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Circular β ensembles, CMV representation, characteristic polynomials

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Abstract

In this note we first briefly review some recent progress in the study of the circular β ensemble on the unit circle, where β > 0 is a model parameter. In the special cases β = 1,2 and 4, this ensemble describes the joint probability density of eigenvalues of random orthogonal, unitary and sympletic matrices, respectively. For general β, Killip and Nenciu discovered a five-diagonal sparse matrix model, the CMV representation. This representation is new even in the case β = 2; and it has become a powerful tool for studying the circular β ensemble. We then give an elegant derivation for the moment identities of characteristic polynomials via the link with orthogonal polynomials on the unit circle.

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References

  1. Diaconis P. Patterns in eigenvalues: the 70th Josiah Williard Gibbs lecture. Bull Amer Math Soc, 40(2): 155–178 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  2. Okounkov A. On N-point correlations in the log-gas at rational temperature. arXiv:hep-th/9702001

  3. Johansson K. On Szegös formula for Toeplitz determinants and generalizations. Bull Sci Math, 112: 257–304 (1988)

    MATH  MathSciNet  Google Scholar 

  4. Forrester P J, Keating J P. Singularity dominated strong fluctuations for some random matrix averages. Comm Math Phys, 250: 119–131 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  5. Dumitriu I, Edelman A. Matrix models for β ensembles. J Math Phys, 43: 5830–5847 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  6. Killip R, Nenciu L. Matrix models for circular ensembles. Int Math Res Not, 50: 2665–2710 (2004)

    Article  MathSciNet  Google Scholar 

  7. Cantero M J, Moral L, Velázquez L. Five-diagonal matrices and zeros of orthogonal polynomials on the unit circle. Linear Algebra Appl, 362: 29–56 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  8. Szegö G. Orthogonal Polynomials. AMS Colloquium Publications, Vol. 23, Providence, RI: American Mathematical Society, 1975

    Google Scholar 

  9. Simon B. Orthogonal Polynomials on the Unit Circle, Part 1: Classical Theory. Colloquim Publications, Vol. 54, Providence, RI: American Mathematical Society, 2004

    Google Scholar 

  10. Simon B. CMV matrices: five years after. J Comput Appl Math, 208(1): 120–154 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  11. Diaconis P, Gamburd A. Random matrices, magic squares and matching polynomials. Electron J Comb, 11(2): 1–26 (2004)

    MathSciNet  Google Scholar 

  12. Haake F, Kus M, Sommers H J, et al. Secular determinant of random unitary matrices. J Phys A, 29: 3641–3658 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  13. Forrester P J, Rains E M. Jacobians and rank 1 perturbations relating to unitary Hessenberg matrices. Int Math Res Not, DOI: 10.1155/IMRN/2006/48306

  14. Killip R. Gaussian fluctuations for β ensembles. Int Math Res Not, 2008, DOI: 10.1093/irmn/rnn007

  15. Killip R, Stoiciu M. Eigenvalue statistics for CMV matrices: from Poisson to Clock via CβE. Duke Math J, 146(3): 361–399 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  16. Keating J P, Snaith N C. Random matrix theory and ζ(1/2+it). Comm Math Phys, 214(1): 57–89 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  17. Arfken G, Weber A. Mathematical Methods for Physicists, 6th edition. Singapore: Elsevier (Singapore) Pte Ltd., 2006

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Zhejiang University, Hangzhou, 310027, China

    ZhongGen Su

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  1. ZhongGen Su
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Correspondence to ZhongGen Su.

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This work was supported by National Natural Science Foundation of China (Grant No. 10671176)

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Su, Z. Circular β ensembles, CMV representation, characteristic polynomials. Sci. China Ser. A-Math. 52, 1467–1477 (2009). https://doi.org/10.1007/s11425-009-0099-2

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  • DOI: https://doi.org/10.1007/s11425-009-0099-2

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