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

TL;DR

This paper proves that eigenvalues of a broad class of random operators exhibit asymptotic ergodicity in the localized phase, confirming Minami's conjecture for the Anderson model and enabling the recovery of spectral statistics.

Contribution

It establishes the asymptotic ergodicity of eigenvalues in the localized phase for general random operators, extending Minami's conjecture to a wider class.

Findings

Eigenvalues are asymptotically ergodic in the localization region.

The results recover level spacing statistics and other spectral statistics.

Provides a local analogue of the ergodicity result.

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 N. Minami (see [Mi:11]). N. 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 abs/1011.1832.