Nodal Count for Orthogonally Invariant Ensembles

TL;DR

This paper studies the distribution of nodal counts of eigenvectors in large orthogonally invariant random matrices, showing it converges to the same law as the eigenvalue distribution, specifically the semicircle law for GOE, contradicting previous Gaussian predictions.

Contribution

It proves that nodal count distributions in large orthogonally invariant ensembles converge to the eigenvalue distribution, challenging prior conjectures of Gaussian behavior.

Findings

Nodal count distribution converges to the semicircle law for GOE.

Contradicts previous conjecture of Gaussian distribution for nodal counts.

Results hold as matrix size tends to infinity.

Abstract

We investigate the nodal count of eigenvectors of random matrices interpreted as operators on signed complete graphs. Our focus is on orthogonally invariant ensembles, with particular attention to the Gaussian Orthogonal Ensemble (GOE). We establish that, as the matrix size tends to infinity, the distribution of nodal counts converges to the same limiting law as the eigenvalue distribution. In the GOE case, this limit is the semicircle law. This result refutes a conjecture, motivated by quantum chaos and quantum graphs, which predicted Gaussian behavior of the nodal count.