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

TL;DR

This paper demonstrates that the Anderson model on certain antitree graphs and deformed Laplacian operators on thin boxes in three-dimensional integer lattices exhibit GOE statistics, with eigenvalue processes converging to the Sine_1 process.

Contribution

It introduces new graph constructions and operator models where GOE eigenvalue statistics emerge without rescaling, expanding understanding of universality in random matrix theory.

Findings

Eigenvalue processes converge to the Sine_1 process

GOE statistics observed without matrix size rescaling

New models on antitrees and deformed Laplacians in 3D lattices

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 $\Delta$.