Electronic Journal of Probability

GOE statistics for Anderson models on antitrees and thin boxes in $\mathbb{Z} ^3$ with deformed Laplacian

Christian Sadel

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Sequences of certain finite graphs - special types of antitrees - are constructed along which the Anderson model shows GOE statistics, i.e. a re-scaled eigenvalue process converges to the ${\mathrm{Sine} }_1$ process. The Anderson model on the graph is a random matrix being the sum of the adjacency matrix and a random diagonal matrix with independent identically distributed entries along the diagonal. The strength of the randomness stays fixed, there is no re-scaling with matrix size. These considered random matrices giving GOE statistics can also be viewed as random Schrödinger operators $\mathcal{P} \Delta +\mathcal{V} $ on thin finite boxes in $\mathbb{Z} ^3$ where the Laplacian $\Delta $ is deformed by a projection $\mathcal{P} $ commuting with $\Delta $.

Article information

Electron. J. Probab., Volume 23 (2018), paper no. 76, 24 pp.

Received: 24 November 2017
Accepted: 11 June 2018
First available in Project Euclid: 8 August 2018

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Zentralblatt MATH identifier

Primary: 60B20: Random matrices (probabilistic aspects; for algebraic aspects see 15B52) 82B44: Disordered systems (random Ising models, random Schrödinger operators, etc.) 60H25: Random operators and equations [See also 47B80] 15B52: Random matrices

Anderson model universality local statistics

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Sadel, Christian. GOE statistics for Anderson models on antitrees and thin boxes in $\mathbb{Z} ^3$ with deformed Laplacian. Electron. J. Probab. 23 (2018), paper no. 76, 24 pp. doi:10.1214/18-EJP187. https://projecteuclid.org/euclid.ejp/1533715241

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