GaussianSymplecticMatrixDistribution

GaussianSymplecticMatrixDistribution[σ,n]

represents a Gaussian symplectic matrix distribution with matrix dimensions {2 n,2 n} over the field of complex numbers and scale parameter σ.

GaussianSymplecticMatrixDistribution[n]

represents a Gaussian symplectic matrix distribution with unit scale parameter.

Details

Background & Context

Examples

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Basic Examples  (3)

Generate a pseudorandom matrix:

Check that the matrix is Hermitian:

Check that is also quaternion selfdual:

Represent the joint distribution of eigenvalues of a random matrix by MatrixPropertyDistribution and sample from it:

Mean and variance:

Scope  (4)

Generate a single pseudorandom matrix:

Generate a set of pseudorandom matrices:

Compute statistical properties numerically:

Distribution parameters estimation:

Estimate the distribution parameters from sample data:

Compare LogLikelihood of the distributions:

Applications  (2)

Sample eigenvalue spacing distribution in a 2by2 GSE matrix:

Compare the histogram with the closed form, also known as Wigner surmise for Dyson index :

Sample the joint distribution of eigenvalues of 2-by-2 GSE matrix:

Generically, eigenvalues have multiplicity 2:

Use RandomSample to randomly permute eigenvalues to compensate for algorithmspecific ordering:

Visualize estimated density:

Compare the estimated density to the known closed form result:

Evaluate the density for the case of 2-by-2 GSE matrices:

Compare the density to the histogram density estimate from the sample:

Confirm the agreement with a goodness-of-fit test:

Properties & Relations  (4)

Each realization of GaussianSymplecticMatrixDistribution is a Hermitian matrix:

Furthermore, a sample matrix from GaussianSymplecticMatrixDistribution satisfies quaternion self-duality condition:

MatrixExp applied to with sampled from GaussianSymplecticMatrixDistribution is unitary symplectic matrix:

Spectral density of large GSE matrix converges to WignerSemicircleDistribution:

Compare the histogram with the PDF:

The distribution of scaled largest eigenvalue of large GSE matrices converges to TracyWidomDistribution:

Compare sample histogram with the PDF of TracyWidomDistribution[4]:

Wolfram Research (2015), GaussianSymplecticMatrixDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/GaussianSymplecticMatrixDistribution.html (updated 2017).

Text

Wolfram Research (2015), GaussianSymplecticMatrixDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/GaussianSymplecticMatrixDistribution.html (updated 2017).

BibTeX

@misc{reference.wolfram_2021_gaussiansymplecticmatrixdistribution, author="Wolfram Research", title="{GaussianSymplecticMatrixDistribution}", year="2017", howpublished="\url{https://reference.wolfram.com/language/ref/GaussianSymplecticMatrixDistribution.html}", note=[Accessed: 18-October-2021 ]}

BibLaTeX

@online{reference.wolfram_2021_gaussiansymplecticmatrixdistribution, organization={Wolfram Research}, title={GaussianSymplecticMatrixDistribution}, year={2017}, url={https://reference.wolfram.com/language/ref/GaussianSymplecticMatrixDistribution.html}, note=[Accessed: 18-October-2021 ]}

CMS

Wolfram Language. 2015. "GaussianSymplecticMatrixDistribution." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2017. https://reference.wolfram.com/language/ref/GaussianSymplecticMatrixDistribution.html.

APA

Wolfram Language. (2015). GaussianSymplecticMatrixDistribution. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/GaussianSymplecticMatrixDistribution.html