Statistical properties of eigenvectors and eigenvalues of structured random matrices

TL;DR

This paper analyzes the spectral properties of structured random matrices, revealing how eigenvector ergodicity and fractal dimensions depend on matrix structure, with implications for understanding complex systems.

Contribution

It derives asymptotic formulas for eigenvalue densities and eigenvector moments of structured random matrices, highlighting conditions for ergodicity and criticality.

Findings

Eigenvectors are generally ergodic under broad conditions.

The degree of ergodicity depends on the structure of W and D.

Eigenvectors can become critical with fractal properties when D=0 and W is random.

Abstract

We study the eigenvalues and the eigenvectors of $N\times N$ structured random matrices of the form $H = W\tilde{H}W+D$ with diagonal matrices $D$ and $W$ and $\tilde{H}$ from the Gaussian Unitary Ensemble. Using the supersymmetry technique we derive general asymptotic expressions for the density of states and the moments of the eigenvectors. We find that the eigenvectors remain ergodic under very general assumptions, but a degree of their ergodicity depends strongly on a particular choice of $W$ and $D$. For a special case of $D=0$ and random $W$, we show that the eigenvectors can become critical and are characterized by non-trivial fractal dimensions.