Abstract
We consider the discrete Anderson model and prove enhanced Wegner and Minami estimates where the interval length is replaced by the IDS computed on the interval. We use these estimates to improve on the description of finite volume eigenvalues and eigenfunctions obtained in Germinet and Klopp (J Eur Math Soc http://arxiv.org/abs/1011.1832, 2010). As a consequence of the improved description of eigenvalues and eigenfunctions, we revisit a number of results on the spectral statistics in the localized regime obtained in Germinet and Klopp (J Eur Math Soc http://arxiv.org/abs/1011.1832, 2010) and Klopp (PTRF http://fr.arxiv.org/abs/1012.0831, 2010) and extend their domain of validity, namely:
-
the local spectral statistics for the unfolded eigenvalues;
-
the local asymptotic ergodicity of the unfolded eigenvalues.
In dimension 1, for the standard Anderson model, the improvement enables us to obtain the local spectral statistics at band edge, that is in the Lifshitz tail regime. In higher dimensions, this works for modified Anderson models.
Article PDF
Similar content being viewed by others
References
Aizenman, M., Schenker, J.H., Friedrich, R.M., Hundertmark, D.: Finite-volume fractional-moment criteria for Anderson localization. Commun. Math. Phys. 224(1), 219–253 (2001). Dedicated to Joel L. Lebowitz
Bellissard J.V., Hislop P.D., Stolz G.: Correlation estimates in the Anderson model. J. Stat. Phys. 129(4), 649–662 (2007)
Combes J.-M., Germinet F., Klein A.: Generalized eigenvalue-counting estimates for the Anderson model. J. Stat. Phys. 135(2), 201–216 (2009)
Costin O., Lebowitz J.L.: Gaussian fluctuation in random matrices. Phys. Rev. Lett. 75(1), 69–72 (1995)
Durrett R.: Probability: theory and examples second edition. Duxbury Press, Belmont (1996)
Germinet F., Klein A.: Bootstrap multiscale analysis and localization in random media. Commun. Math. Phys. 222(2), 415–448 (2001)
Germinet F., Klein A.: A characterization of the Anderson metal-insulator transport transition. Duke Math. J. 124(2), 309–350 (2004)
Germinet F., Klein A.: New characterizations of the region of complete localization for random Schrödinger operators. J. Stat. Phys. 122(1), 73–94 (2006)
Germinet, F., Klein, A.: A comprehensive proof of localization for continuous Anderson models with singular random potentials. J. Eur. Math. Soc. (to appear)
Germinet, F., Klopp, F.: Spectral statistics for random Schrödinger operators in the localized regime. J. Eur. Math. Soc. (2010). http://arxiv.org/abs/1011.1832
Graf G.M., Vaghi A.: A remark on the estimate of a determinant by Minami. Lett. Math. Phys. 79(1), 17–22 (2007)
Grenkova L.N., Molchanov S.A., Sudarev Yu.N.: The structure of the edge of the multidimensional Anderson model spectrum. Teoret. Mat. Fiz. 85(1), 32–40 (1990)
Gustavsson J.: Gaussian fluctuations of eigenvalues in the GUE. Ann. Inst. H. Poincaré Probab. Statist. 41(2), 151–178 (2005)
Kirsch, W.: An invitation to random Schrödinger operators. In: Random Schrödinger Operators. Panor. Synthèses, vol. 25, pp. 1–119. Soc. Math. France, Paris, 2008. With an appendix by Frédéric Klopp
Klein A., Molchanov S.: Simplicity of eigenvalues in the Anderson model. J. Stat. Phys. 122(1), 95–99 (2006)
Kirsch, W., Metzger, B.: The integrated density of states for random Schrödinger operators. In: Spectral theory and mathematical physics: a Festschrift in honor of Barry Simon’s 60th birthday. Proceedings of Symposia in Pure Mathematics, vol. 76, pp. 649–696. American Mathematical Society, Providence, RI (2007)
Klopp F.: Band edge behavior of the integrated density of states of random Jacobi matrices in dimension 1. J. Stat. Phys. 90(3–4), 927–947 (1998)
Klopp F.: Internal Lifshits tails for random perturbations of periodic Schrödinger operators. Duke Math. J. 98(2), 335–396 (1999)
Klopp F.: Correction to: “Internal Lifshits tails for random perturbations of periodic Schrödinger operators” [18]. Duke Math. J. 109 109(2), 411–412 (2001)
Klopp F.: Weak disorder localization and Lifshitz tails. Commun. Math. Phys. 232(1), 125–155 (2002)
Klopp, F.: Asymptotic ergodicity of the eigenvalues of random operators in the localized phase. PTRF (2010). http://fr.arxiv.org/abs/1012.0831
Klopp, F.: Universal joint asymptotic ergodicity of the eigenvalues and localization centers of random operators in the localized phase (2011, in preparation)
Minami N.: Local fluctuation of the spectrum of a multidimensional Anderson tight binding model. Commun. Math. Phys. 177(3), 709–725 (1996)
Minami, N.: Theory of point processes and some basic notions in energy level statistics. In: Probability and mathematical physics. CRM Proceedings and Lecture Notes, vol.~42, pp. 353–398. American Mathematical Society, Providence, RI (2007)
Minami, N.: Energy level statistics: a formulation and some examples. In: Spectra of Random Operators and Related Topics, pp. 79–97. RIMS, Kyoto (2011)
Molchanov S.A.: The local structure of the spectrum of a random one-dimensional Schrödinger operator. Trudy Sem. Petrovsk. 8, 195–210 (1982)
Pastur, L., Figotin, A.: Spectra of random and almost-periodic operators. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 297. Springer, Berlin (1992)
Texier C.: Individual energy level distributions for one-dimensional diagonal and off-diagonal disorder. J. Phys. A 33(35), 6095–6128 (2000)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Jean Bellissard.
The authors are supported by the Grant ANR-08-BLAN-0261-01. The authors would also like to thank the Centre Interfacultaire Bernoulli (EPFL, Lausanne) for its hospitality.
Rights and permissions
About this article
Cite this article
Germinet, F., Klopp, F. Enhanced Wegner and Minami Estimates and Eigenvalue Statistics of Random Anderson Models at Spectral Edges. Ann. Henri Poincaré 14, 1263–1285 (2013). https://doi.org/10.1007/s00023-012-0217-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00023-012-0217-5