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Circular law theorem for random Markov matrices
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  • Published: 12 January 2011

Circular law theorem for random Markov matrices

  • Charles Bordenave1,
  • Pietro Caputo2 &
  • Djalil Chafaï3 

Probability Theory and Related Fields volume 152, pages 751–779 (2012)Cite this article

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  • 37 Citations

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Abstract

Let (X jk ) jk≥1 be i.i.d. nonnegative random variables with bounded density, mean m, and finite positive variance σ 2. Let M be the n × n random Markov matrix with i.i.d. rows defined by \({M_{jk}=X_{jk}/(X_{j1}+\cdots+X_{jn})}\). In particular, when X 11 follows an exponential law, the random matrix M belongs to the Dirichlet Markov Ensemble of random stochastic matrices. Let λ1, . . . , λ n be the eigenvalues of \({\sqrt{n}M}\) i.e. the roots in \({\mathbb{C}}\) of its characteristic polynomial. Our main result states that with probability one, the counting probability measure \({\frac{1}{n}\delta_{\lambda_1}+\cdots+\frac{1}{n}\delta_{\lambda_n}}\) converges weakly as n→∞ to the uniform law on the disk \({\{z\in\mathbb{C}:|z|\leq m^{-1}\sigma\}}\). The bounded density assumption is purely technical and comes from the way we control the operator norm of the resolvent.

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References

  1. Adamczak R., Guédon O., Litvak A., Pajor A., Tomczak-Jaegermann N.: Smallest singular value of random matrices with independent columns. C. R. Math. Acad. Sci. Paris 346(15–16), 853–856 (2008)

    MathSciNet  MATH  Google Scholar 

  2. Akemann G., Kanzieper E.: Integrable structure of Ginibre’s ensemble of real random matrices and a Pfaffian integration theorem. J. Stat. Phys. 129(5–6), 1159–1231 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  3. Aubrun G.: Random points in the unit ball of \({l^n_p}\). Positivity 10(4), 755–759 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  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. Science Press, Beijing (2006)

    Google Scholar 

  6. Bai Z.D., Silverstein J.W., Yin Y.Q.: A note on the largest eigenvalue of a large-dimensional sample covariance matrix. J. Multivar. Anal. 26(2), 166–168 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bai Z.D., Yin Y.Q.: Limiting behavior of the norm of products of random matrices and two problems of Geman–Hwang. Probab. Theory Relat. Fields 73(4), 555–569 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bai Z.D., Yin Y.Q.: Limit of the smallest eigenvalue of a large-dimensional sample covariance matrix. Ann. Probab. 21(3), 1275–1294 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  9. Biane Ph., Lehner F.: Computation of some examples of Brown’s spectral measure in free probability. Colloq. Math. 90(2), 181–211 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  10. Bollobás, B.: Random graphs. Cambridge Studies in Advanced Mathematics, vol. 73, 2nd edn. Cambridge University Press, Cambridge (2001)

  11. Bordenave, Ch., Caputo, P., Chafaï, D.: Spectrum of large random reversible Markov chains: Heavy-tailed weigths on the complete graph. Ann. Probab. (2010). Preprint (accepted). arXiv:0903.3528 [math. PR]

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

  13. Bordenave, Ch., Caputo, P., Chafaï, D.: Spectrum of non-Hermitian heavy tailed random matrices (2010, preprint)

  14. Brown, L.G.: Lidskiĭ’s theorem in the type II case. Geometric Methods in Operator Algebras (Kyoto, 1983). Pitman Res. Notes Math. Ser., vol. 123, pp. 1–35. Longman Sci. Tech. Harlow (1986)

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

    MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  17. Chafaï D.: The Dirichlet Markov ensemble. J. Multivar. Anal. 101, 555–567 (2010)

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  20. Erdős P., Rényi A.: On the evolution of random graphs. Magyar Tud. Akad. Mat. Kutató Int. Közl 5, 17–61 (1960)

    Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  22. Girko, V.L.: Theory of random determinants. Mathematics and its Applications (Soviet Series), vol. 45. Kluwer, Dordrecht (1990, Translated from the Russian)

  23. Girko V.L.: Strong circular law. Random Oper. Stoch. Equ. 5(2), 173–196 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  24. Girko V.L.: The circular law. Twenty years later. III. Random Oper. Stoch. Equ. 13(1), 53–109 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  25. Goldberg G., Neumann M.: Distribution of subdominant eigenvalues of matrices with random rows. SIAM J. Matrix Anal. Appl. 24(3), 747–761 (2003) electronic

    Article  MathSciNet  MATH  Google Scholar 

  26. Götze F., Tikhomirov A.: The circular law for random matrices. Ann. Probab. 38(4), 1444–1491 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  27. Horn A.: On the eigenvalues of a matrix with prescribed singular values. Proc. Am. Math. Soc. 5, 4–7 (1954)

    Article  MathSciNet  MATH  Google Scholar 

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

  29. Hwang, C.-R.: A brief survey on the spectral radius and the spectral distribution of large random matrices with i.i.d. entries. Random Matrices and their Applications (Brunswick, Maine, 1984). Contemp. Math, vol. 50. American Mathematical Society, Providence, RI, pp. 145–152 (1986)

  30. Ledoux, M.: The concentration of measure phenomenon. Mathematical Surveys and Monographs, vol. 89. American Mathematical Society, Providence, RI (2001)

  31. 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 

  32. Mehta M.L.: Random Matrices and the Statistical Theory of Energy Levels. Academic Press, New York (1967)

    MATH  Google Scholar 

  33. 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 

  34. 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 

  35. Saff, E.B., Totik, V.: Logarithmic potentials with external fields, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 316. Springer, Berlin (1997, Appendix B by Thomas Bloom)

  36. Seneta, E.: Non-negative matrices and Markov chains. Springer Series in Statistics. Springer, New York (2006, Revised reprint of the second (1981) edition [Springer-Verlag, New York; MR0719544])

  37. Silverstein J.W.: The spectral radii and norms of large-dimensional non-central random matrices. Commun. Stat. Stoch. Models 10(3), 525–532 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  38. Śniady P.: Random regularization of Brown spectral measure. J. Funct. Anal 193(2), 291–313 (2002)

    Article  MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

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

  43. Trefethen, L.N., Embree, M.: Spectra and pseudospectra. The Behavior of Nonnormal Matrices and Operators. Princeton University Press, Princeton (2005)

  44. Voiculescu D.: Free entropy. Bull. Lond. Math. Soc 34(3), 257–278 (2002)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Institut de Mathématiques de Toulouse, UMR 5219 CNRS, Université de Toulouse, 118 route de Narbonne, 31062, Toulouse, France

    Charles Bordenave

  2. Dipartimento di Matematica, Università Roma Tre, Largo San Murialdo 1, 00146, Rome, Italy

    Pietro Caputo

  3. UMR 8050 CNRS, Université Paris-Est Marne-la-Vallée, 5 boulevard Descartes, 77454, Champs-sur-Marne, France

    Djalil Chafaï

Authors
  1. Charles Bordenave
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  2. Pietro Caputo
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  3. Djalil Chafaï
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Correspondence to Djalil Chafaï.

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Bordenave, C., Caputo, P. & Chafaï, D. Circular law theorem for random Markov matrices. Probab. Theory Relat. Fields 152, 751–779 (2012). https://doi.org/10.1007/s00440-010-0336-1

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  • Received: 12 May 2010

  • Revised: 03 December 2010

  • Published: 12 January 2011

  • Issue Date: April 2012

  • DOI: https://doi.org/10.1007/s00440-010-0336-1

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Keywords

  • Random matrices
  • Eigenvalues
  • Spectrum
  • Stochastic matrices
  • Markov chains

Mathematics Subject Classification (2000)

  • 15A52
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