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

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).