# GaussianSymplecticMatrixDistribution

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

represents a Gaussian symplectic matrix distribution with unit scale parameter.

# 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 :

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).

#### 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

#### BibTeX

@misc{reference.wolfram_2023_gaussiansymplecticmatrixdistribution, author="Wolfram Research", title="{GaussianSymplecticMatrixDistribution}", year="2017", howpublished="\url{https://reference.wolfram.com/language/ref/GaussianSymplecticMatrixDistribution.html}", note=[Accessed: 24-September-2023 ]}

#### BibLaTeX

@online{reference.wolfram_2023_gaussiansymplecticmatrixdistribution, organization={Wolfram Research}, title={GaussianSymplecticMatrixDistribution}, year={2017}, url={https://reference.wolfram.com/language/ref/GaussianSymplecticMatrixDistribution.html}, note=[Accessed: 24-September-2023 ]}