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A solvable mixed charge ensemble on the line: global results
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  • Published: 27 October 2011

A solvable mixed charge ensemble on the line: global results

  • Brian Rider1,
  • Christopher D. Sinclair2 &
  • Yuan Xu2 

Probability Theory and Related Fields volume 155, pages 127–164 (2013)Cite this article

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  • 10 Citations

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Abstract

We consider an ensemble of interacting charged particles on the line consisting of two species of particles with charge ratio 2:1 in the presence of an external field. With the total charge fixed and the system held at temperature corresponding to β = 1, it is proved that the particles form a Pfaffian point process. When the external field is quadratic (the harmonic oscillator potential), we produce the explicit family of skew-orthogonal polynomials necessary to simplify the related matrix kernels. In this setting a variety of limit theorems are proved on the distribution of the number as well as the spatial density of each species of particle as the total charge increases to infinity. Connections to Ginibre’s real ensemble of random matrix theory are highlighted throughout.

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References

  1. Adler M., Forrester P.J., Nagao T., van Moerbeke P.: Classical skew orthogonal polynomials and random matrices. J. Stat. Phys. 99(1–2), 141–170 (2000)

    Article  MATH  Google Scholar 

  2. Ben Arous G., Guionnet A.: Large deviations for Wigner’s law and Voiculescu’s non-commutative entropy. Probab. Theory Relat. Fields 108(4), 517–542 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  3. Ben Arous G., Zeitouni O.: Large deviations from the circular law. ESAIM Probab. Stat. 2, 123–134 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  4. Borodin A., Sinclair C.D.: The Ginibre ensemble of real random matrices and its scaling limits. Commun. Math. Phys. 291(1), 177–224 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  5. Borwein D., Borwein J.M., Crandall R.E.: Effective Laguerre asymptotics. SIAM J. Numer. Anal. 46(6), 3285–3312 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  6. Edelman A., Kostlan E., Shub M.: How many eigenvalues of a random matrix are real?. J. Am. Math. Soc. 7(1), 247–267 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  7. Forrester P.J.: Log-gases and random matrices. London Mathematical Society Monographs. Princeton University Press, Princeton (2010)

    Google Scholar 

  8. Forrester, P.J., Nagao, T.: Eigenvalue statistics of the real Ginibre ensemble. Phys. Rev. Lett. 99 (2007)

  9. Ginibre J.: Statistical ensembles of complex, quaternion, and real matrices. J. Math. Phys. 6, 440–449 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  10. Larsson-Cohn L.: L p norms of Hermite polynomials and an extremal problem on Wiener chaos. Ark. Mat. 40, 133–144 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  11. Majumdar S.N., Schehr G.: Real roots of random polynomials and zero crossing properties of diffusion equation. J. Stat. Phys. 132, 235–273 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  12. Meray C.: Sur un determinant dont celui de Vandermonde n’est qu’un particulier. Revue de Mathématiques Spéciales 9, 217–219 (1899)

    Google Scholar 

  13. Rains, E.M.: Correlation functions for symmetrized increasing subsequences. http://arXiv.org:math/0006097 (2000)

  14. Sinclair C.D.: Averages over Ginibre’s ensemble of random real matrices. Int. Math. Res. Not. 2007, 1–15 (2007)

    Google Scholar 

  15. Sommers, H.-J., Wieczorek, W.: General eigenvalue correlations for the real Ginibre ensemble. J. Phys. A 41(40) (2008)

  16. Stembridge J.R.: Nonintersecting paths, Pfaffians, and plane partitions. Adv. Math. 83(1), 96–131 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  17. Szegő, G.: Orthogonal Polynomials, vol. XXIII, 4th edn. American Mathematical Society, Providence. American Mathematical Society, Colloquium Publications (1975)

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Authors and Affiliations

  1. Department of Mathematics, University of Colorado, Boulder, CO, 80309, USA

    Brian Rider

  2. Department of Mathematics, University of Oregon, Eugene, OR, 97403, USA

    Christopher D. Sinclair & Yuan Xu

Authors
  1. Brian Rider
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  2. Christopher D. Sinclair
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  3. Yuan Xu
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Correspondence to Brian Rider.

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Cite this article

Rider, B., Sinclair, C.D. & Xu, Y. A solvable mixed charge ensemble on the line: global results. Probab. Theory Relat. Fields 155, 127–164 (2013). https://doi.org/10.1007/s00440-011-0394-z

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  • Received: 19 July 2010

  • Revised: 10 October 2011

  • Published: 27 October 2011

  • Issue Date: February 2013

  • DOI: https://doi.org/10.1007/s00440-011-0394-z

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Keywords

  • Random matrix
  • Eigenvalue statistics
  • Pfaffian processes

Mathematics Subject Classification (2000)

  • 60B20
  • 82B05
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