Random Schrodinger operators on long boxes, noise explosion and the GOE
Probability
2015-09-25 v4 Mathematical Physics
math.MP
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.
Cite
@article{arxiv.0912.0097,
title = {Random Schrodinger operators on long boxes, noise explosion and the GOE},
author = {Benedek Valko and Balint Virag},
journal= {arXiv preprint arXiv:0912.0097},
year = {2015}
}
Comments
26 pages. More details added to the proofs. The main result is now proved with a fixed 0<r<=1