The Annals of Probability

The circular law for sparse non-Hermitian matrices

Anirban Basak and Mark Rudelson

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


For a class of sparse random matrices of the form $A_{n}=(\xi_{i,j}\delta_{i,j})_{i,j=1}^{n}$, where $\{\xi_{i,j}\}$ are i.i.d. centered sub-Gaussian random variables of unit variance, and $\{\delta_{i,j}\}$ are i.i.d. Bernoulli random variables taking value $1$ with probability $p_{n}$, we prove that the empirical spectral distribution of $A_{n}/\sqrt{np_{n}}$ converges weakly to the circular law, in probability, for all $p_{n}$ such that $p_{n}=\omega({\log^{2}n}/{n})$. Additionally if $p_{n}$ satisfies the inequality $np_{n}>\exp(c\sqrt{\log n})$ for some constant $c$, then the above convergence is shown to hold almost surely. The key to this is a new bound on the smallest singular value of complex shifts of real valued sparse random matrices. The circular law limit also extends to the adjacency matrix of a directed Erdős–Rényi graph with edge connectivity probability $p_{n}$.

Article information

Ann. Probab., Volume 47, Number 4 (2019), 2359-2416.

Received: July 2017
Revised: June 2018
First available in Project Euclid: 4 July 2019

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Primary: 15B52: Random matrices 60B10: Convergence of probability measures 60B20: Random matrices (probabilistic aspects; for algebraic aspects see 15B52)

Random matrix sparse matrix smallest singular value circular law


Basak, Anirban; Rudelson, Mark. The circular law for sparse non-Hermitian matrices. Ann. Probab. 47 (2019), no. 4, 2359--2416. doi:10.1214/18-AOP1310.

Export citation


  • [1] Adamczak, R. (2011). On the Marchenko–Pastur and circular laws for some classes of random matrices with dependent entries. Electron. J. Probab. 16 1068–1095.
  • [2] Adamczak, R., Chafaï, D. and Wolff, P. (2016). Circular law for random matrices with exchangeable entries. Random Structures Algorithms 48 454–479.
  • [3] Anderson, G. W., Guionnet, A. and Zeitouni, O. (2010). An Introduction to Random Matrices. Cambridge Studies in Advanced Mathematics 118. Cambridge Univ. Press, Cambridge.
  • [4] Bai, Z. and Silverstein, J. W. (2010). Spectral Analysis of Large Dimensional Random Matrices, 2nd ed. Springer Series in Statistics. Springer, New York.
  • [5] Bai, Z. D. (1997). Circular law. Ann. Probab. 25 494–529.
  • [6] Bai, Z. D. (1999). Methodologies in spectral analysis of large-dimensional random matrices, a review. Statist. Sinica 9 611–677.
  • [7] Basak, A., Cook, N. and Zeitouni, O. (2018). Circular law for the sum of random permutation matrices. Electron. J. Probab. 23 Paper No. 33, 51 pp.
  • [8] Basak, A. and Rudelson, M. (2017). The circular law for sparse non-Hermitian matrices. Preprint. Available at arXiv:1707.03675v1.
  • [9] Basak, A. and Rudelson, M. (2017). Invertibility of sparse non-Hermitian matrices. Adv. Math. 310 426–483.
  • [10] Basak, A. and Rudelson, M. (2019). The local circular law for sparse non-Hermitian matrices. In preparation.
  • [11] Benaych-Georges, F. and Knowles, A. (2018). Local semicircle law for Wigner matrices. In Advanced Topics in Random Matrices. Panoramas et Synthéses 53.
  • [12] Bordenave, C., Caputo, P. and Chafaï, D. (2012). Circular law theorem for random Markov matrices. Probab. Theory Related Fields 152 751–779.
  • [13] Bordenave, C., Caputo, P. and Chafaï, D. (2014). Spectrum of Markov generators on sparse random graphs. Comm. Pure Appl. Math. 67 621–669.
  • [14] Bordenave, C. and Chafaï, D. (2012). Around the circular law. Probab. Surv. 9 1–89.
  • [15] Bourgade, P., Yau, H.-T. and Yin, J. (2014). Local circular law for random matrices. Probab. Theory Related Fields 159 545–595.
  • [16] Bryc, W., Dembo, A. and Jiang, T. (2006). Spectral measure of large random Hankel, Markov and Toeplitz matrices. Ann. Probab. 34 1–38.
  • [17] Chatterjee, S. (2005). A simple invariance theorem. Preprint. Available at arXiv:0508.213.
  • [18] Chatterjee, S. (2006). A generalization of the Lindeberg principle. Ann. Probab. 34 2061–2076.
  • [19] Cook, N. (2017). The circular law for random regular digraphs with random edge weights. Random Matrices Theory Appl. 6 1750012, 23.
  • [20] Cook, N. (2017). The circular law for random regular digraphs. Preprint. Available at arXiv:1703.05839.
  • [21] Edelman, A. (1988). Eigenvalues and condition numbers of random matrices. SIAM J. Matrix Anal. Appl. 9 543–560.
  • [22] Esseen, C. G. (1966). On the Kolmogorov–Rogozin inequality for the concentration function. Z. Wahrsch. Verw. Gebiete 5 210–216.
  • [23] Ge, S. (2017). Eigenvalue spacing of i.i.d. random matrices. Preprint.
  • [24] Ginibre, J. (1965). Statistical ensembles of complex, quaternion, and real matrices. J. Math. Phys. 6 440–449.
  • [25] Girko, V. L. (1984). The circular law. Teor. Veroyatn. Primen. 29 669–679.
  • [26] Götze, F. and Tikhomirov, A. (2010). The circular law for random matrices. Ann. Probab. 38 1444–1491.
  • [27] Gross, L. (1975). Logarithmic Sobolev inequalities. Amer. J. Math. 97 1061–1083.
  • [28] Guionnet, A. and Zeitouni, O. (2000). Concentration of the spectral measure for large matrices. Electron. Commun. Probab. 5 119–136.
  • [29] Litvak, A., Lytova, A., Tikhomirov, K., Tomczak-Jaegermann, N. and Youssef, P. (2018). Circular law for sparse random regular digraphs. Preprint. Available at arXiv:1801.05576.
  • [30] Nguyen, H. H. (2014). Random doubly stochastic matrices: The circular law. Ann. Probab. 42 1161–1196.
  • [31] Nguyen, H. H. and Vu, V. H. (2013). Circular law for random discrete matrices of given row sum. J. Comb. 4 1–30.
  • [32] Pastur, L. A. (1972). On the spectrum of random matrices. Theoret. Math. Phys. 10 67–74.
  • [33] Rudelson, M. (2008). Invertibility of random matrices: Norm of the inverse. Ann. of Math. (2) 168 575–600.
  • [34] Rudelson, M. and Vershynin, R. (2008). The Littlewood–Offord problem and invertibility of random matrices. Adv. Math. 218 600–633.
  • [35] Rudelson, M. and Vershynin, R. (2009). Smallest singular value of a random rectangular matrix. Comm. Pure Appl. Math. 62 1707–1739.
  • [36] Rudelson, M. and Vershynin, R. (2016). No-gaps delocalization for general random matrices. Geom. Funct. Anal. 26 1716–1776.
  • [37] Stroock, D. W. (2011). Probability Theory: An Analytic View, 2nd ed. Cambridge Univ. Press, Cambridge.
  • [38] Tao, T. (2012). Topics in Random Matrix Theory. Graduate Studies in Mathematics 132. Amer. Math. Soc., Providence, RI.
  • [39] Tao, T. and Vu, V. (2008). Random matrices: The circular law. Commun. Contemp. Math. 10 261–307.
  • [40] Tao, T. and Vu, V. (2010). Random matrices: Universality of ESDs and the circular law. Ann. Probab. 38 2023–2065.
  • [41] Vershynin, R. (2014). Invertibility of symmetric random matrices. Random Structures Algorithms 44 135–182.
  • [42] Wigner, E. P. (1955). Characteristic vectors of bordered matrices with infinite dimensions. Ann. of Math. (2) 62 548–564.
  • [43] Wigner, E. P. (1958). On the distribution of the roots of certain symmetric matrices. Ann. of Math. (2) 67 325–327.
  • [44] Wood, P. M. (2012). Universality and the circular law for sparse random matrices. Ann. Appl. Probab. 22 1266–1300.