Skip to main content
SpringerLink
Log in

Bulk universality holds in measure for compactly supported measures

  • Published:
Journal d'Analyse Mathématique Aims and scope

Abstract

Let µ be a measure with compact support, with orthonormal polynomials {p n } and associated reproducing kernels {K n }. We show that bulk universality holds in measure in {ξ: µ′(ξ) > 0}. More precisely, given ɛ, r > 0, the linear Lebesgue measure of the set {ξ: µ′(ξ) > 0} and for which

$$\mathop {\sup }\limits_{\left| u \right|,\left| v \right| \leqslant r} \left| {\frac{{K_n (\xi + u/\tilde K_n (\xi ,\xi ),\xi + v/\tilde K_n (\xi ,\xi ))}} {{K_n (\xi ,\xi )}}} \right. - \left. {\frac{{\sin \pi (u - v)}} {{\pi (u - v)}}} \right| \geqslant \varepsilon$$

approaches 0 as n → ∞. There are no local or global regularity conditions on the measure µ.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. N. I. Achieser, Theory of Approximation, Dover, New York, 1992.

    Google Scholar 

  2. A. Avila, Y. Last, and B. Simon, Bulk universality and clock spacing of zeros for ergodic Jacobi matrices with a.c. spectrum, Anal. PDE 3 (2010), 81–105

    Article  MathSciNet  MATH  Google Scholar 

  3. J. Baik, T. Kriecherbauer, K. T-R. McLaughlin, and P. D. Miller, Uniform Asymptotics for Polynomials Orthogonal with Respect to a General Class of Discrete Weights and Universality Results for Associated Ensembles, Princeton University Press, Princeton, NJ, 2007.

    Google Scholar 

  4. C. Bennett and R. Sharpley, Interpolation of Functions, Academic Press, Orlando, 1988.

    Google Scholar 

  5. P. Bleher and A. Its, Random Matrix Models and their Applications, Cambridge University Press, Cambridge, 2001.

    MATH  Google Scholar 

  6. J. Breuer, Y. Last, and B. Simon, The Nevai condition, Constr. Approx. 32 (2010), 221–254.

    Article  MathSciNet  MATH  Google Scholar 

  7. P. Deift, Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert Approach, New York University Press, New York, NY, 1999.

    Google Scholar 

  8. P. Deift and D. Gioev, Random Matrix Theory: Invariant Ensembles and Universality, New York University Press, New York, NY, 2009.

    MATH  Google Scholar 

  9. P. Deift, T. Kriecherbauer, K. T-R. McLaughlin, S. Venakides and X. Zhou, Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory, Comm. Pure Appl. Math. 52 (1999), 1335–1425.

    Article  MathSciNet  MATH  Google Scholar 

  10. L. Erd″os, S. Peche, J. Ramirez, B. Schlein, and H. T. Yau, Bulk universality for Wigner matrices, Comm. Pure Appl. Math. 63 (2010), 895–925.

    MathSciNet  Google Scholar 

  11. L. Erd″os, B. Schlein, H. T. Yau, and J. Yin, The local relaxation flow approach to universality of the local statistics for random matrices, Ann. Inst. Henri Poincaré Probab. Stat. 48 (2012), 1–46.

    Article  Google Scholar 

  12. E. Findley, Universality for regular measures satisfying Szeg″o’s condition, J. Approx. Theory 155 (2008), 136–154.

    Article  MathSciNet  MATH  Google Scholar 

  13. A. Foulquie Moreno, A. Martinez-Finkelshtein, and V. Sousa, Asymptotics of orthogonal polynomials for a weight with a jump on [−1, 1], Constr. Approx. 33 (2011) 219–263.

    Article  MathSciNet  MATH  Google Scholar 

  14. P. J. Forrester, Log-gases and Random Matrices, Princeton University Press, Princeton, NJ, 2010.

    MATH  Google Scholar 

  15. G. Freud, Orthogonal Polynomials, Pergamon Press, Oxford and New York, 1971.

    Google Scholar 

  16. J. B. Garnett, Bounded Analytic Functions, Academic Press, New York — London, 1981.

    MATH  Google Scholar 

  17. I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series and Products, Academic Press, San Diego, 1979.

    Google Scholar 

  18. A. B. Kuijlaars and M. Vanlessen, Universality for eigenvalue correlations from the modified Jacobi unitary ensemble, Int. Math. Res. Not. 2002, 1575–1600.

  19. A. B. Kuijlaars and M. Vanlessen, Universality for eigenvalue correlations at the origin of the spectrum, Comm. Math. Phys. 243 (2003), 163–191.

    Article  MathSciNet  MATH  Google Scholar 

  20. N. S. Landkof, Foundations of Modern Potential Theory, Springer, Berlin, 1972.

    MATH  Google Scholar 

  21. B. Ya. Levin, Lectures on Entire Functions, American Mathematical Society, Providence, 1996.

    MATH  Google Scholar 

  22. Eli Levin and D. S. Lubinsky, Universality limits in the bulk for varying measures, Adv. Math. 219 (2008), 743–779.

    Article  MathSciNet  MATH  Google Scholar 

  23. Eli Levin and D. S. Lubinsky, Universality limits for exponential weights, Constr. Approx. 29 (2009), 247–275.

    Article  MathSciNet  MATH  Google Scholar 

  24. Eli Levin and D. S. Lubinsky, Universality limits at the soft edge of the spectrum via classical complex analysis, in Int. Math. Res. Not. IMRN 2011, 3006–3070.

  25. D. S. Lubinsky, A new approach to universality limits at the edge of the spectrum, in Integrable Systems and Random Matrices, Amer. Math. Soc., Providence, RI, 2008 pp. 281–290.

    Chapter  Google Scholar 

  26. D. S. Lubinsky, Universality limits at the hard edge of the spectrum for measures with compact support, Int. Math. Res. Not. IMRN 2008, Article ID rnn099.

  27. D. S. Lubinsky, Universality limits in the bulk for arbitrary measures on compact sets, J. Anal. Math. 106 (2008), 373–394.

    Article  MathSciNet  MATH  Google Scholar 

  28. D. S. Lubinsky, A new approach to universality limits involving orthogonal polynomials, Ann. of Math. (2) 170 (2009), 915–939.

    Article  MathSciNet  MATH  Google Scholar 

  29. D. S. Lubinsky, Universality limits for random matrices and de Branges spaces of entire functions, J. Funct. Anal. 256 (2009), 3688–3729.

    Article  MathSciNet  MATH  Google Scholar 

  30. D. S. Lubinsky, Universality in the bulk holds close to given points, Approx. Theory 163 (2011), 904–922.

    Article  MathSciNet  MATH  Google Scholar 

  31. D. S. Lubinsky, A maximal function approach to Christoffel functions and Nevai’s operators, Constr. Approx. 34 (2011), 357–369.

    Article  MATH  Google Scholar 

  32. A. Mate, P. Nevai, and V. Totik, Szegő’s extremum problem on the unit circle, Ann. of Math. (2) 134 (1991), 433–453.

    Article  MathSciNet  MATH  Google Scholar 

  33. K. T.-R. McLaughlin and P. Miller, The \(\overline \partial \) steepest descent method and the asymptotic behavior of polynomials orthogonal on the unit circle with fixed and exponentially varying nonanalytic weights, IMRP Int. Math. Res. Pap. 2006, Article ID 48673.

  34. K. T.-R. McLaughlin and P. Miller, The \(\overline \partial \) steepest descent method for orthogonal polynomials on the real line with varying weights, Int. Math. Res. Not. IMRN 2008, Article ID rnn 075.

  35. M. L. Mehta, Random Matrices, 2nd ed., Academic Press, Boston, 1991.

    MATH  Google Scholar 

  36. P. Nevai, Orthogonal Polynomials, Amer. Math. Soc., Providence, RI, 1979.

    Google Scholar 

  37. L. Pastur and M. Shcherbina, Universality of the local eigenvalue statistics for a class of unitary invariant random matrix ensembles, J. Stat. Phys. 86 (1997), 109–147.

    Article  MathSciNet  MATH  Google Scholar 

  38. T. Ransford, Potential Theory in the Complex Plane, Cambridge University Press, Cambridge, 1995.

    Book  MATH  Google Scholar 

  39. W. Rudin, Real and Complex Analysis, 3rd ed., McGraw Hill, New York, 1987.

    MATH  Google Scholar 

  40. E. B. Saff and V. Totik, Logarithmic Potentials with External Fields, Springer, New York, 1997.

    MATH  Google Scholar 

  41. B. Simon, Orthogonal Polynomials on the Unit Circle, Parts 1 and 2, Amer. Math. Soc, Providence, 2005.

    Google Scholar 

  42. B. Simon, The Christoffel-Darboux kernel, in Perspectives in Partial Differential Equations, Harmonic Analysis and Applications, Amer. Math. Soc., Providence, RI, 2008, pp. 295–335.

    Google Scholar 

  43. B. Simon, Two extensions of Lubinsky’s universality theorem, J. Anal. Math. 105 (2008), 345–362.

    Article  MathSciNet  MATH  Google Scholar 

  44. A. Soshnikov, Universality at the edge of the spectrum in Wigner random matrices, Comm. Math. Phys. 207 (1999), 697–733.

    Article  MathSciNet  MATH  Google Scholar 

  45. H. Stahl and V. Totik, General Orthogonal Polynomials, Cambridge University Press, Cambridge, 1992.

    Book  MATH  Google Scholar 

  46. F. Stenger, Numerical Methods Based on Sinc and Analytic Functions, Springer, New York, 1993.

    Book  MATH  Google Scholar 

  47. T. Tao and V. Vu, From the Littlewood-Offord problem to the circular law: universality of the spectral distribution of random matrices, Bull. Amer. Math. Soc. 46 (2009), 377–396.

    Article  MathSciNet  MATH  Google Scholar 

  48. T. Tao and V. Vu, Random covariance matrices: universality of local statistics of eigenvalues, manuscript.

  49. V. Totik, Asymptotics for Christoffel functions for general measures on the real line, J. Anal. Math, 81 (2000), 283–303.

    Article  MathSciNet  MATH  Google Scholar 

  50. V. Totik, Universality and fine zero spacing on general sets, Ark. Mat. 47 (2009), 361–391.

    Article  MathSciNet  MATH  Google Scholar 

  51. V. Totik, Christoffel functions on curves and domains, Trans. Amer. Math. Soc. 362 (2010), 2053–2087.

    Article  MathSciNet  MATH  Google Scholar 

  52. C. A. Tracy and H. Widom, Level-spacing distributions of the Airy kernel, Comm. Math. Phys. 159 (1994), 151–174.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics, Georgia Institute of Technology, Atlanta, GA, 30332-0160, USA

    Doron S. Lubinsky

Authors
  1. Doron S. Lubinsky
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Doron S. Lubinsky.

Additional information

Research supported by NSF grant DMS1001182 and US-Israel BSF grant 2008399.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Lubinsky, D.S. Bulk universality holds in measure for compactly supported measures. JAMA 116, 219–253 (2012). https://doi.org/10.1007/s11854-012-0006-6

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11854-012-0006-6

Keywords

Search

Navigation