Eigenvectors of the square grid plus GUE

TL;DR

This paper investigates how eigenvectors of a discrete torus matrix, when perturbed by GUE noise, transition from product structure to Gaussian waves, identifying the critical point of this phase change.

Contribution

It characterizes the phase transition in eigenvector structure of GUE-perturbed discrete tori, providing precise location of the transition point.

Findings

Eigenvectors retain product structure under small perturbations.

Eigenvectors converge to discrete Gaussian waves under large perturbations.

The phase transition point is explicitly determined.

Abstract

Eigenvectors of the GUE-perturbed discrete torus with uniform boundary conditions retain some product structure for small perturbations but converge to discrete Gaussian waves for large perturbations. We determine where this phase transition happens.