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On the Proof of Universality for Orthogonal and Symplectic Ensembles in Random Matrix Theory

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Abstract

We give a streamlined proof of a quantitative version of a result from P. Deift and D. Gioev, Universality in Random Matrix Theory for Orthogonal and Symplectic Ensembles. IMRP Int. Math. Res. Pap. (in press) which is crucial for the proof of universality in the bulk P. Deift and D. Gioev, Universality in Random Matrix Theory for Orthogonal and Symplectic Ensembles. IMRP Int. Math. Res. Pap. (in press) and also at the edge P. Deift and D. Gioev, {Universality at the edge of the spectrum for unitary, orthogonal and symplectic ensembles of random matrices. Comm. Pure Appl. Math. (in press) for orthogonal and symplectic ensembles of random matrices. As a byproduct, this result gives asymptotic information on a certain ratio of the β=1,2,4 partition functions for log gases.

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References

  1. M. Adler and P. van Moerbeke, Toda versus Pfaff lattice and related polynomials. Duke Math. J. 112:1–58 (2002).

    Article  MATH  MathSciNet  Google Scholar 

  2. P. Deift and D. Gioev, Universality in Random Matrix Theory for Orthogonal and Symplectic Ensembles (IMRP Int. Math. Res. Pap.) (in press), math-ph/0411075.

  3. P. Deift and D. Gioev, Universality at the edge of the spectrum for unitary, orthogonal and symplectic ensembles of random matrices. Comm. Pure Appl. Math. (in press), math-ph/0507023.

  4. P. Deift, D. Gioev, T. Kriecherbauer, and M. Vanlessen, Universality for orthogonal and symplectic Laguerre-type ensembles (submitted), math-ph/0612007.

  5. P. Deift, T. Kriecherbauer, K. T.-R. McLaughlin, S. Venakides and X. Zhou, Strong asymptotics of orthogonal polynomials with respect to exponential weights. Comm. Pure Appl. Math. 52:1491–1552 (1999).

    Article  MATH  MathSciNet  Google Scholar 

  6. N. M. Ercolani and K. D. T.-R. McLaughlin, Asymptotics of the partition function for random matrices via Riemann-Hilbert techniques and applications to graphical enumeration. Int. Math. Res. Not. 2003:755–820 (2003).

    Article  MATH  MathSciNet  Google Scholar 

  7. K. Johansson, On fluctuations of eigenvalues of random Hermitian matrices. Duke Math. J. 91:151–204 (1998).

    Article  MATH  MathSciNet  Google Scholar 

  8. M. Reed and B. Simon, Methods of Modern Mathematical Physics, IV, Academic Press, New York, London (1978).

    MATH  Google Scholar 

  9. A. Stojanovic, Universality in orthogonal and symplectic invariant matrix models with quartic potential. Math. Phys. Anal. Geom. 3:339–373 (2000). Errata: ibid., 7:347–349 (2004).

    Article  MATH  MathSciNet  Google Scholar 

  10. C. A. Tracy and H. Widom, Correlation functions, cluster functions, and spacing distributions for random matrices. J. Statist. Phys. 92:809–835 (1998).

    Article  MATH  MathSciNet  Google Scholar 

  11. H. Widom, On the relation between orthogonal, symplectic and unitary matrix ensembles. J. Statist. Phys. 94:347–363 (1999).

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, The Ohio State University, 231 W. 18th Ave., Columbus, OH, 43210, USA

    Ovidiu Costin

  2. Department of Mathematics, Courant Institute of Mathematical Sciences, New York University, 251 Mercer St., New York, NY, 10012, USA

    Percy Deift

  3. Department of Mathematics, University of Rochester, Hylan Bldg., Rochester, NY, 14627, USA

    Dimitri Gioev

Authors
  1. Ovidiu Costin
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  2. Percy Deift
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  3. Dimitri Gioev
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Correspondence to Ovidiu Costin.

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Costin, O., Deift, P. & Gioev, D. On the Proof of Universality for Orthogonal and Symplectic Ensembles in Random Matrix Theory. J Stat Phys 129, 937–948 (2007). https://doi.org/10.1007/s10955-007-9277-1

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  • DOI: https://doi.org/10.1007/s10955-007-9277-1

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