# A central limit theorem for products of random matrices and GOE statistics for the Anderson model on long boxes

Sadel C, Virág B. 2016. A central limit theorem for products of random matrices and GOE statistics for the Anderson model on long boxes. Communications in Mathematical Physics. 343(3), 881–919.

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Author

Sadel, Christian

^{ISTA}^{}; Virág, BálintDepartment

Abstract

We consider products of random matrices that are small, independent identically distributed perturbations of a fixed matrix (Formula presented.). Focusing on the eigenvalues of (Formula presented.) of a particular size we obtain a limit to a SDE in a critical scaling. Previous results required (Formula presented.) to be a (conjugated) unitary matrix so it could not have eigenvalues of different modulus. From the result we can also obtain a limit SDE for the Markov process given by the action of the random products on the flag manifold. Applying the result to random Schrödinger operators we can improve some results by Valko and Virag showing GOE statistics for the rescaled eigenvalue process of a sequence of Anderson models on long boxes. In particular, we solve a problem posed in their work.

Publishing Year

Date Published

2016-05-01

Journal Title

Communications in Mathematical Physics

Acknowledgement

Open access funding provided by Institute of Science and Technology (IST Austria). The work of C. Sadel was supported by NSERC Discovery Grant 92997-2010 RGPIN and by the People Programme (Marie Curie Actions) of the EU 7th Framework Programme FP7/2007-2013, REA Grant 291734.

Volume

343

Issue

3

Page

881 - 919

IST-REx-ID

### Cite this

Sadel C, Virág B. A central limit theorem for products of random matrices and GOE statistics for the Anderson model on long boxes.

*Communications in Mathematical Physics*. 2016;343(3):881-919. doi:10.1007/s00220-016-2600-4Sadel, C., & Virág, B. (2016). A central limit theorem for products of random matrices and GOE statistics for the Anderson model on long boxes.

*Communications in Mathematical Physics*. Springer. https://doi.org/10.1007/s00220-016-2600-4Sadel, Christian, and Bálint Virág. “A Central Limit Theorem for Products of Random Matrices and GOE Statistics for the Anderson Model on Long Boxes.”

*Communications in Mathematical Physics*. Springer, 2016. https://doi.org/10.1007/s00220-016-2600-4.C. Sadel and B. Virág, “A central limit theorem for products of random matrices and GOE statistics for the Anderson model on long boxes,”

*Communications in Mathematical Physics*, vol. 343, no. 3. Springer, pp. 881–919, 2016.Sadel, Christian, and Bálint Virág. “A Central Limit Theorem for Products of Random Matrices and GOE Statistics for the Anderson Model on Long Boxes.”

*Communications in Mathematical Physics*, vol. 343, no. 3, Springer, 2016, pp. 881–919, doi:10.1007/s00220-016-2600-4.**All files available under the following license(s):**

**Creative Commons Attribution 4.0 International Public License (CC-BY 4.0):**

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