Skip to main content
SpringerLink
Log in

The Altshuler–Shklovskii Formulas for Random Band Matrices I: the Unimodular Case

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

Abstract

We consider the spectral statistics of large random band matrices on mesoscopic energy scales. We show that the correlation function of the local eigenvalue density exhibits a universal power law behaviour that differs from the Wigner–Dyson– Mehta statistics. This law had been predicted in the physics literature by Altshuler and Shklovskii in (Zh Eksp Teor Fiz (Sov Phys JETP) 91(64):220(127), 1986); it describes the correlations of the eigenvalue density in general metallic sampleswith weak disorder. Our result rigorously establishes the Altshuler–Shklovskii formulas for band matrices. In two dimensions, where the leading term vanishes owing to an algebraic cancellation, we identify the first non-vanishing term and show that it differs substantially from the prediction of Kravtsov and Lerner in (Phys Rev Lett 74:2563–2566, 1995). The proof is given in the current paper and its companion (Ann. H. Poincaré. arXiv:1309.5107, 2014).

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. Aizenman M., Molchanov S.: Localization at large disorder and at extreme energies: an elementary derivation. Commun. Math. Phys. 157, 245–278 (1993)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  2. Aizenman M., Sims R., Warzel S.: Absolutely continuous spectra of quantum tree graphs with weak disorder. Commun. Math. Phys. 264, 371–389 (2006)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  3. Altschuler B.L.: Fluctuations in the extrinsic conductivity of disordered conductors. JETP Lett. 41, 648–651 (1985)

    ADS  Google Scholar 

  4. Altshuler, B.L., Shklovskii, B.I.: Repulsion of energy levels and the conductance of small metallic samples. Zh. Eksp. Teor. Fiz. (Sov. Phys. JETP) 91(64), 220(127) (1986)

  5. Anderson G., Zeitouni O.: A CLT for a band matrix model. Probab. Theory Relat. Fields 134, 283–338 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  6. Anderson P.W.: Absence of diffusion in certain random lattices. Phys. Rev. 109, 1492 (1958)

    Article  ADS  Google Scholar 

  7. Ayadi S.: Asymptotic properties of random matrices of long-range percolation model. Random Oper. Stoch. Equ. 17, 295–341 (2009)

    MATH  MathSciNet  Google Scholar 

  8. Boutetde Monvel A., Khorunzhy A.: Asymptotic distribution of smoothed eigenvalue density. I. Gaussian random matrices. Random Oper. Stoch. Equ. 7, 1–22 (1999)

    Article  MathSciNet  Google Scholar 

  9. Boutetde Monvel A., Khorunzhy A.: Asymptotic distribution of smoothed eigenvalue density. II. Wigner random matrices. Random Oper. Stoch. Equ. 7, 149–168 (1999)

    MathSciNet  Google Scholar 

  10. Dyson F.J., Mehta M.L.: Statistical theory of the energy levels of complex systems. IV. J. Math. Phys. 4, 701–713 (1963)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  11. Erdős, L., Knowles, A.: The Altshuler–Shklovskii formulas for random band matrices II: the general case. Ann. H. Poincaré (2014, preprint). arXiv:1309.5107

  12. Erdős L., Knowles A.: Quantum diffusion and delocalization for band matrices with general distribution. Ann. H. Poincaré 12, 1227–1319 (2011)

    Article  Google Scholar 

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

    Article  ADS  Google Scholar 

  14. Erdős L., Knowles A., Yau H.-T., Yin J.: Delocalization and diffusion profile for random band matrices. Commun. Math. Phys. 323(1), 367–416 (2013)

    Article  ADS  Google Scholar 

  15. Erdős L., Knowles A., Yau H.-T., Yin J.: The local semicircle law for a general class of random matrices. Electron. J. Probab. 18, 1–58 (2013)

    MathSciNet  Google Scholar 

  16. Erdős L., Salmhofer M., Yau H.-T.: Quantum diffusion for the Anderson model in the scaling limit. Ann. H. Poincaré 8, 621–685 (2007)

    Article  Google Scholar 

  17. Erdős L., Salmhofer M., Yau H.-T.: Quantum diffusion of the random Schrödinger evolution in the scaling limit II. the recollision diagrams. Commun. Math. Phys. 271, 1–53 (2007)

    Article  ADS  Google Scholar 

  18. Erdős L., Salmhofer M., Yau H.-T.: Quantum diffusion of the random Schrödinger evolution in the scaling limit. Acta Math. 200, 211–277 (2008)

    Article  MathSciNet  Google Scholar 

  19. Erdős L., Schlein B., Yau H.-T.: Local semicircle law and complete delocalization for Wigner random matrices. Commun. Math. Phys. 287, 641–655 (2009)

    Article  ADS  Google Scholar 

  20. Erdős L., Schlein B., Yau H.-T.: Universality of random matrices and local relaxation flow. Invent. Math. 185(1), 75–119 (2011)

    Article  ADS  MathSciNet  Google Scholar 

  21. Erdős L., Yau H.-T.: Linear Boltzmann equation as the weak coupling limit of the random Schrödinger equation. Commun. Math. Phys. 53, 667–735 (2000)

    Google Scholar 

  22. Erdős L., Yau H.-T.: Universality of local spectral statistics of random matrices. Bull. Am. Math. Soc 49, 377–414 (2012)

    Article  Google Scholar 

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

    Article  Google Scholar 

  24. Erdős L., Yau H.-T., Yin J.: Rigidity of eigenvalues of generalized Wigner matrices. Adv. Math. 229, 1435–1515 (2012)

    Article  MathSciNet  Google Scholar 

  25. Feldheim O.N., Sodin S.: A universality result for the smallest eigenvalues of certain sample covariance matrices. Geom. Funct. Anal. 20, 88–123 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  26. Froese R., Hasler D., Spitzer W.: Transfer matrices, hyperbolic geometry and absolutely continuous spectrum for some discrete Schrödinger operators on graphs. J. Funct. Anal. 230, 184–221 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  27. Fröhlich J., de Roeck W.: Diffusion of a massive quantum particle coupled to a quasi-free thermal medium in dimension \({d \geqslant 4}\) . Commun. Math. Phys. 303, 613–707 (2011)

    Article  ADS  MATH  Google Scholar 

  28. Fröhlich J., Martinelli F., Scoppola E., Spencer T.: Constructive proof of localization in the Anderson tight binding model. Commun. Math. Phys. 101, 21–46 (1985)

    Article  ADS  MATH  Google Scholar 

  29. Fröhlich J., Spencer T.: Absence of diffusion in the Anderson tight binding model for large disorder or low energy. Commun. Math. Phys. 88, 151–184 (1983)

    Article  ADS  MATH  Google Scholar 

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

    Article  ADS  MATH  MathSciNet  Google Scholar 

  31. Gradshteyn I.S., Ryzhik I.M.: Table of Integrals, Series, and Products, 7th edn. Academic Press, London (2007)

    MATH  Google Scholar 

  32. Klein A.: Absolutely continuous spectrum in the anderson model on the Bethe lattice. Math. Res. Lett. 1, 399–407 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  33. Kravtsov V.E., Lerner I.V.: Level correlations driven by weak localization in 2d systems. Phys. Rev. Lett. 74, 2563–2566 (1995)

    Article  ADS  Google Scholar 

  34. Lee P.A., Stone A.D.: Universal conductance fluctuations in metals. Phys. Rev. Lett. 55, 1622–1625 (1985)

    Article  ADS  Google Scholar 

  35. Li L., Soshnikov A.: Central limit theorem for linear statistics of eigenvalues of band random matrices. Random Matrices Theory Appl. 02, 1350009 (2013)

    Article  MathSciNet  Google Scholar 

  36. Mehta M.L.: Random Matrices. Academic press, London (2004)

    MATH  Google Scholar 

  37. Minami N.: Local fluctuation of the spectrum of a multidimensional Anderson tight binding model. Commun. Math. Phys. 177, 709–725 (1996)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  38. De Roeck W., Kupiainen A.: Diffusion for a quantum particle coupled to phonons in \({d \geqslant 3}\) . Commun. Math. Phys. 323(3), 889–973 (2013)

    Article  ADS  MATH  MathSciNet  Google Scholar 

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

    Article  ADS  MATH  MathSciNet  Google Scholar 

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

    Article  ADS  MATH  MathSciNet  Google Scholar 

  41. 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)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  42. Silvestrov P.G.: Summing graphs for random band matrices. Phys. Rev. E 55, 6419–6432 (1997)

    Article  ADS  Google Scholar 

  43. Sinai Y., Soshnikov A.: Central limit theorem for traces of large random symmetric matrices with independent matrix elements. Bol. Soc. Bras. Mat. 29, 1–24 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  44. Sodin S.: The spectral edge of some random band matrices. Ann. Math. 172(3), 2223–2251 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  45. Soshnikov A.: The central limit theorem for local linear statistics in classical compact groups and related combinatorial identities. Ann. Probab. 28, 1353–1370 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  46. Sosoe P., Wong P.: Regularity conditions in the CLT for linear eigenvalue statistics of Wigner matrices. Adv. Math. 249(20), 37–87 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  47. Spencer, T.: Random banded and sparse matrices, Chapter 23. In: Akemann, G., Baik, J., Di Francesco, P. (eds.) Oxford Handbook of Random Matrix Theory (2011)

  48. Tao T., Vu V.: Random matrices: universality of local eigenvalue statistics. Acta Math. 206, 1–78 (2011)

    Article  MathSciNet  Google Scholar 

  49. Thouless D.J.: Maximum metallic resistance in thin wires. Phys. Rev. Lett. 39, 1167–1169 (1977)

    Article  ADS  Google Scholar 

  50. Wigner E.P.: Characteristic vectors of bordered matrices with infinite dimensions. Ann. Math. 62, 548–564 (1955)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. IST Austria, Am Campus 1, Klosterneuburg, 3400, Austria

    László Erdős

  2. ETH Zürich, Zurich, Switzerland

    Antti Knowles

Authors
  1. László Erdős
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Antti Knowles
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Antti Knowles.

Additional information

Communicated by H.-T. Yau

László Erdős on leave from Institute of Mathematics, University of Munich. Partially supported by SFB-TR 12 Grant of the German Research Council.

Antti Knowles partially supported by Swiss National Science Foundation Grant 144662.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Erdős, L., Knowles, A. The Altshuler–Shklovskii Formulas for Random Band Matrices I: the Unimodular Case. Commun. Math. Phys. 333, 1365–1416 (2015). https://doi.org/10.1007/s00220-014-2119-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-014-2119-5

Keywords

Search

Navigation