Work

Nodal Count for Orthogonally Invariant Ensembles

Year: 2025

Type: preprint

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.

Source: arXiv (Cornell University)

DOI: 10.48550/arxiv.2511.02784

Authors: Alon, Lior, Mikulincer, Dan, Urschel, John

Institutions: University of Washington, Massachusetts Institute of Technology

Topic: Random Matrices and Applications

Subfield: Statistics and Probability

Field: Mathematics

Domain: Physical Sciences

Open Access status: green

Funders: National Science Foundation, Massachusetts Institute of Technology

Award: Estimating Eigenvalues and Matrix Functions with a Krylov Subspace

OpenAlex ID: W7103996007

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