Skip to main content
SpringerLink
Log in

Characteristic Polynomials for 1D Random Band Matrices from the Localization Side

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

We study the special case of \({n\times n}\) 1D Gaussian Hermitian random band matrices, when the covariance of the elements is determined by \({J=(-W^2\triangle+1)^{-1}}\). Assuming that the band width \({W\ll \sqrt{n}}\), we prove that the limit of the normalized second mixed moment of characteristic polynomials (as \({W, n\to \infty}\)) is equal to one, and so it does not coincide with that for GUE. This complements the result of Shcherbina (J Stat Phys 155(3):466–499, 2014) and proves the existence of the expected crossover for 1D Hermitian random band matrices at \({W\sim \sqrt{n}}\) on the level of characteristic polynomials.

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. Afanasiev I.: On the correlation functions of the characteristic polynomials of the sparse hermitian random matrices. J. Stat. Phys. 163, 324–356 (2016)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  2. Arfken, G.: Mathematical Methods for Physicists, 3rd edn. Academic Press, London (1985)

  3. Bao, J., Erdős, L.: Delocalization for a class of random block band matrices. Probab. Theory Relat. Fields (2016). doi:10.1007/s00440-015-0692-y

  4. Bogachev, L.V., Molchanov, S.A., Pastur, L.A.: On the level density of random band matrices. Mat. Zametki 50:6, 31–42 (1991)

  5. Bourgade, P., Erdős, L., Yau, H.-T., Yin, J.: Universality for a class of random band matrices. arXiv:1602.02312

  6. Brézin E., Hikami S.: Characteristic polynomials of random matrices. Commun. Math. Phys. 214, 111–135 (2000)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  7. Brézin E., Hikami S.: Characteristic polynomials of real symmetric random matrices. Commun. Math. Phys. 223, 363–382 (2001)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  8. Casati G., Molinari L., Israilev F.: Scaling properties of band random matrices. Phys. Rev. Lett. 64, 1851–1854 (1990)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  9. Disertori, M., Sodin, S.: Semi-classical analysis of non self-adjoint transfer matrices in statistical mechanics. I, Annales Henri Poincaré. http://dx.doi.org/10.1007/s00023-015-0397-x (2015)

  10. Erdős L., Knowles A.: Quantum diffusion and eigenfunction delocalization in a random band matrix model. Commun. Math. Phys. 303, 509–554 (2011)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  11. Erdős L., Knowles A., Yau H.T., Yin J.: Delocalization and diffusion profile for random band matrices. Commun. Math. Phys. 323, 367–416 (2013)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  12. Erdős L., Yau H.T., Yin J.: Bulk universality for generalized Wigner matrices. Probab. Theory Relat. Fields 154, 341–407 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  13. Fyodorov Y.V., Mirlin A.D.: Scaling properties of localization in random band matrices: a \({\sigma}\)-model approach. Phys. Rev. Lett. 67, 2405–2409 (1991)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  14. Molchanov S.A., Pastur L.A., Khorunzhii A.M.: Distribution of the eigenvalues of random band matrices in the limit of their infinite order. Theor. Math. Phys. 90, 108–118 (1992)

    Article  MathSciNet  Google Scholar 

  15. Pastur, L.A., Shcherbina, M.: Eigenvalue distribution of large random matrices. American Mathematical Society, Providence (2011)

  16. Peled, R., Schenker, J., Shamis, M., Sodin, A.: On the Wegner orbital model. arXiv:1608.02922

  17. Schenker J.: Eigenvector localization for random band matrices with power law band width. Commun. Math. Phys. 290, 1065–1097 (2009)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  18. Shcherbina T.: On the correlation function of the characteristic polynomials of the hermitian Wigner ensemble. Commun. Math. Phys. 308, 1–21 (2011)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  19. Shcherbina T.: On the correlation functions of the characteristic polynomials of the hermitian sample covariance ensemble. Probab. Theory Relat. Fields 156, 449–482 (2013)

    Article  MATH  Google Scholar 

  20. Shcherbina T.: On the second mixed moment of the characteristic polynomials of the 1D band matrices. Commun. Math. Phys. 328, 45–82 (2014)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  21. Shcherbina, T.: Universality of the local regime for the block band matrices with a finite number of blocks. J. Stat. Phys. 155(3), 466–499 (2014)

  22. Tao T., Vu V.: Random matrices: universality of the local eigenvalue statistics. Acta Math. 206, 127–204 (2011)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute for Low Temperature Physics, Kharkiv, Ukraine

    Mariya Shcherbina

  2. Karazin Kharkiv National University, Kharkiv, Ukraine

    Mariya Shcherbina

  3. School of Mathematics, Institute for Advanced Study, Princeton, NJ, USA

    Tatyana Shcherbina

  4. Department of Mathematics, Princeton University, Princeton, NJ, USA

    Tatyana Shcherbina

Authors
  1. Mariya Shcherbina
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Tatyana Shcherbina
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Tatyana Shcherbina.

Additional information

Communicated by L. Erdös

Tatyana Shcherbina was partially supported by NSF Grant DMS-1128155.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Shcherbina, M., Shcherbina, T. Characteristic Polynomials for 1D Random Band Matrices from the Localization Side. Commun. Math. Phys. 351, 1009–1044 (2017). https://doi.org/10.1007/s00220-017-2849-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-017-2849-2

Search

Navigation