Nodal Count for Orthogonally Invariant Ensembles
Mathematical Physics
2025-11-05 v1 math.MP
Spectral Theory
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.
Keywords
Cite
@article{arxiv.2511.02784,
title = {Nodal Count for Orthogonally Invariant Ensembles},
author = {Lior Alon and Dan Mikulincer and John Urschel},
journal= {arXiv preprint arXiv:2511.02784},
year = {2025}
}