## Random Schrödinger operators on long boxes, noise explosion and the GOE

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- by Benedek Valkó and Bálint Virág PDF
- Trans. Amer. Math. Soc.
**366**(2014), 3709-3728 Request permission

## Abstract:

It is conjectured that the eigenvalues of random Schrödinger operators at the localization transition in dimensions $d\ge 2$ behave like the eigenvalues of the Gaussian Orthogonal Ensemble (GOE). We show that there are sequences of $n\times m$ boxes with $1\ll m\ll n$ so that the eigenvalues in low disorder converge to Sine$_1$, 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.## References

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## Additional Information

**Benedek Valkó**- Affiliation: Department of Mathematics, University of Wisconsin-Madison, Madison,Wisconsin 53706
- Email: valko@math.wisc.edu
**Bálint Virág**- Affiliation: Department of Mathematics, University of Toronto, Ontario, Canada M5S 2E4
- MR Author ID: 641409
- Email: balint@math.toronto.edu
- Received by editor(s): December 13, 2011
- Received by editor(s) in revised form: October 2, 2012
- Published electronically: February 6, 2014
- © Copyright 2014
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc.
**366**(2014), 3709-3728 - MSC (2010): Primary 60B20, 81Q10
- DOI: https://doi.org/10.1090/S0002-9947-2014-05974-6
- MathSciNet review: 3192614