Global Poissonian behavior of the eigenvalues and localization centers of random operators in the localized phase

Klopp, Frédéric

Séminaire Laurent Schwartz — EDP et applications (2011-2012), Exposé no. 9, 12 p.

Dans cette note, nous passerons en revue les résultats récents sur l’ergodicité asymptotique des valeurs propres et des centres de localisation des opérateurs aléatoires dans la phase localisées, obtenus dans [12]. Ces résultats généralisent ceux de [10] en prenant en compte les centres de localisation. Plus précisémenet, on démontre que, pour une classe générale d’opérateurs aléatoires, dans la région de localisation, les couples constitués des valeurs propres “dépliées” et du centre de localisation associé sont asymptotiquement ergodiques. On démontre également un résultat similaire lorsqu’on se restreint à de petits intervalles d’énergie.

In the present note, we review some recent results on the spectral statistics of random operators in the localized phase obtained in [12]. For a general class of random operators, we show that the family of the unfolded eigenvalues in the localization region considered jointly with the associated localization centers is asymptotically ergodic. This can be considered as a generalization of [10]. The benefit of the present approach is that one can vary the scaling of the unfolded eigenvalues covariantly with that of the localization centers. The convergence result then holds for all the scales that are asymptotically larger than the localization scale. We also provide a similar result that is localized in energy. Full proofs of the results presented here will be published elsewhere ([12]).

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