Eigenvalue independence

In this tutorial, we highlight the fact that eigenvalues of a random matrix sample are not independent. However, random numbers sampled using the PDF of a spectral law (e.g. Wigner’s Semicircle law) are actually drawn independently.

# Author: Alejandro Santorum Varela
# License: BSD 3-Clause

Independent random samples vs. eigenvalues of a random matrix sample

Using scikit-rmt to simulate the behaviour of the ensembles, it is possible to illustrate how the eigenvalues of a random matrix sample are not independent, since they are draw from the same matrix sample. However, random variates sampled using the PDF of a spectral law (for example, Wigner’s Semicircle law) are drawn independently.

We can observe this phenomenon by plotting the histogram of a random matrix alongside the PDF of the corresponding spectral law. This is easily done by using the function plot_spectral_hist_and_law in skrmt.ensemble.utils:

from skrmt.ensemble.gaussian_ensemble import GaussianEnsemble
from skrmt.ensemble.utils import plot_spectral_hist_and_law

goe = GaussianEnsemble(beta=1, n=3000, tridiagonal_form=True)
plot_spectral_hist_and_law(ensemble=goe, bins=60)

In the previous example, the histogram of the spectrum of a sample from the Gaussian Ensemble was plotted next to the PDF of the Wigner’s Semicircle law.

It can be observed that the eigenvalues of a single sample of a random matrix are not independent since the fluctuations of the histogram compared to Wigner’s Semicircle PDF are really small. In contrast, if we compare independent random samples from the Wigner Semicircle law and the actual PDF we observe higher fluctuations for the same sample size (in this example, 3000).

from skrmt.ensemble.spectral_law import WignerSemicircleDistribution

wsd = WignerSemicircleDistribution(beta=1)
wsd.plot_empirical_pdf(
    sample_size=3000,
    bins=60,
    density=True,
    plot_law_pdf=True
)

Similarly, this can be seen for other ensembles. For example, with the Wishart Ensemble.

from skrmt.ensemble.wishart_ensemble import WishartEnsemble
from skrmt.ensemble.utils import plot_spectral_hist_and_law

wre = WishartEnsemble(beta=1, p=1000, n=3000, tridiagonal_form=True)
plot_spectral_hist_and_law(ensemble=wre, bins=60)

The fluctuations with respect the PDF of the Marchenko-Pastur law are larger if we directly draw independent random samples from that distribution:

from skrmt.ensemble.spectral_law import MarchenkoPasturDistribution

mpd = MarchenkoPasturDistribution(beta=1, ratio=1/3)
mpd.plot_empirical_pdf(
    sample_size=1000,
    bins=60,
    density=True,
    plot_law_pdf=True
)