Corner eigenvalues of GOEs and orthogonally invariant random matrices with fixed spectrum random-matrices

Connor MacMahon, Leonid Petrov

Simulation Info

Corner eigenvalues of GOEs and orthogonally invariant random matrices with fixed spectrum random-matrices

Connor MacMahon, Leonid Petrov

This simulation computes the eigenvalues of successive corners of a random matrix. You can choose between two regimes:

GOE: The matrix is generated as a random Gaussian Orthogonal Ensemble (GOE) matrix.
Discrete Top Eigenvalue Profile: A diagonal matrix with 10 distinct eigenvalues (each with high multiplicity) is conjugated by a random Haar matrix. You can adjust the 10 discrete eigenvalues by dragging the red points or by using the numeric fields below.

Use the slider to set the matrix size \(N\) (maximum 300), then click “Resample” to generate a new simulation.

Simulation Regime:

Discrete Top Eigenvalue Profile GOE

Discrete Top Eigenvalue Profile (Drag the red points):

Drag the 10 red points horizontally to set the 10 distinct eigenvalues.

Discrete Profile Values

Matrix size \(N\): 50

Corner Eigenvalue Dot Plot:

code

(note: parameters in the code might differ from the ones in simulation results below)

Link to code (This simulation is interactive, written in JavaScript – see the source code of this page at the link)
Link to code (C++ code for the simulation)

Dear colleagues:

Feel free to use code (unless otherwise specified next to the corresponding link), data, and visualizations to illustrate your research in talks and papers, with attribution (CC BY-SA 4.0). Some images are available in very high resolution upon request. I can also produce other simulations upon request - email me at lenia.petrov@gmail.com

This material is based upon work supported by the National Science Foundation under Grant DMS-2153869