Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Location of the spectrum of Kronecker random matrices

Johannes Alt, László Erdős, Torben Krüger, and Yuriy Nemish

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For a general class of large non-Hermitian random block matrices ${\boldsymbol X}$ we prove that there are no eigenvalues away from a deterministic set with very high probability. This set is obtained from the Dyson equation of the Hermitization of ${\boldsymbol X}$ as the self-consistent approximation of the pseudospectrum. We demonstrate that the analysis of the matrix Dyson equation from (Probab. Theory Related Fields (2018)) offers a unified treatment of many structured matrix ensembles.


Pour une classe générale de grandes matrices aléatoires par blocs non hermitiennes ${\boldsymbol X}$, nous montrons qu’avec très grande probabilité, il n’y a pas de valeurs propres en dehors d’un ensemble déterministe. Cet ensemble est obtenu à partir de l’équation de Dyson pour l’hermitisation de ${\boldsymbol X}$ comme l’approximation auto-cohérente du pseudo-spectre. Nous démontrons que l’analyse de l’équation de Dyson provenant de (Probab. Theory Related Fields (2018)) permet d’étudier de façon unifiée de nombreux ensembles de matrices structurées.

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 55, Number 2 (2019), 661-696.

Received: 15 July 2017
Revised: 25 January 2018
Accepted: 23 February 2018
First available in Project Euclid: 14 May 2019

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Outliers Block matrices Local law Non-Hermitian random matrix Self-consistent pseudospectrum


Alt, Johannes; Erdős, László; Krüger, Torben; Nemish, Yuriy. Location of the spectrum of Kronecker random matrices. Ann. Inst. H. Poincaré Probab. Statist. 55 (2019), no. 2, 661--696. doi:10.1214/18-AIHP894.

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  • [1] Y. Ahmadian, F. Fumarola and K. D. Miller. Properties of networks with partially structured and partially random connectivity. Phys. Rev. E 91 (2015) 012820.
  • [2] O. Ajanki, L. Erdős and T. Krüger. Universality for general Wigner-type matrices. Probab. Theory Related Fields 169 (3–4) (2017) 667–727.
  • [3] O. Ajanki, L. Erdős and T. Krüger. Stability of the matrix Dyson equation and random matrices with correlations. Probab. Theory Related Fields (2018). doi:10.1007/s00440-018-0835-z.
  • [4] A. Aljadeff, D. Renfrew and M. Stern. Eigenvalues of block structured asymmetric random matrices. J. Math. Phys. 56 (10) (2015) 103502.
  • [5] J. Aljadeff, M. Stern and T. Sharpee. Transition to chaos in random networks with cell-type-specific connectivity. Phys. Rev. Lett. 114 (2015) 088101.
  • [6] J. Alt, L. Erdős and T. Krüger. Local inhomogeneous circular law. Ann. Appl. Probab. 28 (1) (2018) 148–203.
  • [7] J. Alt, L. Erdős and T. Krüger. Local law for random Gram matrices. Electron. J. Probab. 22 (2017) no. 25, 41 pp.
  • [8] G. W. Anderson. Convergence of the largest singular value of a polynomial in independent Wigner matrices. Ann. Probab. 41 (2013) 2103–2181.
  • [9] G. W. Anderson, A. Guionnet and O. Zeitouni. An Introduction to Random Matrices. Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2010.
  • [10] Z. Bai and J. W. Silverstein. No eigenvalues outside the support of the limiting spectral distribution of information-plus-noise type matrices. Random Matrices Theory Appl. 1 (1) (2012) 1150004, 44.
  • [11] Z. D. Bai and Y. Q. Yin. Limiting behavior of the norm of products of random matrices and two problems of Geman–Hwang. Probab. Theory Related Fields 73 (4) (1986) 555–569.
  • [12] Z. D. Bai and Y. Q. Yin. Necessary and sufficient conditions for almost sure convergence of the largest eigenvalue of a Wigner matrix. Ann. Probab. 16 (4) (1988) 1729–1741.
  • [13] S. T. Belinschi and M. Capitaine. Spectral properties of polynomials in independent Wigner and deterministic matrices. J. Funct. Anal. 273 (12) (2017) 3901–3963.
  • [14] C. Bordenave and M. Capitaine. Outlier eigenvalues for deformed i.i.d. random matrices. Comm. Pure Appl. Math. 69 (11) (2016) 2131–2194.
  • [15] C. Bordenave, P. Caputo, D. Chafaï and K. Tikhomirov. On the spectral radius of a random matrix. Preprint, 2016. Available at arXiv:1607.05484.
  • [16] C. Bordenave and D. Chafaï. Around the circular law. Probab. Surv. 9 (2012) 1–89.
  • [17] P. Bourgade, H.-T. Yau and J. Yin. Local circular law for random matrices. Probab. Theory Related Fields 159 (3–4) (2014) 545–595.
  • [18] M. Capitaine and C. Donati-Martin. Strong asymptotic freeness for Wigner and Wishart matrices. Indiana Univ. Math. J. 56 (2007) 767–804.
  • [19] L. Erdős, A. Knowles, H.-T. Yau and J. Yin. The local semicircle law for a general class of random matrices. Electron. J. Probab. 18 (2013) no. 59, 1–58.
  • [20] L. Erdős, T. Krüger and Yu. Nemish. Local spectral analysis of polynomials in random matrices. In preparation, 2018.
  • [21] L. Erdős, T. Krüger and D. Schröder. Random matrices with slow correlation decay. Preprint, 2017. Available at arXiv:1705.10661.
  • [22] L. Erdős, H.-T. Yau and J. Yin. Universality for generalized Wigner matrices with Bernoulli distribution. J. Comb. 2 (1) (2011) 15–82.
  • [23] S. Geman. The spectral radius of large random matrices. Ann. Probab. 14 (4) (1986) 1318–1328.
  • [24] V. L. Girko. Theory of Stochastic Canonical Equations, Vols. I and II. Mathematics and Its Applications. Springer, Netherlands, 2012.
  • [25] U. Haagerup and S. Thorbjørnsen. A new application of random matrices: $\operatorname{Ext}(C^{*}_{\mathrm{red}}(F_{2}))$ is not a group. Ann. of Math. 162 (2) (2005) 711–775.
  • [26] H. M. Hastings, F. Juhasz and M. A. Schreiber. Stability of structured random matrices. Proc. Biol. Sci. 249 (1326) (1992) 223–225.
  • [27] J. W. Helton, R. Rashidi Far and R. Speicher. Operator-valued semicircular elements: Solving a quadratic matrix equation with positivity constraints. Int. Math. Res. Not. 2007 (2007) Art. ID rnm086.
  • [28] B. Khoruzhenko. Large-$N$ eigenvalue distribution of randomly perturbed asymmetric matrices. J. Phys. A 29 (7) (1996) L165–L169.
  • [29] R. M. May. Will a large complex system be stable? Nature 238 (1972) 413–414.
  • [30] K. Rajan and L. F. Abbott. Eigenvalue spectra of random matrices for neural networks. Phys. Rev. Lett. 97 (2006) 188104.
  • [31] T. Tao. Outliers in the spectrum of iid matrices with bounded rank perturbations. Probab. Theory Related Fields 155 (1) (2013) 231–263.
  • [32] T. Tao, V. Vu and M. Krishnapur. Random matrices: Universality of ESDs and the circular law. Ann. Probab. 38 (5) (2010) 2023–2065.
  • [33] E. P. Wigner. Characteristic vectors of bordered matrices with infinite dimensions. Ann. of Math. 62 (3) (1955) 548–564.