The Annals of Applied Probability

Local inhomogeneous circular law

Johannes Alt, László Erdős, and Torben Krüger

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

Abstract

We consider large random matrices $X$ with centered, independent entries, which have comparable but not necessarily identical variances. Girko’s circular law asserts that the spectrum is supported in a disk and in case of identical variances, the limiting density is uniform. In this special case, the local circular law by Bourgade et al. [Probab. Theory Related Fields 159 (2014) 545–595; Probab. Theory Related Fields 159 (2014) 619–660] shows that the empirical density converges even locally on scales slightly above the typical eigenvalue spacing. In the general case, the limiting density is typically inhomogeneous and it is obtained via solving a system of deterministic equations. Our main result is the local inhomogeneous circular law in the bulk spectrum on the optimal scale for a general variance profile of the entries of $X$.

Article information

Source
Ann. Appl. Probab., Volume 28, Number 1 (2018), 148-203.

Dates
Received: February 2017
Revised: April 2017
First available in Project Euclid: 3 March 2018

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1520046086

Digital Object Identifier
doi:10.1214/17-AAP1302

Mathematical Reviews number (MathSciNet)
MR3770875

Zentralblatt MATH identifier
06873682

Subjects
Primary: 60B20: Random matrices (probabilistic aspects; for algebraic aspects see 15B52)
Secondary: 15B52: Random matrices

Keywords
Circular law local law variance profile

Citation

Alt, Johannes; Erdős, László; Krüger, Torben. Local inhomogeneous circular law. Ann. Appl. Probab. 28 (2018), no. 1, 148--203. doi:10.1214/17-AAP1302. https://projecteuclid.org/euclid.aoap/1520046086


Export citation

References

  • [1] Ajanki, O., Erdős, L. and Krüger, T. (2016). Universality for general Wigner-type matrices. Probab. Theory Related Fields. To appear. DOI:10.1007/s00440-016-0740-2.
  • [2] Ajanki, O., Erdős, L. and Krüger, T. (2015). Quadratic vector equations on complex upper half-plane. Available at arXiv:1506.05095v4.
  • [3] Ajanki, O., Erdős, L. and Krüger, T. (2016). Stability of the matrix Dyson equation and random matrices with correlations. Available at arXiv:1604.08188.
  • [4] Ajanki, O., Erdős, L. and Krüger, T. (2017). Singularities of solutions to quadratic vector equations on the complex upper half-plane. Comm. Pure Appl. Math. 70 1672–1705.
  • [5] Aljadeff, J., Renfrew, D. and Stern, M. (2015). Eigenvalues of block structured asymmetric random matrices. J. Math. Phys. 56 103502.
  • [6] Aljadeff, J., Stern, M. and Sharpee, T. (2015). Transition to chaos in random networks with cell-type-specific connectivity. Phys. Rev. Lett. 114 088101.
  • [7] Alt, J., Erdős, L. and Krüger, T. (2017). Local law for random Gram matrices. Electron. J. Probab. 22 paper no. 25, 41 pp.
  • [8] Alt, J., Erdős, L. Krüger, T. and Nemish, Y. (2017). Location of the spectrum of Kronecker random matrices. Available at arXiv:1706.08343.
  • [9] Bai, Z. D. (1997). Circular law. Ann. Probab. 25 494–529.
  • [10] Bao, Z., Erdős, L. and Schnelli, K. (2016). Local single ring theorem on optimal scale. Available at arXiv:1612.05920.
  • [11] Bordenave, C. and Chafaï, D. (2012). Around the circular law. Probab. Surv. 9 1–89.
  • [12] Bourgade, P., Yau, H.-T. and Yin, J. (2014). Local circular law for random matrices. Probab. Theory Related Fields 159 545–595.
  • [13] Bourgade, P., Yau, H.-T. and Yin, J. (2014). The local circular law II: The edge case. Probab. Theory Related Fields 159 619–660.
  • [14] Cook, N., Hachem, W., Najim, J. and Renfrew, D. (2016). Limiting spectral distribution for non-Hermitian random matrices with a variance profile. Available at arXiv:1612.04428.
  • [15] Cook, N. A. (2016). Lower bounds for the smallest singular value of structured random matrices. Available at arXiv:1608.07347v3.
  • [16] Erdős, L. and Yau, H.-T. (2012). Universality of local spectral statistics of random matrices. Bull. Amer. Math. Soc. (N.S.) 49 377–414.
  • [17] Erdős, L., Yau, H.-T. and Yin, J. (2011). Universality for generalized Wigner matrices with Bernoulli distribution. J. Comb. 2 15–81.
  • [18] Erdős, L., Yau, H.-T. and Yin, J. (2012). Bulk universality for generalized Wigner matrices. Probab. Theory Related Fields 154 341–407.
  • [19] Fey, A., van der Hofstad, R. and Klok, M. J. (2008). Large deviations for eigenvalues of sample covariance matrices, with applications to mobile communication systems. Adv. in Appl. Probab. 40 1048–1071.
  • [20] Girko, V. L. (1984). The circular law. Teor. Veroyatn. Primen. 29 669–679.
  • [21] Girko, V. L. (2001). Theory of Stochastic Canonical Equations: Vol. I and II. Mathematics and Its Applications 535. Kluwer Academic, Dordrecht.
  • [22] Götze, F. and Tikhomirov, A. (2010). The circular law for random matrices. Ann. Probab. 38 1444–1491.
  • [23] Guionnet, A., Krishnapur, M. and Zeitouni, O. (2011). The single ring theorem. Ann. of Math. (2) 174 1189–1217.
  • [24] Helton, J. W., Rashidi Far, R. and Speicher, R. (2007). Operator-valued semicircular elements: Solving a quadratic matrix equation with positivity constraints. Int. Math. Res. Not. IMRN 2007.
  • [25] May, R. M. (1972). Will a large complex system be stable? Nature 238 413–414.
  • [26] Pan, G. and Zhou, W. (2010). Circular law, extreme singular values and potential theory. J. Multivariate Anal. 101 645–656.
  • [27] Rudelson, M. and Vershynin, R. (2015). Delocalization of eigenvectors of random matrices with independent entries. Duke Math. J. 164 2507–2538.
  • [28] Sompolinsky, H., Crisanti, A. and Sommers, H.-J. (1988). Chaos in random neural networks. Phys. Rev. Lett. 61 259–262.
  • [29] Tao, T. and Vu, V. (2008). Random matrices: The circular law. Commun. Contemp. Math. 10 261–307.
  • [30] Tao, T. and Vu, V. (2010). Random matrices: Universality of ESDs and the circular law. Ann. Probab. 38 2023–2065.
  • [31] Tao, T. and Vu, V. (2015). Random matrices: Universality of local spectral statistics of non-Hermitian matrices. Ann. Probab. 43 782–874.
  • [32] Xi, H., Yang, F. and Yin, J. (2017). Local circular law for the product of a deterministic matrix with a random matrix. Electron. J. Probab. 22 paper no. 60, 77 pp.
  • [33] Yin, J. (2014). The local circular law III: General case. Probab. Theory Related Fields 160 679–732.