Plotting spectral laws

In this section, we will briefly explain what is the spectral distribution of a random matrix and how to plot it in order to study its properties. Most ensembles spectral distribution are explained through spectral laws, that are going to be detailed as well.

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

Spectral distribution

Most of the information of a random matrix is explained through their eigenvalues. Random matrix eigenvalues or random matrix spectrum follows a certain distribution depending on the ensemble. We would like to study that distribution in order to have insight of the random matrix limiting behaviour.

We can plot the eigenvalue density of any random matrix ensemble implemented in scikit-rmt. However, some technical details must be taken into account.

As an example, we can try to plot the spectral density of the Gaussian Orthogonal Ensemble (GOE) in the interval (-10, 10).

from skrmt.ensemble import GaussianEnsemble

goe = GaussianEnsemble(beta=1, n=1000)
goe.plot_eigval_hist(bins=80, interval=(-10, 10), normalize=False)

As we can see, the histogram does not explain barely anything about the ensemble. That is because we are not using correctly the theory behind the model.

Most of the ensembles need their eigenvalues to be normalized by a certain constant in order to perceive their limiting density properly. By default, the scikit-rmt is implemented to normalize eigenvalues with the more suitable constant depending on the ensemble.

In the former example, GOE matrices of size \(n \times n\) should be normalized by \(1/\sqrt{n}\), which is done by default or by setting normalize=True.

import numpy as np
from skrmt.ensemble import GaussianEnsemble

goe = GaussianEnsemble(beta=1, n=1000)
goe.plot_eigval_hist(bins=80, interval=(-10, 10), normalize=True)

Additionally, we can control the plotting interval to get a better picture of the spectral density. In the GOE case, the spectral distribution is concentrated in the (-2,2) interval. Note that the user does not need to worry about the interval, scikit-rmt uses an adequate interval by default.

import numpy as np
from skrmt.ensemble import GaussianEnsemble

goe = GaussianEnsemble(beta=1, n=1000)
goe.plot_eigval_hist(bins=80, interval=(-2, 2), normalize=True)

This example would be equivalent for Gaussian Unitary Ensemble (GUE) and Gaussian Symplectic Ensemble (GSE).

Another example would be using the Wishart Ensemble. In the following snippet of code Wishart Real Ensemble (WRE) is sampled.

from skrmt.ensemble import WishartEnsemble

wre = WishartEnsemble(beta=1, p=1000, n=3000)
wre.plot_eigval_hist(bins=80, interval=(-5,5))

By default, scikit-rmt takes charge of using the most suitable plotting  interval and eigenvalue normalization constant.

from skrmt.ensemble import WishartEnsemble

wre = WishartEnsemble(beta=1, p=1000, n=3000)
wre.plot_eigval_hist(bins=80)

This example would be equivalent for Wishart Complex Ensemble (WCE) and Wishart Quaternion Ensemble (WQE), with the detail of controlling its representation interval.

So far we have analyzed spectral distributions of GOE and WRE. In the first example it is easy to see that its eigenvalue distribution is quite similar to a semicircle. Thats because Gaussian Ensemble random matrices spectrum follow Wigner’s Semicircle Law. In the second example, the shown density has a very particular shape too, it is known as the Marchenko-Pastur Law. But, what is a spectral law?

Spectral laws

Spectral laws define random matrix eigenvalue distribution when its size goes to infinity. So the spectral laws describes the limiting behavior of the spectrum of a random matrix ensemble.

The most known spectral laws, and implemented by scikit-rmt, are the following:

• Winger Semicircle Law: describes the limiting behaviour of Gaussian Ensemble random matrix spectrum.

• Marchenko-Pastur Law: describes the limiting behaviour of Wishart Ensemble random matrix spectrum.

• Tracy-Widom Law: describes the limiting behaviour of the largest eigenvalue of a Gaussian Ensemble random matrix.

Anyone can forget about the specific plotting details and directly study those laws by using the functions provided in scikit-rmt.

We can easily analyze Wigner’s Semicircle Law as follows.

from skrmt.ensemble.spectral_law import WignerSemicircleDistribution

wsd = WignerSemicircleDistribution(beta=1)
wsd.plot_empirical_pdf(sample_size=2000, bins=80)

We can also study Marchenko-Pastur Law as by:

from skrmt.ensemble.spectral_law import MarchenkoPasturDistribution

mpd = MarchenkoPasturDistribution(beta=1, ratio=1/3)
mpd.plot_empirical_pdf(sample_size=2000, bins=80)

Finally, Tracy-Widom Law can be represented using:

from skrmt.ensemble.spectral_law import TracyWidomDistribution

twd = TracyWidomDistribution(beta=1)
twd.plot_empirical_pdf(sample_size=2000, bins=80)

Spectral laws analytical expression

The spectral laws described so far have been proven to converge to certain analytical functions that defines the limiting behaviour of the eigenvalue distribution of the random matrices. The functions of scikit-rmt that are capable of plotting the spectral laws also support the representation of the theoretical eigenvalue pdf.

The analytical probability function for the Gaussian Ensemble, known as Wigner Semicircle Law, supported on \([-R, R]\) and centered at \((0,0)\) is \(f(x) = \frac{2}{\pi R^2} \sqrt{R^2 - x^2}\).

from skrmt.ensemble.spectral_law import WignerSemicircleDistribution

wsd = WignerSemicircleDistribution(beta=1)
wsd.plot_empirical_pdf(sample_size=2000, bins=80, density=True, plot_law_pdf=True)

The analytical probability function for the Wishart Ensemble known as Marchenko-Pastur Law with parameter \(\lambda = p/n \in (0,1]\) is \(f_{\lambda}(x) = \frac{1}{2\pi \sigma^2}\frac{\sqrt{(\lambda_+ - x)(x - \lambda_-)}}{\lambda x}\), where \(\lambda_{\pm} = \sigma^2 (1 \pm \sqrt{\lambda})^2\). If \(\lambda > 1\) then the limiting distribution has an additional mass probability point in the origin of size \(1 - \frac{1}{\lambda}\).

from skrmt.ensemble.spectral_law import MarchenkoPasturDistribution

mpd = MarchenkoPasturDistribution(beta=1, ratio=1/3)
mpd.plot_empirical_pdf(sample_size=2000, bins=80, density=True, plot_law_pdf=True)

In the other hand, the Tracy-Widom Law has a complex analytical expression, that is the solution of a particular non-linear differential equation, described in detail in: S. Bauman. “The Tracy-Widom Distribution and its Application to Statistical Physics”. MIT Department of Physics. 2017. The package scikit-rmt represents a precise approximation of the theoretical Tracy-Widom pdf.

from skrmt.ensemble.spectral_law import TracyWidomDistribution

twd = TracyWidomDistribution(beta=1)
twd.plot_empirical_pdf(sample_size=2000, bins=80, density=True, plot_law_pdf=True)

Finally, the limiting distribution of the Manova Ensemble is not described by any famous Law, but its expression has been determined. This library provides functionality to show it on top of the empirical histogram of the eigenvalue spectrum.

from skrmt.ensemble.spectral_law import ManovaSpectrumDistribution

msd = ManovaSpectrumDistribution(beta=1, ratio_a=3, ratio_b=3)
msd.plot_empirical_pdf(sample_size=2000, bins=80, density=True, plot_law_pdf=True)