Skip to main content
SpringerLink
Log in

Polynomial Ensembles and Recurrence Coefficients

  • Published:
Constructive Approximation Aims and scope

Abstract

Polynomial ensembles are determinantal point processes associated with (not necessarily orthogonal) projections onto polynomial subspaces. The aim of this survey article is to put forward the use of recurrence coefficients to obtain the global asymptotic behavior of such ensembles in a rather simple way. We provide a unified approach to recover well-known convergence results for real OP ensembles. We study the mutual convergence of the polynomial ensemble and the zeros of its average characteristic polynomial; we discuss in particular the complex setting. We also control the variance of linear statistics of polynomial ensembles and derive comparison results, as well as asymptotic formulas for real OP ensembles. Finally, we reinterpret the classical algorithm to sample determinantal point processes so as to cover the setting of nonorthogonal projection kernels. A few open problems are also suggested.

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. Bardenet, R., Hardy, A.: Monte Carlo with Determinantal Point Processes, p. 48 (2016). arXiv:1605.00361

  2. Berman, R.J.: Determinantal point processes and fermions on complex manifolds: Bulk universality. To appear in Algebraic and Analytic Microlocal Analysis (2016). arXiv:0811.3341v2

  3. Borodin, A.: Biorthogonal ensembles. Nucl. Phys. B 536(3), 704–732 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  4. Breuer, J., Duits, M.: Central limit theorems for biorthogonal ensembles and asymptotics of recurrence coefficients. J. Am. Math. Soc. 30(1), 27–66 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  5. Haagerup, U., Thorbjørnsen, S.: Asymptotic expansions for the Gaussian unitary ensemble. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 15(1), 1250003 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  6. Hardy, A.: Average characteristic polynomials of determinantal point processes. Ann. Inst. Henri Poincare Probab. Stat. 51(1), 283–303 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  7. Hough, J.B., Krishnapur, M., Peres, Y., Virág, B.: Determinantal processes and independence. Probab. Surv. 3, 206–229 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  8. Johansson, K.: Random Matrices and Determinantal Processes, Mathematical Statistical Physics. Elsevier B.V., Amsterdam (2006)

    MATH  Google Scholar 

  9. Köning, W.: Orthogonal polynomial ensembles in probability theory. Probab. Surv. 2, 385–447 (2005)

    Article  MathSciNet  Google Scholar 

  10. Kuijlaars, A.B.J.: Multiple orthogonal polynomial ensembles. Recent Trends Orthogonal Polyn. Approx. Theory Contemp. Math. Number 507, 155–176 (2010)

    MathSciNet  MATH  Google Scholar 

  11. Kuijlaars, A.B.J.: Transformations of polynomials ensembles. In: Hardin, D.P., Lubinsky, D.S., Simanek, B. (eds.) Modern Trends in Constructive Function Theory, vol. 661. Contemporary Mathematics, Singapore (2016)

    Chapter  Google Scholar 

  12. Kuijlaars, A.B.J., Van Assche, W.: The asymptotic zero distribution of orthogonal polynomials with varying recurrence coefficients. J. Approx. Theory 99, 167–197 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  13. Lambert, G.: CLT for biorthogonal ensembles and related combinatorial identities (2015) arXiv:1511.06121

  14. Lavancier, F., Møller, J., Rubak, E.: Determinantal point process models and statistical inference. J. R. Stat. Soc. Ser. B (Stat. Methodol.) 77, 853–877 (2015)

    Article  MathSciNet  Google Scholar 

  15. Ledoux, M.: Differential operators and spectral distributions of invariant ensembles from the classical orthogonal polynomials the continuous case. Electron. J. Probab. 9(7), 177–208 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  16. Ledoux, M.: Differential operators and spectral distributions of invariant ensembles from the classical orthogonal polynomials: the discrete case. Electron. J. Probab. 10(34), 1116–1146 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  17. Lyons, R.: Determinantal probability measures. Publ. Math. Inst. Ht. Etud. Sci. 98, 167–212 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  18. Pastur, L., Shcherbina, M.: Eigenvalue Distribution of Large Random Matrices, Mathematical Surveys and Monographs, vol. 171. American Mathematical Society, Providence (2011)

    MATH  Google Scholar 

  19. Scardicchio, A., Zachary, C.E., Torquato, S.: Statistical properties of determinantal point processes in high-dimensional euclidean spaces. Phys. Rev. E (3) 79(4, 041108), 19 (2009)

    MathSciNet  Google Scholar 

  20. Simon, B.: Weak convergence of CD kernels and applications. Duke Math. J. 146, 305–330 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  21. Simon, B.: Szegő’s Theorem and its Descendants: Spectral Theory for \(L^2\) Perturbations of Orthogonal Polynomials. M. B. Porter Lecture Series. Princeton University Press, Princeton (2011)

    Google Scholar 

  22. Soshnikov, A.: Determinantal random point fields. Russ. Math. Surv. 55, 923–975 (2000)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

This work has been written for the special issue of Constructive Approximation on the theme “Approximation and Statistical Physics” related to the workshop “Optimal and Random Point Configurations,” which took place at the Institut Henri Poincaré in June 2016. I would like to thank the organizers for giving me the opportunity to present the work [1] there. I also would like to thank Rémi Bardenet and Thomas Bloom for enriching discussions related to the present article. I acknowledge the support from CNRS through PEPS JCJC DppMc and from ANR through the grant ANR JCJC BoB (ANR-16-CE23-0003) and Labex CEMPI (ANR-11-LABX-0007-01).

Author information

Authors and Affiliations

  1. Laboratoire Paul Painlevé, Université de Lille, Cité Scientifique, 59655, Villeneuve d’Ascq Cedex, France

    Adrien Hardy

Authors
  1. Adrien Hardy
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Adrien Hardy.

Additional information

Communicated by Peter J. Forrester.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hardy, A. Polynomial Ensembles and Recurrence Coefficients. Constr Approx 48, 137–162 (2018). https://doi.org/10.1007/s00365-017-9413-3

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00365-017-9413-3

Keywords

Mathematics Subject Classification

Search

Navigation