Poisson eigenvalue statistics for random Schr\"odinger operators on regular graphs

TL;DR

This paper investigates the spectral statistics of random Schr"odinger operators on finite regular graphs, demonstrating Poisson eigenvalue statistics in localized regimes, and introduces a novel eigenvector comparison method.

Contribution

It provides the first proof of Poisson eigenvalue statistics for such operators on regular graphs, adapting techniques to the graph's geometry.

Findings

Poisson eigenvalue statistics in localized regimes

Convergence of rescaled eigenvalues to a Poisson process

Development of a new eigenvector comparison approach

Abstract

For random operators it is conjectured that spectral properties of an infinite-volume operator are related to the distribution of spectral gaps of finite-volume approximations. In particular, localization and pure point spectrum in infinite volume is expected to correspond to Poisson eigenvalue statistics. Motivated by results about the Anderson model on the infinite tree we consider random Schr\"odinger operators on finite regular graphs. We study local spectral statistics: We analyze the number of eigenvalues in intervals with length comparable to the inverse of the number of vertices of the graph, in the limit where this number tends to infinity. We show that the random point process generated by the rescaled eigenvalues converges in certain spectral regimes of localization to a Poisson process. The corresponding result on the lattice was proved by Minami. However, due to the geometric structure of regular graphs the known methods turn out to be difficult to adapt. Therefore we develop a new approach based on direct comparison of eigenvectors.