Skip to main content
SpringerLink
Log in

On the Integrable Structure of Deformed Sine Kernel Determinants

  • Published:
Mathematical Physics, Analysis and Geometry Aims and scope Submit manuscript

Abstract

We study a family of Fredholm determinants associated to deformations of the sine kernel, parametrized by a weight function w. For a specific choice of w, this kernel describes bulk statistics of finite temperature free fermions. We establish a connection between these determinants and a system of integro-differential equations generalizing the fifth Painlevé equation, and we show that they allow us to solve an integrable PDE explicitly for a large class of initial data.

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

Data Availability

Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

References

  1. Amir, G., Corwin, I., Quastel, J.: Probability distribution of the free energy of the continuum directed random polymer in \(1+1\) dimensions. Comm. Pure Appl. Math. 64(4), 466–537 (2011)

    Article  MathSciNet  Google Scholar 

  2. Anderson, G., Guionnet, A., Zeitouni, O.: An introduction to random matrices, vol. 118. Cambridge University Press, Cambridge (2010)

    Google Scholar 

  3. Bertola, M., Cafasso, M.: The Transition between the Gap Probabilities from the Pearcey to the Airy Process - a Riemann-Hilbert Approach. Int. Math. Res. Notices (7), 1519–1568 (2012)

  4. Bothner, T., Little, A.: The complex elliptic Ginibre ensemble at weak non-hermiticity: bulk spacing distributions. Preprint arXiv:2212.00525 (2022)

  5. Bothner, T., Little, A.: The complex elliptic Ginibre ensemble at weak non-Hermiticity: edge spacing distributions. Preprint arXiv:2208.04684 (2022)

  6. Bothner, T.: On the origins of Riemann-Hilbert problems in mathematics. Nonlinearity 34, R1–R73 (2021)

    Article  ADS  MathSciNet  Google Scholar 

  7. Bothner, T., Cafasso, M., Tarricone, S.: Momenta spacing distributions in anharmonic oscillators and the higher order finite temperature Airy kernel. Ann. Inst. Henri Poincaré Probab. Stat. 58(3), 1505–1546 (2022)

    Article  MathSciNet  Google Scholar 

  8. Cafasso, M., Claeys, T.: A Riemann-Hilbert approach to the lower tail of the Kardar-Parisi-Zhang equation. Comm. Pure Appl. Math. 75(3), 493–540 (2022)

    Article  MathSciNet  Google Scholar 

  9. Cafasso, M., Claeys, T., Ruzza, G.: Airy kernel determinant solutions to the KdV equation and integro-differential Painlevé equations. Comm. Math. Phys. 386(2), 1107–1153 (2021)

    Article  ADS  MathSciNet  Google Scholar 

  10. Claeys , T., Glesner, G.: Determinantal point processes conditioned on randomly incomplete configurations. To appear in Ann. Inst. H. Poincaré

  11. Claeys, T., Glesner, G., Ruzza, G., Tarricone, S.: Jáinossy densities and Darboux transformations for the Stark and cylindrical KdV equations. Preprint arXiv:2303.09848 (2023)

  12. Claeys, T., Its, A., Krasovsky, I.: Emergence of a singularity for Toeplitz determinant and Painlevé V. Duke Math. J. 160, 2 (2011)

    Article  Google Scholar 

  13. Dean, D.S., Le Doussal, P., Majumdar, S.N., Schehr, G.: Non-interacting fermions at finite temperature in a d-dimensional trap: universal correlations. Phys. Rev. A 94 (2016)

  14. Deift, P., Trubowitz, E.: Inverse scattering on the line. Comm. Pure Appl. Math. 32(2), 121–251 (1979)

    Article  MathSciNet  Google Scholar 

  15. Deift, P., Its, A., Zhou, X.: A Riemann-Hilbert approach to asymptotic problems arising in the theory of random matrix models, and also in the theory of integrable statistical mechanics. Annal. Math. 146, 149–235 (1997)

    Article  MathSciNet  Google Scholar 

  16. Fokas, A., Its, A., Kapaev, A., Novokshenov., V.: Painlevé transcendents: the Riemann–Hilbert approach. Math. Surv. Monogr. AMS 128 (2006)

  17. Gouraud, G., Le Doussal, P., Schehr, G.: Quench dynamics of noninteracting fermions with a delta impurity. J. Phys. A: Math. Theo. 55, 39 (2022)

    Article  MathSciNet  Google Scholar 

  18. Gouraud, G., Le Doussal, P., Schehr, G.: Stationary time correlations for fermions after a quench in the presence of an impurity. Europhys. Lett. 142, 4 (2023)

    Article  Google Scholar 

  19. Gurevich, A.G., Pitaevskii, L.P.: Non stationary structure of collisionless shock waves. JEPT Lett. 17, 193–195 (1973)

    ADS  Google Scholar 

  20. Its, A., Izergin, A., Korepin, A., Slavnov, N.: Differential equations for quantum correlation functions. Int. J. Modern Phys. B 4(5), 1003–1037 (1990)

    Article  ADS  MathSciNet  Google Scholar 

  21. Jimbo, M., Miwa, T., Mori, Y., Sato, M.: Density matrix of an impenetrable Bose gas and the fifth Painlevé transcendent. Phys. D: Nonlinear Phenom. 1(1), 80–158 (1980)

    Article  ADS  Google Scholar 

  22. Johansson, K.: From Gumbel to Tracy-Widom. Probab. Theo. Related Fields 138(1–2), 75–112 (2007)

    Article  MathSciNet  Google Scholar 

  23. Krajenbrink, A.: From Painlevé to Zakharov-Shabat and beyond: Fredholm determinants and integro-differential hierarchies. J. Phys. A: Math. Theo. 54, 3 (2020)

    MathSciNet  Google Scholar 

  24. P. D. Lax & C. D. Levermore. “The small dispersion limit of the Korteweg de Vries equation, I,II,III”. Comm. Pure Appl. Math. 36, 253–290, 571–593, 809–830 (1983)

  25. Le Doussal, P., Majumdar, S., Rosso, A., Schehr, G.: Exact short-time height distribution in 1D KPZ equation and edge fermions at high temperature. Phys. Rev. Lett. 117, 070403 (2016)

    Article  Google Scholar 

  26. Liechty, K., Wang, D.: Asymptotics of free fermions in a quadratic well at finite temperature and the Moshe-Neuberger-Shapiro random matrix model. Ann. Inst. Henri Poincaré Probab. Statist. 56, 2 (2020)

    Article  MathSciNet  Google Scholar 

  27. Moshe, M., Neuberger, H., Shapiro, B.: Generalized ensemble of random matrices. Phys. Rev. Lett. 73(11), 1497–1500 (1994)

    Article  ADS  MathSciNet  Google Scholar 

  28. Shabat, A., Zakharov, V.: Exact theory of twodimensional self-focusing and one-dimensional selfmodulation of waves in nonlinear media. Soviet Phys. JETP 34, 1 (1972)

    Google Scholar 

  29. Soshnikov, A.: “Determinantal random point fields”. Uspekhi Mat. Nauk 55(5) (335), 107–160 (2000); translation in Russian Math. Surveys, 55(5), 923–975 (2000)

  30. Tracy, C.A., Widom, H.: Level-spacing distributions and the Airy kernel. Comm. Math. Phys. 159(1), 151–174 (1994)

    Article  ADS  MathSciNet  Google Scholar 

  31. Tracy, C.A., Widom, H.: Fredholm determinants, differential equations and matrix models. Comm. Math. Phys. 163(1), 33–72 (1994)

    Article  ADS  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors were supported by Fonds de la Recherche Scientifique-FNRS under EOS project O013018F, and by CNRS International Research Network PIICQ. TC was also supported by FNRS Research Project T.0028.23. The authors are grateful to Thomas Bothner for drawing their attention to the PDE (2.22) in [20], and to Alexander Its, Peter Miller and Gregory Schehr for useful comments.

Author information

Authors and Affiliations

  1. Institut de Recherche en Mathématique et Physique, Université catholique de Louvain, Chemin du Cyclotron 2, 1348, Louvain-la-Neuve, Belgium

    Tom Claeys

  2. Institut de Physique Théorique, Université Paris-Saclay, CEA, CNRS, F-91191, Gif-sur-Yvette, France

    Sofia Tarricone

Authors
  1. Tom Claeys
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Sofia Tarricone
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sofia Tarricone.

Ethics declarations

Statements and Declarations

The authors have no competing interests to declare that are relevant to the content of this article.

Additional information

Publisher's Note

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

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Claeys, T., Tarricone, S. On the Integrable Structure of Deformed Sine Kernel Determinants. Math Phys Anal Geom 27, 3 (2024). https://doi.org/10.1007/s11040-024-09476-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11040-024-09476-x

Keywords

Mathematics Subject Classification

Search

Navigation