Products of Independent non-Hermitian Random Matrices

Sean O'Rourke (Rutgers University)
Alexander B. Soshnikov (University of California Davis)

Abstract


We consider the product of a finite number of non-Hermitian random matrices with i.i.d. centered entries of growing size. We assume that the entries have a finite moment of order bigger than two. We show that the empirical spectral distribution of the properly normalized product converges, almost surely, to a non-random, rotationally invariant distribution with compact support in the complex plane. The limiting distribution is a power of the circular law.

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Pages: 2219-2245

Publication Date: November 15, 2011

DOI: 10.1214/EJP.v16-954

References

  1. N. Alexeev, F. G\"{o}tze, A. Tikhomirov. On the singular spectrum of powers and products of random matrices, Doklady Mathematics 82 (2010), 505-507. Math. Review number not available.
  2. N. Alexeev, F. G\"{o}tze, A. Tikhomirov. On the asymptotic distribution of singular values of power of random matrices. Lithuanian Math. Journal, 50 (2010), 121-132. Math. Review 2011c:15102
  3. Z. D. Bai. Circular law. Ann. Probab. 25 (1997), 494--529. Math. Review 98k:60040
  4. Z. D. Bai, Y.Q. Yin. Limiting behavior of the norm of products of random matrices and two problems of Geman-Hwang. Prob. Th. Rel. Fields 73 (1986), 555-569. Math. Review 88e:60041
  5. Z. D. Bai, B. Miao, B. Jin. On the limit theorem for the eigenvalues of product of two matrices. J. Multivariate Anal. 98 (2007), 76-101. Math. Review 2008d:60041
  6. Z. D. Bai, J. Silverstein. Spectral analysis of large dimensional random matrices. Springer Series in Statistics (2010) Springer. Math. Review 2011d:60014
  7. T. Banica, S. Belinschi, M. Capitaine, B. Collins. Free Bessel laws. available at arXiv:0710.5931. Math. Review number not available.
  8. Z. Burda, R. A. Janik,and B. Waclaw. Spectrum of the product of independent random Gaussian matrices. Phys. Rev. E 81 (2010), 041132. Math. Review 2011g:60013
  9. Z. Burda et.al. Eigenvalues and singular values of products of rectangular Gaussian random matrices. Phys. Rev. E 82 (2010), 061114. Math. Review MR2787469
  10. J. Ginibre. Statistical Ensembles of Complex, Quaternion, and Real Matrices. Journal of Mathematical Physics 6 (1965), 440-449. Math. Review MR0173726
  11. V. L. Girko. Circular law. Theory Probab. Appl. 29 (1984), 694--706. Math. Review 87c:15042
  12. V. L. Girko. The strong circular law. Twenty years later. II. Random Oper. Stochastic Equations 12 (2004), 255--312. Math. Review 2006e:60045
  13. V. L. Girko, A.I. Vladimirova. Spectral analysis of stochastic recurrence systems of growing dimension under G-condition. Canonical equation $K_{91}$. Random Oper. Stochastic Equations 17 (2009), 243--274. Math. Review 2010m:60020
  14. F. G\"{o}tze, A. Tikhomirov. The circular law for random matrices. Annals of Probab. 38 (2010), 1444-1491. Math. Review MR2663633
  15. F. G\"{o}tze, A. Tikhomirov. On the circular law. available at arXiv:0702386. Math. Review number not available.
  16. F. G\"{o}tze, A. Tikhomirov. On the asymptotic of spectrum of products of independent random matrices. available at arXiv:1012.2710v1 math.PR. Math. Review number not available.
  17. R. Horn, Ch. Johnson. Topics in Matrix analysis (1991) Cambridge University Press. Math. Review 92e:15003
  18. A.M.Khorunzhy, B.A. Khoruzhenko, and L.A. Pastur. Asymptotic properties of large random matrices with independent entries. J. Math. Phys. 37 (1996), 5033--5059. Math. Review 97j:82082
  19. R. Latala. Some estimates of norms of random matrices. Proc. Amer. Math. Soc. 133 (2005), 1273?1282. Math. Review 2005i:15041
  20. V. Marchenko, L. Pastur. Distribution of eigenvalues of some sets of random matrices. Math USSR-Sb. 1 (1967), 457--486. Math. Review MR0208649
  21. C. McDiarmid. On the Method of Bounded Differences. Surveys in Combinatorics 141 (1989), 148-188. Math. Review 91e:05077
  22. M.L. Mehta. Random Matrices and Statistical Theory of Energy Levels (1967) Academic Press. Math. Review MR0220494
  23. M.L. Mehta. Random Matrices Pure and Applied Mathematics 142 (2004) Academic Press. Math. Review 2006b:82001
  24. J.A. Mingo, R.Speicher. Sharp bounds for sums associated to graphs of matrices. available at arXiv:0909.4277. Math. Review number not available.
  25. G. Pan, W. Zhou. Circular law, extreme singular values and potential theory. J. Multivariate Anal. 101 (2010), 645-656. Math. Review 2011a:60033
  26. M. Rudelson. Invertibility of random matrices: Norm of the inverse. Ann. of Math. 168 (2008), 575-600 (2008). Math. Review 2009i:60104
  27. M. Rudelson, R. Vershynin. The Littlewood-Offord problem and invertibility of random matrices. Adv. Math. 218 (2008), 600-633. Math. Review 2010g:60048
  28. T. Tao, V. Vu. Random Matrices: the Circular Law. Commun. Contemp. Math. 10, (2008), 261-307 (2008). Math. Review 2009d:60091
  29. T. Tao, V. Vu. Random Matrices: Universality of ESDs and the Circular Law. Annals of Probab. 38 (2010), 2023-2065. Math. Review 2011e:60017
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