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Rectangular random matrices, related convolution
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  • Published: 08 April 2008

Rectangular random matrices, related convolution

  • Florent Benaych-Georges1 

Probability Theory and Related Fields volume 144, pages 471–515 (2009)Cite this article

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Abstract

We characterize asymptotic collective behavior of rectangular random matrices, the sizes of which tend to infinity at different rates. It appears that one can compute the limits of all noncommutative moments (thus all spectral properties) of the random matrices we consider because, when embedded in a space of larger square matrices, independent rectangular random matrices are asymptotically free with amalgamation over a subalgebra. Therefore, we can define a “rectangular-free convolution”, which allows to deduce the singular values of the sum of two large independent rectangular random matrices from the individual singular values. This convolution is linearized by cumulants and by an analytic integral transform, that we called the “rectangular R-transform”.

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References

  1. Akhiezer, N.I.: The classical moment problem. Moscou (1961)

  2. Belinschi, S.T.: The Lebesgue decomposition of the free additive convolution of two probability distributions, preprint (2006), ArXiv math.OA/0603104

  3. Belinschi, S.T., Benaych-Georges, F., Guionnet, A.: Regularization by free additive convolution, square and rectangular cases, preprint (2007), ArXiv

  4. Benaych-Georges, F.: Infinitely divisible distributions for rectangular free convolution: classification and matricial interpretation. Probability Theory and Related Fields, vol. 139, nos. 1–2/September 2007, pp. 143–189

  5. Benaych-Georges, F.: Rectangular random matrices, related free entropy and free Fisher’s information. J. Oper. Theory (2008, in press)

  6. Bercovici, H., Pata, V.: with an appendix by Biane, P. Stable laws and domains of attraction in free probability theory. Ann. Math. 149, 1023–1060 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  7. Bercovici, H., Voiculescu, D.: Free convolution of measures with unbounded supports. Indiana Univ. Math. J. 42, 733–773 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  8. Billingsley, P.: Convergence of Probability Measures. Wiley, London (1968)

    MATH  Google Scholar 

  9. Capitaine, M., Casalis, M.: Asymptotic freeness by generalized moments for Gaussian and Wishart matrices. Application to beta random matrices. Indiana Univ. Math. J. 53(2), 397–431 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  10. Capitaine, M., Donati-Martin, C.: Strong asymptotic freeness for Wigner and Wishart matrices, preprint (2005)

  11. Donoghue, W.: Monotone Matrix Functions and Analytic Continuation. Springer, New York (1974)

    MATH  Google Scholar 

  12. Dozier, B., Silverstein, J.: Analysis of the limiting distribution of large dimensional information-plus-noise-type matrices, preprint (2004)

  13. Guionnet, A., Zeitouni, O.: Concentration of the spectral measure for large matrices. Electron. Comm. Probab. 5, 119–136 (2000)

    MATH  MathSciNet  Google Scholar 

  14. Haagerup, U., Larsen, F.: Brown’s spectral distribution measure for R-diagonal elements in finite von Neumann algebras. J. Funct. Anal. 176, 331–367 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  15. Hachem, W., Loubaton, P., Najim, J.: The empirical distribution of the eigenvalues of a Gram matrix with a given variance profile, preprint (2004)

  16. Hachem, W., Loubaton, P., Najim, J.: Deterministic equivalents for certain functionals of large random matrices, preprint, July (2005)

  17. Hiai, F., Petz, D.: The semicircle law, free random variables, and entropy. Am. Math. Soc., Mathematical Surveys and Monographs, vol. 77 (2000)

  18. Horn, R.A., Johnson, C.R.: Topics in Matrix Analysis. Cambridge University Press, Cambridge (1991)

    MATH  Google Scholar 

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

    MATH  Google Scholar 

  20. Nelson, E.: Notes on non-commutative integration. J. Funct. Anal. 15, 103–116 (1974)

    Article  MATH  Google Scholar 

  21. Nica, A., Shlyakhtenko, D.: Speicher, Roland Some minimization problems for the free analogue of the Fisher information. Adv. Math. 141(2), 282–321 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  22. Nica, A., Shlyakhtenko, D.: Speicher, Roland Operator-valued distributions. I. Characterizations of freeness. Int. Math. Res. Notes 29, 1509–1538 (2002)

    Article  MathSciNet  Google Scholar 

  23. Pastur, L., Lejay, A.: Matrices aléatoires: statistique asymptotique des valeurs propres. Séminaire de Probabilités XXXVI, Lectures Notes in Mathematics, vol. 1801, Springer, Heidelberg (2002)

  24. Rota, Gian-Carlo On the foundations of combinatorial theory. I. Theory of Möbius functions. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, vol. 2, pp. 340–368 (1964)

  25. Shlyakhtenko, D.: Random Gaussian band matrices and freeness with amalgamation. Int. Math. Res. Notices 20, 1013–1025 (1996)

    Article  MathSciNet  Google Scholar 

  26. Speicher, R.: Multiplicative functions on the lattice of non-crossing partitions and free convolution. Math. Ann. 298, 611–628 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  27. Speicher, R.: Combinatorial theory of the free product with amalgamation and operator-valued free probability theory. Mem. Am. Math. Soc. 132, no. 627 (1998)

    Google Scholar 

  28. Speicher, R.: Notes of my lectures on Combinatorics of Free Probability (IHP, Paris, 1999). Available on http://www.mast.queensu.ca/~speicher (1999)

  29. Śniady, P., Speicher, R.: Continuous family of invariant subspaces for R-diagonal operators. Invent. Math. 146, 329–363 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  30. Voiculescu, D.: Limit laws for random matrices and free products. Invent. Math. 104(1), 201–220 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  31. Voiculescu, D.: Operations on certain non-commutative operator-valued random variables. Recent advances in operator algebras (Orléans, 1992). AstéRisque 232, 243–275 (1995)

    MathSciNet  Google Scholar 

  32. Voiculescu, D.: A strengthened asymptotic freeness result for random matrices with applications to free entropy. Int. Math. Res. Notices 1, 41–63 (1998)

    Article  MathSciNet  Google Scholar 

  33. Voiculescu, D.V., Dykema, K., Nica, A.: Free random variables. CRM Monograghs Series No. 1, Am. Math. Soc., Providence, RI (1992)

  34. Wigner, E.: On the distribution of the roots of certain symmetric matrices. Ann. Math. 67, 325–327 (1958)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. LPMA, UPMC Univ Paris 6, Case Courier 188, 4 Place Jussieu, 75252, Paris Cedex 05, France

    Florent Benaych-Georges

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  1. Florent Benaych-Georges
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Correspondence to Florent Benaych-Georges.

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Benaych-Georges, F. Rectangular random matrices, related convolution. Probab. Theory Relat. Fields 144, 471–515 (2009). https://doi.org/10.1007/s00440-008-0152-z

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  • Received: 01 August 2007

  • Revised: 04 March 2008

  • Published: 08 April 2008

  • Issue Date: July 2009

  • DOI: https://doi.org/10.1007/s00440-008-0152-z

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Keywords

  • Random matrices
  • Free probability
  • Free convolution

Mathematics Subject Classification (2000)

  • 15A52
  • 46L54
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