## Gaussian fluctuations for non-Hermitian random matrix ensembles.(English)Zbl 1122.15022

An ensemble of $$N \times N$$ non-Hermitian matrices in which all entries are independent identically distributed complex random variables of mean zero and mean-square one is considered. Z. D. Bai [Ann. Probab. 25, 494–529 (1997; Zbl 0871.62018)] has proved so far the most general result that the circular law holds for ensembles of independent entries such that the common entry distribution possesses a bounded density and finite $$({\text 4}+ \epsilon)$$ moment.
In this paper the fluctuations from the circular law are studied in the class of non-Hermitian matrices for which higher moments of the entries obey a growth condition. Attention is paid to linear spectral statistics of the form $$X_N(f) = \sum_{k=1}^N f(\lambda_k)$$, where $$\lambda_1, \lambda_2, \dots, \lambda_N$$ are ensemble eigenvalues. $$X_N(f)$$ is given by $$X_N(f) = N \int f(\lambda) \mu_N (d \lambda)$$, where $$\mu_N (d \lambda)$$ is the empirical spectral distribution and the test test function $$f$$ is analytic in a neighborhood of the disk $$|z| \leq {\text 4}$$. The central limit theorem for such an ensemble is established.
The proof is based on the work of Z. D. Bai and J. W. Silverstein [Ann. Probab. 32, 553–605 (2004; Zbl 1063.60022)] and the used method is fairly robust in terms of the underlying distribution of the matrix entries but does not require the analyticity of the test function $$f$$ in $$X_N(f)$$.

### MSC:

 15B52 Random matrices (algebraic aspects) 60F05 Central limit and other weak theorems

### Citations:

Zbl 0871.62018; Zbl 1063.60022
Full Text:

### References:

 [1] Bai, Z. D. (1997). Circular law. Ann. Probab. 25 494–529. · Zbl 0871.62018 [2] Bai, Z. D. and Silverstein, J. W. (1998). No eigenvalues outside the support of the limiting spectral distribution of large-dimensional sample covariance matrices. Ann. Probab. 26 316–345. · Zbl 0937.60017 [3] Bai, Z. D. and Silverstein, J. W. (2004). CLT for linear spectral statistics of large-dimensional sample covariance matrices. Ann. Probab. 32 533–605. · Zbl 1063.60022 [4] Bell, S. R. (1992). The Cauchy Transform , Potential Theory , and Conformal Mapping. CRC Press, Boca Raton, FL. [5] Billingsley, P. (1968). Convergence of Probability Measures . Wiley, New York. · Zbl 0172.21201 [6] Billingsley, P. (1995). Probability and Measure , 3rd ed. Wiley, New York. · Zbl 0822.60002 [7] Costin, O. and Lebowtz, J. (1995). Gaussian fluctuations in random matrices. Phys. Rev. Lett. 75 69–72. [8] Diaconis, P. and Evans, S. N. (2001). Linear functionals of eigenvalues of random matrices. Trans. Amer. Math. Soc. 353 2615–2633. JSTOR: · Zbl 1008.15013 [9] Edelman, A. (1997). The probability that a random real matrix has $$k$$ real eigenvalues, related distributions, and the circular law. J. Multivariate Anal. 60 203–232. · Zbl 0886.15024 [10] Forrester, P. J. (1999). Fluctuation formula for complex random matrices. J. Phys. A 32 159–163. · Zbl 0936.82001 [11] Fyodorov, Y. V. and Sommers, H.-J. (2003). Random matrices close to Hermitian and unitary: Overview of methods and results. J. Phys. A 36 3303–3347. · Zbl 1069.82006 [12] Geman, S. (1980). A limit theorem for the norm of random matrices. Ann. Probab. 8 252–261. · Zbl 0428.60039 [13] Geman, S. (1986). The spectral radius of large random matrices. Ann. Probab. 14 1318–1328. · Zbl 0605.60037 [14] Girko, V. L. (1984). Circle law. Theory Probab. Appl. 29 694–706. [15] Ginibre, J. (1965). Statistical ensembles of complex, quaternion, and real matrices. J. Math. Phys. 6 440–449. · Zbl 0127.39304 [16] Guionnet, A. (2002). Large deviations and upper bounds for non-commutative functionals of Gaussian large random matrices. Ann. Inst. H. Poincaré Probab. Statist. 38 341–384. · Zbl 0995.60028 [17] Horn, R. A. and Johnson, C. R. (1985). Matrix Analysis . Cambridge Univ. Press. · Zbl 0576.15001 [18] Hughes, C. P., Keating, J. P. and O’Connell, N. (2000). On the characteristic polynomial of a random unitary matrix. Comm. Math. Phys. 220 429–451. · Zbl 0987.60039 [19] Hwang, C. R. (1986). A brief survey on the spectral radius and the spectral distribution of large dimensional random matrices with iid entries. In Random Matrices and Their Applications (M. L. Mehta, ed.) 50 145–152. Amer. Math. Soc., Providence, RI. · Zbl 0584.60019 [20] Israelson, S. (2001). Asymptotic fluctuations of a particle system with singular interaction. Stochastic Process. Appl. 93 25–56. · Zbl 1053.60104 [21] Keating, J. P. and Snaith, N. C. (2000). Random matrix theory and $$\zeta(1/2 + i t)$$. Comm. Math. Phys. 214 57–89. · Zbl 1051.11048 [22] Johansson, K. (1997). On random matrices from the classical compact groups. Ann. of Math. ( 2 ) 145 519–545. JSTOR: · Zbl 0883.60010 [23] Johansson, K. (1998). On the fluctuation of eigenvalues of random Hermitian matrices. Duke Math. J. 91 151–204. · Zbl 1039.82504 [24] Lebœuf, P. (1999). Random matrices, random polynomials and coulomb systems. In Proceedings of the International Conference on Strongly Coupled Coulomb Systems , Saint-Malo. · Zbl 1156.81396 [25] Peres, Y. and Vir$$\acute\mboxa$$g, B. (2005). Zeros of the i.i.d. Gaussian power series: A conformally invariant determinantal process. Acta Math. 194 1–35. · Zbl 1099.60037 [26] Rider, B. (2004). Deviations from the circular law. Probab. Theory Related Fields 130 337–367. · Zbl 1071.82029 [27] Sinai, Ya. and Soshnikov, A. (1998). Central limit theorems for traces of large random matrices with independent entries. Bol. Soc. Brasil. Mat. 29 1–24. · Zbl 0912.15027 [28] Soshnikov, A. (2002). Gaussian limits for determinantal random point fields. Ann. Probab. 30 171–181. · Zbl 1033.60063 [29] Titchmarsh, E. C. (1939). The Theory of Functions , 2nd ed. Oxford Univ. Press. · JFM 65.0302.01 [30] Wieand, K. (2002). Eigenvalue distributions of random unitary matrices. Probab. Theory Related Fields 123 202–224. · Zbl 1044.15016 [31] Yin, Y. Q., Bai, Z. D. and Krishnaiah, P. R. (1988). On the limit of the largest eigenvalue of the large dimensional sample covariance matrix. Probab. Theory Related Fields 78 509–521. · Zbl 0627.62022
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