Electronic Journal of Probability

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

Abstract

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

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

Dates
Accepted: 11 June 2018
First available in Project Euclid: 8 August 2018

https://projecteuclid.org/euclid.ejp/1533715241

Digital Object Identifier
doi:10.1214/18-EJP187

Mathematical Reviews number (MathSciNet)
MR3858904

Zentralblatt MATH identifier
06924688

Citation

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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