Corner processes with unitary invariance and outliers     random-matrices

This page computes eigenvalues of successive top-left corners of three different complex Hermitian random-matrix ensembles (each with an option to add up to five outliers).

• 10-Point Atomic: a diagonal matrix with 10 distinct eigenvalues (each repeated proportionally in size \(N\)), plus 5 outliers in the last 5 diagonal entries, all conjugated by a random complex unitary.
• GUE: a complex Hermitian GUE matrix + a rank $\le 5$ perturbation in the first 5 diagonal entries.
• Rotated GUE: a random complex Hermitian GUE matrix + a rank-5 diagonal perturbation \(U D U^\dagger\), where \(D\) has up to 5 outliers. The difference with the previous ensemble is that the perturbation is free with respect to the original GUE matrix.

Adjust \(N\) (up to 500) and outlier values, then click “Resample” to see the corner eigenvalue scatter plot.