Random Schrodinger operators on long boxes, noise explosion and the GOE

Abstract

It is conjectured that the eigenvalues of random Schrodinger operators at the localization transition in dimensions d>=2 behave like the eigenvalues of the Gaussian Orthogonal Ensemble (GOE). We show that there are sequences of n by m boxes with 1<<m<<n so that the eigenvalues in low disorder converge to Sine1, the limiting eigenvalue process of the GOE. For the GOE case, this is the first example where Wigner's famous prediction is proven rigorously: we exhibit a complex system whose eigenvalues behave like those of random matrices.

26 pages. More details added to the proofs. The main result is now proved with a fixed 0<r<=1

References in corpus (3)

• Universality of Random Matrices and Local Relaxation Flow
• The local relaxation flow approach to universality of the local statistics for random matrices
• From the Anderson model on a strip to the DMPK equation and random matrix theory

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• A Universality Property of Gaussian Analytic Functions