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Journal of Theoretical Probability

, Volume 29, Issue 4, pp 1199–1239 | Cite as

Limit Theorems for Orthogonal Polynomials Related to Circular Ensembles

  • Joseph Najnudel
  • Ashkan Nikeghbali
  • Alain Rouault
Article

Abstract

For a natural extension of the circular unitary ensemble of order n, we study as \(n\rightarrow \infty \) the asymptotic behavior of the sequence of monic orthogonal polynomials \((\varPhi _{k,n}, k=0, \ldots , n)\) with respect to the spectral measure associated with a fixed vector, the last term being the characteristic polynomial. We show that, as \(n\rightarrow \infty \), the sequence of processes \((\log \varPhi _{\lfloor nt\rfloor ,n}(1), t \in [0,1])\) converges to a deterministic limit, and we describe the fluctuations and the large deviations.

Keywords

Random matrices Unitary ensemble Orthogonal polynomials  Large deviation principle Invariance principle 

Mathematics Subject Classification (2010)

15B52 42C05 60F10 60F17 

References

  1. 1.
    Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th edn, pp. 258–259. Dover, New York (1972)zbMATHGoogle Scholar
  2. 2.
    Borot, G., Guionnet, A.: Asymptotic expansion of \(\beta \) matrix models in the one-cut regime. Comm. Math. Phys. 317(2), 447–483 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bourgade, P., Hughes, C.P., Nikeghbali, A., Yor, M.: The characteristic polynomial of a random unitary matrix: a probabilistic approach. Duke Math. J. 145(1), 45–69 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bourgade, P., Nikeghbali, A., Rouault, A.: The characteristic polynomial on compact groups with Haar measure: some equalities in law. C. R. Math. Acad. Sci. Paris 345(4), 229–232 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bourgade, P., Nikeghbali, A., Rouault, A.: Circular Jacobi ensembles and deformed Verblunsky coefficients. Int. Math. Res. Not. IMRN 23, 4357–4394 (2009)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Bourgade, P., Nikeghbali, A., Rouault, A.: Ewens Measures on Compact Groups and Hypergeometric Kernels. In: Séminaire de probabilités XLIII, volume 2006 of Lecture Notes in Mathematics, pp. 351–378. Springer, Berlin (2010)Google Scholar
  7. 7.
    Dembo, A., Zeitouni, O.: Large Deviations Techniques and Applications, 2nd edn. Springer, Berlin (1998)CrossRefzbMATHGoogle Scholar
  8. 8.
    Dette, H., Gamboa, F.: Asymptotic properties of the algebraic moment range process. Acta Math. Hung. 116, 247–264 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Erdelyi, A., Magnus, W., Oberhettinger, F., Tricomi, F.G.: Higher Transcendental Functions, vol. I. Krieger, New York (1981)zbMATHGoogle Scholar
  10. 10.
    Forrester, P.J.: Log-Gases and Random Matrices, volume 34 of London Mathematical Society Monographs Series. Princeton University Press, Princeton (2010)Google Scholar
  11. 11.
    Hiai, F., Petz, D.: A large deviation theorem for the empirical eigenvalue distribution of random unitary matrices. Ann. Inst. H. Poincaré Probab. Stat. 36(1), 71–85 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Hughes, C.P., Keating, J.P., O’Connell, N.: On the characteristic polynomial of a random unitary matrix. Comm. Math. Phys. 220(2), 429–451 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Jacod, J., Shiryaev, A.N.: Limit Theorems for Stochastic Processes. Springer, Berlin (1987)CrossRefzbMATHGoogle Scholar
  14. 14.
    Keating, J.P., Snaith, N.C.: Random matrix theory and \(\zeta (1/2+it)\). Comm. Math. Phys. 214(1), 57–89 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Killip, R.: Gaussian fluctuations for \(\beta \) ensembles. Int. Math. Res. Not. 2008(8) (2008); Art. ID rnn007, 19Google Scholar
  16. 16.
    Killip, R., Nenciu, I.: Matrix models for circular ensembles. Int. Math. Res. Not. 50, 2665–2701 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Killip, R., Stoiciu, M.: Eigenvalue statistics for CMV matrices: from Poisson to clock via random matrix ensembles. Duke Math. J. 146(3), 361–399 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Neretin, Y.A.: Hua-type integrals over unitary groups and over projective limits of unitary groups. Duke Math. J. 114(2), 239–266 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Neretin, YuA: Matrix analogues of the \(B\)-function, and the Plancherel formula for Berezin kernel representations. Math. Sb. 191(5), 67–100 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Olver, F.W.J.: Asymptotics and Special Functions. Academic Press, New York (1974)zbMATHGoogle Scholar
  21. 21.
    Rockafellar, R.T.: Integrals which are convex functionals, II. Pac. J. Math. 39(2), 439–469 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Rouault, A.: Asymptotic behavior of random determinants in the Laguerre, Gram and Jacobi ensembles. ALEA Lat. Am. J. Probab. Math. Stat. 3, 181–230 (2007). (electronic)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Ryckman, E.: Linear statistics of point processes via orthogonal polynomials. J. Stat. Phys. 132(3), 473–486 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Saff, E.B., Totik, V.: Logarithmic Potentials with External Fields, Volume 316 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer, Berlin (1997). Appendix B by Thomas BloomGoogle Scholar
  25. 25.
    Simon, B.: Orthogonal Polynomials on the Unit Circle. Part 1: Classical Theory. Colloquium Publications. American Mathematical Society 54, Part 1. American Mathematical Society (AMS), Providence, RI (2005)Google Scholar
  26. 26.
    Simon, B.: CMV matrices: five years after. J. Comput. Appl. Math. 208, 120–154 (2007)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Joseph Najnudel
    • 1
  • Ashkan Nikeghbali
    • 2
  • Alain Rouault
    • 3
  1. 1.Institut de Mathématiques de ToulouseUniversité Paul SabatierToulouse Cedex 9France
  2. 2.Institut für MathematikUniversität ZürichZürichSwitzerland
  3. 3.Université Versailles-Saint Quentin, LMV, Bâtiment FermatVersailles CedexFrance

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