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2-Matrix versus Complex Matrix Model, Integrals over the Unitary Group as Triangular Integrals

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An Erratum to this article was published on 15 November 2007

Abstract

We prove that the 2-hermitian matrix model and the complex-matrix model obey the same loop equations, and as a byproduct, we find a formula for Itzykzon-Zuber type integrals over the unitary group. Integrals over U(n) are rewritten as gaussian integrals over triangular matrices and then computed explicitly. That formula is an efficient alternative to the former Shatashvili's formula.

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References

  1. Bergère, M.: Biorthogonal polynomials for potentials of two variables and external sources at the denominator. http://arxiv.org/list/hep-th/0404126, 2004

  2. Bergère, M.: Biorthogonal polynomials for potentials of two variables and external sources at the denominator. 4th Eurogrid meeting, Les Houches (March 2004), and Capri meeting (Sept 2004), http://www.na.infn.it/congr/Bohr/, 2004

  3. Bertola, M.: Bilinear semi–classical moment functionals and their integral representation. J. App. Theory 121, 71–99 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  4. Bertola, M., Eynard, B.: The PDEs of biorthogonal polynomials arising in the 2-matrix model. http://arxiv.org/list/nlin.SI/0311033, 2003

  5. Bertola, M., Eynard, B.: Mixed Correlation Functions of the Two-Matrix Model. J. Phys. A36, 7733–7750 (2003)

    Google Scholar 

  6. Bertola, M., Eynard, B., Harnad, J.: Semiclassical orthogonal polynomials, matrix models and isomonodromic tau functions. http://arxiv.org/list/nlin.SI/0410043, 2004

  7. Bertola, M., Eynard, B., Harnad, J.: Differential systems for biorthogonal polynomials appearing in 2–matrix models and the associated Riemann–Hilbert problem. Commun. Math. Phys. 243, 193–240 (2003)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  8. Bertola, M., Eynard, B., Harnad, J.: Duality, Biorthogonal Polynomials and Multi–Matrix Models. Commun. Math. Phys. 229, 73–120 (2002)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  9. Di Francesco, P., Ginsparg, P., Zinn-Justin, J.: 2D Gravity and Random Matrices. Phys. Rep. 254, 1 (1995)

    Article  ADS  Google Scholar 

  10. Duistermaat, J.J., Heckman, G.J.: Inv. Math. 69, 259 (1982)

    MATH  MathSciNet  Google Scholar 

  11. Eynard, B.: Eigenvalue distribution of large random matrices, from one matrix to several coupled matrices. Nucl. Phys. B 506, 3, 633–664 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  12. Eynard, B.: A short note about Morozov's formula. http://arxiv.org/list/math-ph/0406063, 2004

  13. Eynard, B.: Master loop equations, free energy and correlations for the chain of matrices. JHEP 03101, 018 (2003)

    Article  MathSciNet  Google Scholar 

  14. Eynard, B.: Large N expansion of the 2-matrix model. JHEP 0301, 051, (2003)

    Article  ADS  MathSciNet  Google Scholar 

  15. Eynard, B., Mehta, M.L.: Matrices coupled in a chain: eigenvalue correlations. J. Phys. A: Math. Gen. 31, 4449 (1998)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  16. Fulton, W., Harris, J.: Representation theory. Series Graduate Texts in Mathematics, Vol. 129, Berlin-Heidelberg-New York: Springer, 1991

  17. Itzykson, C., Zuber, J.B.: The planar approximation (II). J. Math. Phys. 21, 411 (1980)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  18. Harish-Chandra,: Amer. J. Math. 80, 241 (1958)

    MathSciNet  Google Scholar 

  19. Kazakov, V.A.: Ising model on a dynamical planar random lattice: exact solution. Phys. Lett. A 119, 140 (1986)

    Article  ADS  MathSciNet  Google Scholar 

  20. Knapp, A.W.: Representation theory of semisimple groups. Princeton, NJ: Princeton University Press, 1986

  21. Mehta, M.L.: A method of integration over matrix variables. Commun. Math. Phys. 79, 327 (1981)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  22. Mehta, M.L.: Random Matrices. 2nd edition, New York: Academic Press, 1991

  23. Morozov, A.: Pair correlator in the Itzykson–Zuber Integral. Modern Phys. Lett. A 7, no. 37, 3503–3507 (1992)

    Google Scholar 

  24. Shatashvili, S.L.: Correlation Functions in the Itzykson–Zuber Model. Commun. Math. Phys. 154, 421–432 (1993)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  25. Staudacher, M.: Combinatorial Solution of the Two–Matrix Model. Phys. Lett. B 305, 332–338 (1993)

    Article  ADS  MathSciNet  Google Scholar 

  26. Ginibre, J.: Statistical ensembles of complex, quaternion and real matrices. J. Math. Phys. 6, 440 (1965)

    Article  MATH  MathSciNet  Google Scholar 

  27. Kostov, I.K., Krichever, I., Mineev-Weinstein, M., Wiegmann, P., Zabrodin, A.: τ-function for analytic curves. In: Random matrices and their applications, MSRI publications, Vol. 40, Cambridge: Cambridge University Press, 2001, p. 285

  28. Zabrodin, A.: Matrix models and growth processes: from viscous flows to the quantum Hall effect. Lectures given at the School Applications of Random Matrices in Physics, Les Houches, June 2004, http://arxiv.org/list/hep-th/0412219, 2004

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Authors and Affiliations

  1. Service de Physique Théorique de Saclay, CEA/DSM/SPhT - CNRS/SPM/URA 2306, 91191, Gif-sur-Yvette Cedex, France

    B. Eynard

  2. Departament d'Estructura i constituents de la Matèria, Universitad Barcelona, Av. Diagonal 647, 08028, Barcelona, Spain

    A. Prats Ferrer

Authors
  1. B. Eynard
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  2. A. Prats Ferrer
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Correspondence to B. Eynard.

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Communicated by L. Takhtajan

An erratum to this article is available at http://dx.doi.org/10.1007/s00220-007-0364-6.

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Eynard, B., Ferrer, A. 2-Matrix versus Complex Matrix Model, Integrals over the Unitary Group as Triangular Integrals. Commun. Math. Phys. 264, 115–144 (2006). https://doi.org/10.1007/s00220-006-1541-8

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