Skip to main content
SpringerLink
Log in

Spectrum of Non-Hermitian Heavy Tailed Random Matrices

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

Let (X jk ) j,k ≥ 1 be i.i.d. complex random variables such that |X jk | is in the domain of attraction of an α-stable law, with 0 < α < 2. Our main result is a heavy tailed counterpart of Girko’s circular law. Namely, under some additional smoothness assumptions on the law of X jk , we prove that there exist a deterministic sequence a n ~ n 1/α and a probability measure μ α on \({\mathbb{C}}\) depending only on α such that with probability one, the empirical distribution of the eigenvalues of the rescaled matrix \({(a_n^{-1}X_{jk})_{1\leq j,k\leq n}}\) converges weakly to μ α as n → ∞. Our approach combines Aldous & Steele’s objective method with Girko’s Hermitization using logarithmic potentials. The underlying limiting object is defined on a bipartized version of Aldous’ Poisson Weighted Infinite Tree. Recursive relations on the tree provide some properties of μ α . In contrast with the Hermitian case, we find that μ α is not heavy tailed.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Aldous D.: Asymptotics in the random assignment problem. Probab. Th. Rel. Fields 93(4), 507– 534 (1982)

    Article  MathSciNet  Google Scholar 

  2. Aldous D., Lyons R.: Processes on unimodular random networks. Electron. J. Probab 12(54), 1454–1508 (2007) (electronic)

    MathSciNet  MATH  Google Scholar 

  3. Aldous, D., Steele, J.M.: The objective method: probabilistic combinatorial optimization and local weak convergence. Probability on discrete structures, Encyclopaedia Math. Sci., Vol. 110, Berlin: Springer, 2004, pp. 1–72

  4. Bai Z.D.: Circular law. Ann. Probab. 25(1), 494–529 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bai, Z.D., Silverstein, J.W.: Spectral Analysis of Large Dimensional Random Matrices. Mathematics Monograph Series 2, Beijing: Science Press, 2006

  6. Belinschi S., Dembo A., Guionnet A.: Spectral measure of heavy tailed band and covariance random matrices. Commun. Math. Phys. 289(3), 1023–1055 (2009)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  7. Ben Arous G., Guionnet A.: The spectrum of heavy tailed random matrices. Commun. Math. Phys. 278(3), 715–751 (2008)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  8. Benjamini I., Schramm O.: Recurrence of distributional limits of finite planar graphs. Electron. J. Probab 6(23), 13 (2001) (electronic)

    MathSciNet  Google Scholar 

  9. Bingham N.H., Goldie C.M., Teugels J.L.: Regular variation. Encyclopedia of Mathematics and its Applications, Vol. 27. Cambridge University Press, Cambridge (1989)

    Google Scholar 

  10. Bordenave Ch., Caputo P., Chafaï D.: Spectrum of large random reversible Markov chains: two examples. ALEA Lat. Am. J. Probab. Math. Stat. 7, 41–64 (2010)

    MathSciNet  Google Scholar 

  11. Bordenave, Ch., Caputo, P., Chafaï, D.: Spectrum of large random reversible Markov chains: heavy tailed weigths on the complete graph. http://arXiv.org/abs/0903.3528v4 Ann. Prob. 39(4), 1544–1590 (2011).

  12. Bordenave, Ch., Caputo, P., Chafaï, D.: Circular Law Theorem for Random Markov Matrices. Prob. Th. Rel. Fields, doi:10.1007/s00440-010-0336-1, 2011

  13. Bordenave Ch., Lelarge M., Salez J.: The rank of diluted random graphs. Ann. Prob. 39(3), 1097–1121 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  14. Bouchaud J., Cizeau P.: Theory of Lévy matrices. Phys. Rev. E 3, 1810–1822 (1994)

    Google Scholar 

  15. Brown, L.G.: Lidskiĭ’s theorem in the type II case. In: Geometric methods in operator algebras (Kyoto, 1983), Pitman Res. Notes Math. Ser., Vol. 123, Harlow: Longman Sci. Tech., 1986, pp. 1–35

  16. Chafaï D.: Aspects of large random Markov kernels. Stochastics 81(3-4), 415–429 (2009)

    MathSciNet  MATH  Google Scholar 

  17. Chafaï D.: Circular law for noncentral random matrices. J. Theoret. Probab. 23(4), 945–950 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  18. Chafaï D.: The Dirichlet Markov ensemble. J. Multivariate Anal. 101(3), 555–567 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  19. Dozier R.B., Silverstein J.W.: Analysis of the limiting spectral distribution of large dimensional information-plus-noise type matrices. J. Multivariate Anal. 98(6), 1099–1122 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  20. Dozier R.B., Silverstein J.W.: On the empirical distribution of eigenvalues of large dimensional information-plus-noise-type matrices. J. Multivariate Anal. 98(4), 678–694 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  21. Edelman A.: The probability that a random real Gaussian matrix has k real eigenvalues, related distributions, and the circular law. J. Multivariate Anal. 60(2), 203–232 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  22. Feinberg J., Zee A.: Non-Hermitian random matrix theory: Method of Hermitian reduction. Nucl. Phys. B 504(3), 579–608 (1997)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  23. Feller, W.: An introduction to probability theory and its applications. Vol. II. Second edition, New York: John Wiley & Sons Inc., 1971

  24. Girko V.L.: The circular law. Teor. Veroyatnost. i Primenen. 29(4), 669–679 (1984)

    MathSciNet  MATH  Google Scholar 

  25. Girko V.L.: Strong circular law. Random Oper. Stochastic Eqs. 5(2), 173–196 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  26. Girko V.L.: The circular law. Twenty years later. III. Random Oper. Stochastic Eqs. 13(1), 53–109 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  27. Goldsheid I.Y., Khoruzhenko B.A.: The Thouless formula for random non-Hermitian Jacobi matrices. Israel J. Math. 148, 331–346 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  28. Götze F., Tikhomirov A.: The Circular Law for Random Matrices. Ann. Probab. 38(4), 1444–1491 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  29. Gudowska-Nowak E., Jarosz A., Nowak M., Pappe G.: Towards non-Hermitian random Lévy matrices. Acta Physica Polonica B 38(13), 4089–4104 (2007)

    MathSciNet  ADS  Google Scholar 

  30. Guntuboyina A., Leeb H.: Concentration of the spectral measure of large Wishart matrices with dependent entries. Electron. Commun. Probab. 14, 334–342 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  31. Haagerup U., Schultz H.: Brown measures of unbounded operators affiliated with a finite von Neumann algebra. Math. Scand. 100(2), 209–263 (2007)

    MathSciNet  MATH  Google Scholar 

  32. Horn, R.A., Johnson, Ch.R.: Topics in matrix analysis. Cambridge: Cambridge University Press, 1994 (corrected reprint of the 1991 original)

  33. Hwang, C.-R.: A brief survey on the spectral radius and the spectral distribution of large random matrices with i.i.d. entries. In: Random matrices and their applications (Brunswick, Maine, 1984), Contemp. Math., Vol. 50, Providence, RI: Amer. Math. Soc., 1986, pp. 145–152

  34. Ledoux, M.: The concentration of measure phenomenon. Mathematical Surveys and Monographs, Vol. 89, Providence, RI: Amer. Math. Soc., 2001

  35. LePage R., Woodroofe M., Zinn J.: Convergence to a stable distribution via order statistics. Ann. Probab. 9(4), 624–632 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  36. Lyons R.: Identities and Inequalities for Tree Entropy. Combin. Probab. Comput. 19(2), 303–313 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  37. Marchenko V.A., Pastur L.A.: The distribution of eigenvalues in sertain sets of random matrices. Mat. Sb. 72, 507–536 (1967)

    MathSciNet  Google Scholar 

  38. McDiarmid, C.: On the method of bounded differences. Surveys in combinatorics, (Norwich, 1989), London Math. Soc. Lecture Note Ser., Vol. 141, Cambridge: Cambridge Univ. Press, 1989, pp. 148–188

  39. Mehta M.L.: Random matrices and the statistical theory of energy levels. Academic Press, New York (1967)

    MATH  Google Scholar 

  40. Pan G.M., Zhou W.: Circular law, extreme singular values and potential theory. J. Multivar. Anal. 101(3), 645–656 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  41. Panchenko D., Talagrand M.: On one property of Derrida-Ruelle cascades. C. R. Math. Acad. Sci. Paris 345(11), 653–656 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  42. Reed M., Simon B.: Methods of modern mathematical physics I Second ed. Academic Press Inc, New York (1980) [Harcourt Brace Jovanovich Publishers]

    Google Scholar 

  43. Rogers T.: Universal sum and product rules for random matrices. J. Math. Phys. 51, 093304 (2010)

    Article  MathSciNet  ADS  Google Scholar 

  44. Rogers T., Castillo I.P.: Cavity approach to the spectral density of non-Hermitian sparse matrices. Phys. Rev. E 79, 012101 (2009)

    Google Scholar 

  45. Rudelson M., Vershynin R.: The Littlewood-Offord problem and invertibility of random matrices. Adv. Math. 218(2), 600–633 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  46. Talagrand M.: Concentration of measure and isoperimetric inequalities in product spaces. Inst. Hautes Études Sci. Publ. Math. 81(1), 73–205 (1995)

    Article  MATH  Google Scholar 

  47. Tao, T.: Outliers in the spectrum of iid matrices with bounded rank perturbations. http://arXiv.org/abs/1012.4818v3 [math.PR], 2011

  48. Tao T., Vu V.: Random matrices: the circular law. Commun. Contemp. Math. 10(2), 261–307 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  49. Tao T., Vu V.: Smooth analysis of the condition number and the least singular value. Math. Comp. 79(272), 2333–2352 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  50. Tao T., Vu V.: Random matrices: universality of ESDs and the circular law, with an appendix by Manjunath Krishnapur. Ann. Probab 38(5), 2023–2065 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  51. Thompson R.C.: The behavior of eigenvalues and singular values under perturbations of restricted rank. Linear Algebra and Appl 13(1/2), 69–78 (1976) (collection of articles dedicated to Olga Taussky Todd)

    Article  MathSciNet  MATH  Google Scholar 

  52. Wachter K.W.: The strong limits of random matrix spectra for sample matrices of independent elements. Ann. Prob. 6(1), 1–18 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  53. Weyl H.: Inequalities between the two kinds of eigenvalues of a linear transformation. Proc. Nat. Acad. Sci. U. S. A. 35, 408–411 (1949)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  54. Yin Y.Q.: Limiting spectral distribution for a class of random matrices. J. Multivariate Anal. 20(1), 50–68 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  55. Zhan, X.: Matrix inequalities. Lecture Notes in Mathematics, Vol. 1790, Berlin: Springer-Verlag, 2002

  56. Zolotarev, V.M.: One-dimensional stable distributions. In: Translations of Mathematical Monographs, Vol. 65, Providence, RI: Amer. Math. Soc., 1986, Translated from the Russian by H. H. McFaden, Translation edited by Ben Silver

Download references

Author information

Authors and Affiliations

  1. IMT UMR 5219 CNRS and Université Paul-Sabatier Toulouse III, Toulouse, France

    Charles Bordenave

  2. Dipartimento di Matematica, Università Roma Tre, Rome, Italy

    Pietro Caputo

  3. LAMA UMR 8050 CNRS and Université Paris-Est Marne-la-Vallée, Paris, France

    Djalil Chafaï

Authors
  1. Charles Bordenave
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Pietro Caputo
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Djalil Chafaï
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Djalil Chafaï.

Additional information

Communicated by P. Forrester

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bordenave, C., Caputo, P. & Chafaï, D. Spectrum of Non-Hermitian Heavy Tailed Random Matrices. Commun. Math. Phys. 307, 513–560 (2011). https://doi.org/10.1007/s00220-011-1331-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-011-1331-9

Keywords

Search

Navigation