Skip to main content
SpringerLink
Log in

Gaussian Unitary Ensembles with Pole Singularities Near the Soft Edge and a System of Coupled Painlevé XXXIV Equations

  • Published:
Annales Henri Poincaré Aims and scope Submit manuscript

Abstract

In this paper, we study the singularly perturbed Gaussian unitary ensembles defined by the measure

$$\begin{aligned} \frac{1}{C_n} \mathrm{e}^{- n\text {tr}\, V(M;\lambda ,\mathbf {t}\;)}\mathrm{d}M, \end{aligned}$$

over the space of \(n \times n\) Hermitian matrices M, where \(V(x;\lambda ,\mathbf {t}\;):= 2x^2 + \sum _{k=1}^{2m}t_k(x-\lambda )^{-k}\) with \(\mathbf {t}= (t_1, t_2, \ldots , t_{2m})\in {\mathbb {R}}^{2m-1} \times (0,\infty )\), in the multiple scaling limit, where \(\lambda \rightarrow 1\) together with \(\mathbf {t} \rightarrow \mathbf {0}\) as \(n\rightarrow \infty \) at appropriate related rates. We obtain the asymptotics of the partition function, which is described explicitly in terms of an integral involving a smooth solution to a new coupled Painlevé system generalizing the Painlevé XXXIV equation. The large n limit of the correlation kernel is also derived, which leads to a new universal class built out of the \(\Psi \)-function associated with the coupled Painlevé system.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others

References

  1. Akemann, G., Villamaina, D., Vivo, P.: A singular-potential random matrix model arising in mean-field glassy systems. Phys. Rev. E 89, 062146 (2014)

    Article  ADS  Google Scholar 

  2. Atkin, M.: The Lenard recursion relation and a family of singularly perturbed matrix models. Acta Phys. Polon. B 46(9), 1825–1832 (2015)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  3. Atkin, M., Claeys, T., Mezzadri, F.: Random matrix ensembles with singularities and a hierarchy of Painlevé III equations. Int. Math. Res. Not. 2016, 2320–2375 (2016)

    Article  MATH  Google Scholar 

  4. Berry, M.V., Shukla, P.: Tuck’s incompressibility function: statistics for zeta zeros and eigenvalues. J. Phys. A 41, 385202 (2008)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  5. Bleher, P., Its, A.: Semiclassical asymptotics of orthogonal polynomials, Riemann–Hilbert problem, and universality in the matrix model. Ann. Math. 150, 185–266 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  6. Brightmore, L., Mezzadri, F., Mo, M.Y.: A matrix model with a singular weight and Painlevé III. Commun. Math. Phys. 333, 1317–1364 (2015)

    Article  ADS  MATH  Google Scholar 

  7. Brouwer, P.W., Frahm, K.M., Beenakker, C.W.J.: Quantum mechanical time-delay matrix in chaotic scattering. Phys. Rev. Lett. 78, 4737–4740 (1997)

    Article  ADS  Google Scholar 

  8. Clarkson, P.A., Joshi, N., Pickering, A.: Bäcklund transformations for the second Painlevé hierarchy: a modified truncation approach. Inverse Probl. 15, 175–187 (1999)

    Article  ADS  MATH  Google Scholar 

  9. Charlier, C., Doeraene, A.: The generating function for the Bessel point process and a system of coupled Painlevé V equations. Appear Random Matrices Theory Appl. 08(03), 1950008 (2019)

    Article  Google Scholar 

  10. Chen, Y., Its, A.: Painlevé III and a singular linear statistics in Hermitian random matrix ensembles, I. J. Approx. Theory 162, 270–297 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  11. Claeys, T., Doeraene, A.: The generating function for the Airy point process and a system of coupled Painlevé II equations. Stud. Appl. Math. 140, 403–437 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  12. Dai, D., Xu, S.-X., Zhang, L.: Gap probability at the hard edge for random matrix ensembles with pole singularities in the potential. SIAM J. Math. Anal. 50, 2233–2279 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  13. Deift, P.: Orthogonal Polynomials and Random Matrices: A Riemann–Hilbert Approach, Courant Lecture Notes 3. New York University (1999)

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

    Article  MathSciNet  MATH  Google Scholar 

  15. Deift, P., Kriecherbauer, T., McLaughlin, K.T.-R., Venakides, S., Zhou, X.: Strong asymptotics of orthogonal polynomials with respect to exponential weights. Commun. Pure Appl. Math. 52, 1491–1552 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  16. Deift, P., Zhou, X.: A steepest descent method for oscillatory Riemann–Hilbert problems. Asymptotics for the MKdV equation. Ann. Math. 137(2), 295–368 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  17. Fokas, A.S., Its, A.R., Kapaev, A.A., Novokshenov, V.Yu.: Painlevé Transcendents: The Riemann–Hilbert Approach, AMS Mathematical Surveys and Monographs, vol. 128. American Mathematical Society, Providence (2006)

  18. Fokas, A.S., Its, A.R., Kitaev, A.V.: The isomonodromy approach to matrix models in 2D quantum gravity. Commun. Math. Phys. 147, 395–430 (1992)

    Article  ADS  MATH  Google Scholar 

  19. Ince, E.L.: Ordinary Differential Equations. Dover, New York (1956)

    Google Scholar 

  20. Its, A.R., Kuijlaars, A.B.J., Östensson, J.: Critical edge behavior in unitary random matrix ensembles and the thirty fourth Painlevé transcendent. Int. Math. Res. Not. 2008(9), Art. ID rnn017, 67 pp (2008)

  21. Mehta, M.L.: Random Matrices, 3rd edn. Elsevier/Academic Press, Amsterdam (2004)

    MATH  Google Scholar 

  22. Mezzadri, F., Mo, M.Y.: On an average over the Gaussian unitary ensemble. Int. Math. Res. Not. 2009, 3486–3515 (2009)

    MathSciNet  MATH  Google Scholar 

  23. Mezzadri, F., Simm, N.J.: Tau-function theory of chaotic quantum transport with \(\beta =1,2,4\). Commun. Math. Phys. 324, 465–513 (2013)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  24. Olver, F., Lozier, D., Boisvert, R., Clark, C.: NIST Handbook of Mathematical Functions. Cambridge University Press, Cambridge (2010)

    MATH  Google Scholar 

  25. Osipov, V.A., Kanzieper, E.: Are bosonic replicas faulty? Phys. Lett. Rev. 99, 050602 (2007)

    Article  ADS  Google Scholar 

  26. Szegö, G.: Orthogonal Polynomials, 4th edn. American Mathematical Society, Providence (1975)

    MATH  Google Scholar 

  27. Texier, C., Majumdar, S.N.: Wigner time-delay distribution in chaotic cavities and freezing transition. Phys. Rev. Lett. 110, 250602 (2013)

    Article  ADS  Google Scholar 

  28. Xu, S.-X., Dai, D.: Tracy–Widom distributions in critical unitary random matrix ensembles and the coupled Painlevé II system. Commun. Math. Phys. 365, 515–567 (2019)

    Article  ADS  MATH  Google Scholar 

  29. Xu, S.-X., Dai, D., Zhao, Y.-Q.: Painlevé III asymptotics of Hankel determinants for a singularly perturbed Laguerre weight. J. Approx. Theory 192, 1–18 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  30. Xu, S.-X., Dai, D., Zhao, Y.-Q.: Critical edge behavior and the Bessel to Airy transition in the singularly perturbed Laguerre unitary ensemble. Commun. Math. Phys. 332, 1257–1296 (2014)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  31. Xu, S.-X., Zhao, Y.-Q.: Painlevé XXXIV asymptotics of orthogonal polynomials for the Gaussian weight with a jump at the edge. Stud. Appl. Math. 127, 67–105 (2011)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The work of Dan Dai was partially supported by Grants from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project Nos. CityU 11300115, CityU 11303016), and by Grants from City University of Hong Kong (Project Nos. 7004864, 7005032). The work of Shuai-Xia Xu was partially supported by National Natural Science Foundation of China under Grant number 11571376 and GuangDong Natural Science Foundation under Grant number 2014A030313176. The work of Lun Zhang was partially supported by National Natural Science Foundation of China under Grant numbers 11822104 and 11501120, by The Program for Professor of Special Appointment (Eastern Scholar) at Shanghai Institutions of Higher Learning and by Grant EZH1411513 from Fudan University.

Author information

Authors and Affiliations

  1. Department of Mathematics, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong

    Dan Dai

  2. Institut Franco-Chinois de l’Energie Nucléaire, Sun Yat-sen University, Guangzhou, 510275, China

    Shuai-Xia Xu

  3. School of Mathematical Sciences and Shanghai Key Laboratory for Contemporary Applied Mathematics, Fudan University, Shanghai, 200433, China

    Lun Zhang

Authors
  1. Dan Dai
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Shuai-Xia Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Lun Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Shuai-Xia Xu.

Additional information

Communicated by Vadim Gorin.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Dai, D., Xu, SX. & Zhang, L. Gaussian Unitary Ensembles with Pole Singularities Near the Soft Edge and a System of Coupled Painlevé XXXIV Equations. Ann. Henri Poincaré 20, 3313–3364 (2019). https://doi.org/10.1007/s00023-019-00834-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00023-019-00834-y

Mathematics Subject Classification

Search

Navigation