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Journal of Theoretical Probability

, Volume 32, Issue 2, pp 1051–1075 | Cite as

Asymptotic Expansion of Spherical Integral

  • Jiaoyang HuangEmail author
Article
  • 28 Downloads

Abstract

We consider the spherical integral of real symmetric or Hermitian matrices when the rank of one matrix is one. We prove the existence of the full asymptotic expansions of these spherical integrals and derive the first and the second term in the asymptotic expansion.

Keywords

Random matrices Spherical integral Asymptotic expansion Free probability 

Mathematics Subject Classification (2010)

15B52 46L54 

Notes

Acknowledgements

This research was conducted at the Undergraduate Research Opportunities Program of the MIT Mathematics Department, under the direction of Prof. Alice Guionnet. I would like to express to her my warmest thanks both for introducing me to this problem and for her dedicated guidance throughout the research process. I want to also thank the anonymous reviewer for careful reading of the manuscript and helpful suggestions.

References

  1. 1.
    Anderson, G.W., Guionnet, A., Zeitouni, O.: An Introduction to Random Matrices, volume 118 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (2010)Google Scholar
  2. 2.
    Duistermaat, J.J., Heckman, G.J.: On the variation in the cohomology of the symplectic form of the reduced phase space. Invent. Math. 69(2), 259–268 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Gorin, V., Panova, G.: Asymptotics of symmetric polynomials with applications to statistical mechanics and representation theory. Ann. Probab. 43(6), 3052–3132 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Guionnet, A., Maï da, M.: A Fourier view on the \(R\)-transform and related asymptotics of spherical integrals. J. Funct. Anal. 222(2), 435–490 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Guionnet, A., Zeitouni, O.: Large deviations asymptotics for spherical integrals. J. Funct. Anal. 188(2), 461–515 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Guionnet, A., Zeitouni, O.: Addendum to: “Large deviations asymptotics for spherical integrals” [J. Funct. Anal. 188 (2002), no. 2, 461–515; mr1883414]. J. Funct. Anal. 216(1), 230–241 (2004)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Itzykson, C., Zuber, J.B.: The planar approximation. II. J. Math. Phys. 21(3), 411–421 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Maassen, H.: Addition of freely independent random variables. J. Funct. Anal. 106(2), 409–438 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Mehta, M.L.: Random Matrices, volume 142 of Pure and Applied Mathematics (Amsterdam), third edn. Elsevier/Academic Press, Amsterdam (2004)Google Scholar
  10. 10.
    Voiculescu, D.: Addition of certain noncommuting random variables. J. Funct. Anal. 66(3), 323–346 (1986)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsHarvardCambridgeUSA

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