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

Abstract

Sequences of certain finite graphs, antitrees, are constructed along which the Anderson model shows GOE statistics, i.e. a re-scaled eigenvalue process converges to the ${\rm Sine}_1$ 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 Schr\"odinger 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 $\Delta$.