## The Annals of Probability

### Random matrices: Universality of local spectral statistics of non-Hermitian matrices

#### Abstract

It is a classical result of Ginibre that the normalized bulk $k$-point correlation functions of a complex $n\times n$ Gaussian matrix with independent entries of mean zero and unit variance are asymptotically given by the determinantal point process on $\mathbb{C}$ with kernel $K_{\infty}(z,w):=\frac{1}{\pi}e^{-|z|^{2}/2-|w|^{2}/2+z\bar{w}}$ in the limit $n\to\infty$. In this paper, we show that this asymptotic law is universal among all random $n\times n$ matrices $M_{n}$ whose entries are jointly independent, exponentially decaying, have independent real and imaginary parts and whose moments match that of the complex Gaussian ensemble to fourth order. Analogous results at the edge of the spectrum are also obtained. As an application, we extend a central limit theorem for the number of eigenvalues of complex Gaussian matrices in a small disk to these more general ensembles.

These results are non-Hermitian analogues of some recent universality results for Hermitian Wigner matrices. However, a key new difficulty arises in the non-Hermitian case, due to the instability of the spectrum for such matrices. To resolve this issue, we the need to work with the log-determinants $\log|\det(M_{n}-z_{0})|$ rather than with the Stieltjes transform $\frac{1}{n}\operatorname{tr}(M_{n}-z_{0})^{-1}$, in order to exploit Girko’s Hermitization method. Our main tools are a four moment theorem for these log-determinants, together with a strong concentration result for the log-determinants in the Gaussian case. The latter is established by studying the solutions of a certain nonlinear stochastic difference equation.

With some extra consideration, we can extend our arguments to the real case, proving universality for correlation functions of real matrices which match the real Gaussian ensemble to the fourth order. As an application, we show that a real $n\times n$ matrix whose entries are jointly independent, exponentially decaying and whose moments match the real Gaussian ensemble to fourth order has $\sqrt{\frac{2n}{\pi}}+o(\sqrt{n})$ real eigenvalues asymptotically almost surely.

#### Article information

Source
Ann. Probab., Volume 43, Number 2 (2015), 782-874.

Dates
First available in Project Euclid: 2 February 2015

https://projecteuclid.org/euclid.aop/1422885575

Digital Object Identifier
doi:10.1214/13-AOP876

Mathematical Reviews number (MathSciNet)
MR3306005

Zentralblatt MATH identifier
1316.15042

Subjects
Primary: 15A52

#### Citation

Tao, Terence; Vu, Van. Random matrices: Universality of local spectral statistics of non-Hermitian matrices. Ann. Probab. 43 (2015), no. 2, 782--874. doi:10.1214/13-AOP876. https://projecteuclid.org/euclid.aop/1422885575

#### References

• [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.
• [2] Alon, N. and Spencer, J. H. (2008). The Probabilistic Method, 3rd ed. Wiley, Hoboken, NJ.
• [3] Bai, Z. D. (1997). Circular law. Ann. Probab. 25 494–529.
• [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.
• [5] Bordenave, C. and Chafaï, D. (2012). Around the circular law. Probab. Surv. 9 1–89.
• [6] Borodin, A. and Sinclair, C. D. Correlation functions of ensembles of asymmetric real matrices. Preprint. Available at arXiv:0706.2670v1.
• [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.
• [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.
• [9] Bourgade, P., Yau, H.-T. and Yin, J. (2014). Local circular law for random matrices. Probab. Theory Related Fields 159 545–595.
• [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.
• [12] Costin, A. and Lebowitz, J. L. (1995). Gaussian fluctuations in random matrices. Phys. Rev. Lett. 75 69–72.
• [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.
• [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.
• [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.
• [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.
• [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.
• [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.
• [20] Erdős, L., Schlein, B. and Yau, H.-T. (2011). Universality of random matrices and local relaxation flow. Invent. Math. 185 75–119.
• [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.
• [22] Erdős, L., Yau, H.-T. and Yin, J. (2012). Bulk universality for generalized Wigner matrices. Probab. Theory Related Fields 154 341–407.
• [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.
• [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.
• [26] Ginibre, J. (1965). Statistical ensembles of complex, quaternion, and real matrices. J. Math. Phys. 6 440–449.
• [27] Girko, V. L. (1984). The circular law. Teor. Veroyatnost. i Primenen. 29 669–679.
• [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.
• [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.
• [31] Knowles, A. and Yin, J. (2013). Eigenvector distribution of Wigner matrices. Probab. Theory Related Fields 155 543–582.
• [32] Kostlan, E. (1992). On the spectra of Gaussian matrices. Linear Algebra Appl. 162/164 385–388.
• [33] Krishnapur, M. and Virág, B. (2014). The Ginibre ensemble and Gaussian analytic functions. Int. Math. Res. Not. IMRN 6 1441–1464.
• [34] Lehmann, N. and Sommers, H.-J. (1991). Eigenvalue statistics of random real matrices. Phys. Rev. Lett. 67 941–944.
• [35] Mehta, M. L. (1967). Random Matrices and the Statistical Theory of Energy Levels. Academic Press, New York.
• [36] Nguyen, H. H. and Vu, V. (2014). Random matrices: Law of the determinant. Ann. Probab. 42 146–167.
• [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.
• [38] Rider, B. (2003). A limit theorem at the edge of a non-Hermitian random matrix ensemble. J. Phys. A 36 3401–3409.
• [39] Rider, B. (2004). Deviations from the circular law. Probab. Theory Related Fields 130 337–367.
• [40] Rider, B. and Silverstein, J. W. (2006). Gaussian fluctuations for non-Hermitian random matrix ensembles. Ann. Probab. 34 2118–2143.
• [41] Rudelson, M. and Vershynin, R. (2008). The Littlewood–Offord problem and invertibility of random matrices. Adv. Math. 218 600–633.
• [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.
• [43] Schlein, B. (2011). Spectral properties of Wigner matrices. In Mathematical Results in Quantum Physics 79–94. World Sci. Publ., Hackensack, NJ.
• [44] Sinclair, C. D. (2009). Correlation functions for $\beta=1$ ensembles of matrices of odd size. J. Stat. Phys. 136 17–33.
• [45] Sommers, H.-J. and Wieczorek, W. (2008). General eigenvalue correlations for the real Ginibre ensemble. J. Phys. A 41 405003, 24.
• [46] Soshnikov, A. (2002). Gaussian limit for determinantal random point fields. Ann. Probab. 30 171–187.
• [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.
• [48] Talagrand, M. (1996). A new look at independence. Ann. Probab. 24 1–34.
• [49] Tao, T. and Vu, V. Random matrices: The universality phenomenon for Wigner ensembles. Preprint. Available at arXiv:1202.0068.
• [50] Tao, T. and Vu, V. (2006). Additive Combinatorics. Cambridge Univ. Press, Cambridge.
• [51] Tao, T. and Vu, V. (2007). The condition number of a randomly perturbed matrix. In STOC’07—Proceedings of the 39th Annual ACM Symposium on Theory of Computing 248–255. ACM, New York.
• [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.
• [53] Tao, T. and Vu, V. (2010). Random matrices: Universality of local eigenvalue statistics up to the edge. Comm. Math. Phys. 298 549–572.
• [54] Tao, T. and Vu, V. (2010). Smooth analysis of the condition number and the least singular value. Math. Comp. 79 2333–2352.
• [55] Tao, T. and Vu, V. (2011). Random matrices: Universality of local eigenvalue statistics. Acta Math. 206 127–204.
• [56] Tao, T. and Vu, V. (2011). The Wigner–Dyson–Mehta bulk universality conjecture for Wigner matrices. Electron. J. Probab. 16 2104–2121.
• [57] Tao, T. and Vu, V. (2012). Random covariance matrices: Universality of local statistics of eigenvalues. Ann. Probab. 40 1285–1315.
• [58] Tao, T. and Vu, V. (2012). A central limit theorem for the determinant of a Wigner matrix. Adv. Math. 231 74–101.
• [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.
• [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.
• [61] Vu, V. H. (2002). Concentration of non-Lipschitz functions and applications. Random Structures Algorithms 20 262–316.
• [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.