0508519
[12] Costin, A. and Lebowitz, J. L. (1995). Gaussian fluctuations in random matrices. Phys. Rev. Lett. 75 69-72. · doi:10.1103/PhysRevLett.75.69
[13] Edelman, A. (1997). The probability that a random real Gaussian matrix has \(k\) real eigenvalues, related distributions, and the circular law. J. Multivariate Anal. 60 203-232. · Zbl 0886.15024 · doi:10.1006/jmva.1996.1653
[14] Edelman, A., Kostlan, E. and Shub, M. (1994). How many eigenvalues of a random matrix are real? J. Amer. Math. Soc. 7 247-267. · Zbl 0790.15017 · doi:10.2307/2152729
[15] Erdësh, L. (2011). Universality of Wigner random matrices: A survey of recent results. Russian Math. Surveys 66 67-198.
[16] Erdős, L., Ramírez, J., Schlein, B., Tao, T., Vu, V. and Yau, H.-T. (2010). Bulk universality for Wigner Hermitian matrices with subexponential decay. Math. Res. Lett. 17 667-674. · Zbl 1277.15027 · doi:10.4310/MRL.2010.v17.n4.a7
[17] Erdős, L., Schlein, B. and Yau, H.-T. (2009). Semicircle law on short scales and delocalization of eigenvectors for Wigner random matrices. Ann. Probab. 37 815-852. · Zbl 1175.15028 · doi:10.1214/08-AOP421 · arxiv:0711.1730
[18] Erdős, L., Schlein, B. and Yau, H.-T. (2009). Local semicircle law and complete delocalization for Wigner random matrices. Comm. Math. Phys. 287 641-655. · Zbl 1186.60005 · doi:10.1007/s00220-008-0636-9 · arxiv:0803.0542
[19] Erdős, L., Schlein, B. and Yau, H.-T. (2010). Wegner estimate and level repulsion for Wigner random matrices. Int. Math. Res. Not. IMRN 3 436-479. · Zbl 1204.15043 · doi:10.1093/imrn/rnp136
[20] Erdős, L., Schlein, B. and Yau, H.-T. (2011). Universality of random matrices and local relaxation flow. Invent. Math. 185 75-119. · Zbl 1225.15033 · doi:10.1007/s00222-010-0302-7
[21] Erdős, L., Schlein, B., Yau, H.-T. and Yin, J. (2012). The local relaxation flow approach to universality of the local statistics for random matrices. Ann. Inst. Henri Poincaré Probab. Stat. 48 1-46. · Zbl 1285.82029 · doi:10.1214/10-AIHP388
[22] Erdős, L., Yau, H.-T. and Yin, J. (2012). Bulk universality for generalized Wigner matrices. Probab. Theory Related Fields 154 341-407. · Zbl 1277.15026 · doi:10.1007/s00440-011-0390-3
[23] Forrester, P. J. and Mays, A. (2009). A method to calculate correlation functions for \(\beta=1\) random matrices of odd size. J. Stat. Phys. 134 443-462. · Zbl 1172.82010 · doi:10.1007/s10955-009-9684-6 · arxiv:0809.5116
[24] Forrester, P. J. and Nagao, T. (2007). Eigenvalue statistics of the real Ginibre ensemble. Phys. Rev. Lett. 99 050603.
[25] Geman, S. (1986). The spectral radius of large random matrices. Ann. Probab. 14 1318-1328. · Zbl 0605.60037 · doi:10.1214/aop/1176992372
[26] Ginibre, J. (1965). Statistical ensembles of complex, quaternion, and real matrices. J. Math. Phys. 6 440-449. · Zbl 0127.39304 · doi:10.1063/1.1704292
[27] Girko, V. L. (1984). The circular law. Teor. Veroyatnost. i Primenen. 29 669-679. · Zbl 0565.60034
[28] Guionnet, A. (2009-2010). Grandes matrices aléatoires et théorèmes d’universalité, Séminaire BOURBAKI . Avril 2010. 62ème année, 2009-2010, no 1019.
[29] Hanson, D. L. and Wright, F. T. (1971). A bound on tail probabilities for quadratic forms in independent random variables. Ann. Math. Statist. 42 1079-1083. · Zbl 0216.22203 · doi:10.1214/aoms/1177693335
[30] Kanzieper, E. and Akemann, G. (2005). Statistics of real eigenvalues in Ginibre’s ensemble of random real matrices. Phys. Rev. Lett. 95 230201, 4. · doi:10.1103/PhysRevLett.95.230201
[31] Knowles, A. and Yin, J. (2013). Eigenvector distribution of Wigner matrices. Probab. Theory Related Fields 155 543-582. · Zbl 1268.15033 · doi:10.1007/s00440-011-0407-y · arxiv:1102.0057
[32] Kostlan, E. (1992). On the spectra of Gaussian matrices. Linear Algebra Appl. 162/164 385-388. · Zbl 0748.15024 · doi:10.1016/0024-3795(92)90386-O
[33] Krishnapur, M. and Virág, B. (2014). The Ginibre ensemble and Gaussian analytic functions. Int. Math. Res. Not. IMRN 6 1441-1464. · Zbl 1316.60017 · doi:10.1093/imrn/rns255 · arxiv:1112.2457
[34] Lehmann, N. and Sommers, H.-J. (1991). Eigenvalue statistics of random real matrices. Phys. Rev. Lett. 67 941-944. · Zbl 0990.82528 · doi:10.1103/PhysRevLett.67.941
[35] Mehta, M. L. (1967). Random Matrices and the Statistical Theory of Energy Levels . Academic Press, New York. · Zbl 0925.60011
[36] Nguyen, H. H. and Vu, V. (2014). Random matrices: Law of the determinant. Ann. Probab. 42 146-167. · Zbl 1299.60005 · doi:10.1214/12-AOP791 · euclid:aop/1389278522 · arxiv:1112.0752
[37] Nourdin, I. and Peccati, G. (2010). Universal Gaussian fluctuations of non-Hermitian matrix ensembles: From weak convergence to almost sure CLTs. ALEA Lat. Am. J. Probab. Math. Stat. 7 341-375. · Zbl 1276.60026 · alea.impa.br · arxiv:1002.1212
[38] Rider, B. (2003). A limit theorem at the edge of a non-Hermitian random matrix ensemble. J. Phys. A 36 3401-3409. · Zbl 1039.60037 · doi:10.1088/0305-4470/36/12/331
[39] Rider, B. (2004). Deviations from the circular law. Probab. Theory Related Fields 130 337-367. · Zbl 1071.82029 · doi:10.1007/s00440-004-0355-x
[40] Rider, B. and Silverstein, J. W. (2006). Gaussian fluctuations for non-Hermitian random matrix ensembles. Ann. Probab. 34 2118-2143. · Zbl 1122.15022 · doi:10.1214/009117906000000403 · arxiv:math/0502400
[41] Rudelson, M. and Vershynin, R. (2008). The Littlewood-Offord problem and invertibility of random matrices. Adv. Math. 218 600-633. · Zbl 1139.15015 · doi:10.1016/j.aim.2008.01.010 · arxiv:math/0703503
[42] Rudelson, M. and Vershynin, R. (2010). Non-asymptotic theory of random matrices: Extreme singular values. In Proceedings of the International Congress of Mathematicians. Volume III 1576-1602. Hindustan Book Agency, New Delhi. · Zbl 1227.60011 · ebooks.worldscinet.com · arxiv:1003.2990
[43] Schlein, B. (2011). Spectral properties of Wigner matrices. In Mathematical Results in Quantum Physics 79-94. World Sci. Publ., Hackensack, NJ. · Zbl 1238.81184 · doi:10.1142/9789814350365_0006 · ebooks.worldscinet.com · arxiv:1009.5027
[44] Sinclair, C. D. (2009). Correlation functions for \(\beta=1\) ensembles of matrices of odd size. J. Stat. Phys. 136 17-33. · Zbl 1220.82066 · doi:10.1007/s10955-009-9771-8 · arxiv:0811.1276
[45] Sommers, H.-J. and Wieczorek, W. (2008). General eigenvalue correlations for the real Ginibre ensemble. J. Phys. A 41 405003, 24. · Zbl 1149.82005 · doi:10.1088/1751-8113/41/40/405003 · arxiv:0806.2756
[46] Soshnikov, A. (2002). Gaussian limit for determinantal random point fields. Ann. Probab. 30 171-187. · Zbl 1033.60063 · doi:10.1214/aop/1020107764 · arxiv:math/0006037
[47] Soshnikov, A. B. (2000). Gaussian fluctuation for the number of particles in Airy, Bessel, sine, and other determinantal random point fields. J. Stat. Phys. 100 491-522. · Zbl 1041.82001 · doi:10.1023/A:1018672622921 · arxiv:math-ph/9907012
[48] Talagrand, M. (1996). A new look at independence. Ann. Probab. 24 1-34. · Zbl 0858.60019 · doi:10.1214/aop/1042644705
[49] Tao, T. and Vu, V. Random matrices: The universality phenomenon for Wigner ensembles. Preprint. Available at . arXiv:1202.0068 · Zbl 1310.15077 · arxiv:1202.0068
[50] Tao, T. and Vu, V. (2006). Additive Combinatorics . Cambridge Univ. Press, Cambridge. · Zbl 1127.11002
[51] Tao, T. and Vu, V. (2007). The condition number of a randomly perturbed matrix. In STOC’ 07 -Proceedings of the 39 th Annual ACM Symposium on Theory of Computing 248-255. ACM, New York. · Zbl 1232.15030 · arxiv:math/0703307
[52] Tao, T. and Vu, V. (2009). From the Littlewood-Offord problem to the circular law: Universality of the spectral distribution of random matrices. Bull. Amer. Math. Soc. ( N.S. ) 46 377-396. · Zbl 1168.15018 · doi:10.1090/S0273-0979-09-01252-X · arxiv:0810.2994
[53] Tao, T. and Vu, V. (2010). Random matrices: Universality of local eigenvalue statistics up to the edge. Comm. Math. Phys. 298 549-572. · Zbl 1202.15038 · doi:10.1007/s00220-010-1044-5 · arxiv:0908.1982
[54] Tao, T. and Vu, V. (2010). Smooth analysis of the condition number and the least singular value. Math. Comp. 79 2333-2352. · Zbl 1253.65067 · doi:10.1090/S0025-5718-2010-02396-8
[55] Tao, T. and Vu, V. (2011). Random matrices: Universality of local eigenvalue statistics. Acta Math. 206 127-204. · Zbl 1217.15043 · doi:10.1007/s11511-011-0061-3 · arxiv:0906.0510
[56] Tao, T. and Vu, V. (2011). The Wigner-Dyson-Mehta bulk universality conjecture for Wigner matrices. Electron. J. Probab. 16 2104-2121. · Zbl 1245.15041 · doi:10.1214/EJP.v16-944 · arxiv:1101.5707
[57] Tao, T. and Vu, V. (2012). Random covariance matrices: Universality of local statistics of eigenvalues. Ann. Probab. 40 1285-1315. · Zbl 1247.15036 · doi:10.1214/11-AOP648 · euclid:aop/1336136064 · arxiv:0912.0966
[58] Tao, T. and Vu, V. (2012). A central limit theorem for the determinant of a Wigner matrix. Adv. Math. 231 74-101. · Zbl 1248.60011 · doi:10.1016/j.aim.2012.05.006 · arxiv:1111.6300
[59] Tao, T. and Vu, V. H. (2009). Inverse Littlewood-Offord theorems and the condition number of random discrete matrices. Ann. of Math. (2) 169 595-632. · Zbl 1250.60023 · doi:10.4007/annals.2009.169.595 · annals.math.princeton.edu · arxiv:math/0511215
[60] Trotter, H. F. (1984). Eigenvalue distributions of large Hermitian matrices; Wigner’s semicircle law and a theorem of Kac, Murdock, and Szegő. Adv. Math. 54 67-82. · Zbl 0562.15005 · doi:10.1016/0001-8708(84)90037-9
[61] Vu, V. H. (2002). Concentration of non-Lipschitz functions and applications. Random Structures Algorithms 20 262-316. · Zbl 0999.60027 · doi:10.1002/rsa.10032
[62] Wright, F. T. (1973). A bound on tail probabilities for quadratic forms in independent random variables whose distributions are not necessarily symmetric. Ann. Probab. 1 1068-1070. · Zbl 0271.60033 · doi:10.1214/aop/1176996815
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