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Asymptotic ergodicity of the eigenvalues of random operators in the localized phase
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  • Published: 22 February 2012

Asymptotic ergodicity of the eigenvalues of random operators in the localized phase

  • Frédéric Klopp1 

Probability Theory and Related Fields volume 155, pages 867–909 (2013)Cite this article

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Abstract

We prove that, for a general class of random operators, the family of the unfolded eigenvalues in the localization region is asymptotically ergodic in the sense of Minami (Spectra of random operators and related topics, 2011). Minami conjectured this to be the case for discrete Anderson model in the localized regime. We also provide a local analogue of this result. From the asymptotics ergodicity, one can recover the statistics of the level spacings as well as a number of other spectral statistics. Our proofs rely on the analysis developed in Germinet and Klopp (Spectral statistics for random Schrödinger operators in the localized regime, 2010).

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Authors and Affiliations

  1. LAGA, U.M.R. 7539 C.N.R.S, Institut Galilée, Université Paris-Nord, 99 Avenue J.-B. Clément, 93430, Villetaneuse, France

    Frédéric Klopp

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  1. Frédéric Klopp
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Correspondence to Frédéric Klopp.

Additional information

The author is partially supported by the grant ANR-08-BLAN-0261-01. He thanks the anonymous referees for useful suggestions that helped him improve the paper.

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Klopp, F. Asymptotic ergodicity of the eigenvalues of random operators in the localized phase. Probab. Theory Relat. Fields 155, 867–909 (2013). https://doi.org/10.1007/s00440-012-0415-6

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  • Received: 09 December 2010

  • Accepted: 09 July 2011

  • Published: 22 February 2012

  • Issue Date: April 2013

  • DOI: https://doi.org/10.1007/s00440-012-0415-6

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Mathematics Subject Classification (2000)

  • 47B80
  • 60H25
  • 82B44
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