## Annals of Probability

### Random matrices: Universality of ESDs and the circular law

#### Abstract

Given an n×n complex matrix A, let $$\mu_{A}(x,y):=\frac{1}{n}|\{1\le i\le n,\operatorname{Re}\lambda_{i}\le x,\operatorname{Im}\lambda_{i}\le y\}|$$ be the empirical spectral distribution (ESD) of its eigenvalues λi∈ℂ, i=1, …, n.

We consider the limiting distribution (both in probability and in the almost sure convergence sense) of the normalized ESD $\mu_{{1}/{\sqrt{n}}A_{n}}$ of a random matrix An=(aij)1≤i, jn, where the random variables aijE(aij) are i.i.d. copies of a fixed random variable x with unit variance. We prove a universality principle for such ensembles, namely, that the limit distribution in question is independent of the actual choice of x. In particular, in order to compute this distribution, one can assume that x is real or complex Gaussian. As a related result, we show how laws for this ESD follow from laws for the singular value distribution of $\frac{1}{\sqrt{n}}A_{n}-zI$ for complex z.

As a corollary, we establish the circular law conjecture (both almost surely and in probability), which asserts that $\mu_{{1}/{\sqrt{n}}A_{n}}$ converges to the uniform measure on the unit disc when the aij have zero mean.

#### Article information

Source
Ann. Probab., Volume 38, Number 5 (2010), 2023-2065.

Dates
First available in Project Euclid: 17 August 2010

https://projecteuclid.org/euclid.aop/1282053780

Digital Object Identifier
doi:10.1214/10-AOP534

Mathematical Reviews number (MathSciNet)
MR2722794

Zentralblatt MATH identifier
1203.15025

Subjects
Primary: 15A52 60F17: Functional limit theorems; invariance principles
Secondary: 60F15: Strong theorems

#### Citation

Tao, Terence; Vu, Van; Krishnapur, Manjunath. Random matrices: Universality of ESDs and the circular law. Ann. Probab. 38 (2010), no. 5, 2023--2065. doi:10.1214/10-AOP534. https://projecteuclid.org/euclid.aop/1282053780

#### References

• [1] Bai, Z. D. (1997). Circular law. Ann. Probab. 25 494–529.
• [2] Bai, Z. D. and Silverstein, J. (2006). Spectral Analysis of Large Dimensional Random Matrices. Mathematics Monograph Series 2. Science Press, Beijing.
• [3] Dozier, R. B. and Silverstein, J. W. (2007). On the empirical distribution of eigenvalues of large dimensional information-plus-noise-type matrices. J. Multivariate Anal. 98 678–694.
• [4] Chatterjee, S. (2005). A simple invariance principle. Available at arXiv:math/0508213.
• [5] Chafai, D. (2008). Circular law for non-central random matrices. Preprint.
• [6] Edelman, A. (1988). Eigenvalues and condition numbers of random matrices. SIAM J. Matrix Anal. Appl. 9 543–560.
• [7] Girko, V. L. (1984). The circular law. Theory Probab. Appl. 29 694–706.
• [8] Girko, V. L. (2004). The strong circular law. Twenty years later. II. Random Oper. Stochastic Equations 12 255–312.
• [9] Götze, F. and Tikhomirov, A. N. (2007). On the circular law. Preprint.
• [10] Götze, F. and Tikhomirov, A. N. (2007). The circular law for random matrices. Preprint.
• [11] Krishnapour, M. and Vu, V. Manuscript in preparation.
• [12] Ledoux, M. (2001). The Concentration of Measure Phenomenon. Mathematical Surveys and Monographs 89. Amer. Math. Soc., Providence, RI.
• [13] Mehta, M. L. (1967). Random Matrices and the Statistical Theory of Energy Levels. Academic Press, New York.
• [14] Pan, G. and Zhou, W. (2010). Circular law, extreme singular values and potential theory. J. Multivariate Anal. 101 645–656.
• [15] Pastur, L. A. (1972). The spectrum of random matrices. Teoret. Mat. Fiz. 10 102–112.
• [16] Rudelson, M. (2008). Invertibility of random matrices: Norm of the inverse. Ann. of Math. (2) 168 575–600.
• [17] Rudelson, M. and Vershynin, R. (2008). The least singular value of a random square matrix is O(n−1/2). C. R. Math. Acad. Sci. Paris 346 893–896.
• [18] Rudelson, M. and Vershynin, R. (2009). Smallest singular value of a random rectangular matrix. Comm. Pure Appl. Math. 62 1707–1739.
• [19] Rudelson, M. and Vershynin, R. (2010). The Littlewood-Offord problem and the condition number of random matrices. Adv. Math. 218 600–633.
• [20] Speicher, R. Survey in preparation.
• [21] Tao, T. and Vu, V. (2006). On random ±1 matrices: Singularity and determinant. Random Structures Algorithms 28 1–23.
• [22] Tao, T. and Vu, V. (2006). Additive Combinatorics. Cambridge Studies in Advanced Mathematics 105. Cambridge Univ. Press, Cambridge.
• [23] 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.
• [24] 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.
• [25] Tao, T. and Vu, V. (2008). Random matrices: The circular law. Commun. Contemp. Math. 10 261–307.
• [26] Tao, T. and Vu, V. (2010). Random matrices: The distribution of the smallest singular values. Geom. Funct. Anal. 20 260–297.
• [27] Wigner, E. P. (1958). On the distribution of the roots of certain symmetric matrices. Ann. of Math. (2) 67 325–327.