# CircularSymplecticMatrixDistribution

represents a circular symplectic matrix distribution with matrix dimensions {2 n,2 n} over the field of complex numbers.

# Examples

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## Basic Examples(2)

Generate a pseudorandom matrix from unitary symplectic group:

The random matrix is unitary:

It also verifies the symplectic self-duality condition:

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

## Scope(3)

Generate a single pseudorandom matrix:

Generate a set of pseudorandom matrices:

Compute statistical properties numerically:

## Applications(1)

The joint distribution of the eigenvalues for CircularSymplecticMatrixDistribution is also Boltzmann distribution of Dyson's Coulomb gas on a circle with inverse temperature . The average Hamiltonian per particle of the system is (without kinetic terms):

Define the distribution of the value of the Hamiltonian on random CSE matrix:

Compute the sample mean of the Hamiltonian for systems of different size:

Plot the sample means and compare them with thermodynamic limit:

## Properties & Relations(2)

Distribution of phase angle of the eigenvalues:

Compute the spacing between eigenvalues, taking into account that they come in pairs:

Compare the histogram of sample level spacings with the closed form, also known as Wigner surmise for Dyson index 4:

For eigenvectors of CircularSymplecticMatrixDistribution with dimension large, the scaled modulus of the quaternion elements is distributed:

Compare the histogram with PDF of ChiSquareDistribution:

## Possible Issues(1)

A matrix from CircularSymplecticMatrixDistribution need not be symplectic:

Use CircularQuaternionMatrixDistribution to randomly generate a unitary symplectic matrix:

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

#### Text

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

#### CMS

Wolfram Language. 2015. "CircularSymplecticMatrixDistribution." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/CircularSymplecticMatrixDistribution.html.

#### APA

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

#### BibTeX

@misc{reference.wolfram_2024_circularsymplecticmatrixdistribution, author="Wolfram Research", title="{CircularSymplecticMatrixDistribution}", year="2015", howpublished="\url{https://reference.wolfram.com/language/ref/CircularSymplecticMatrixDistribution.html}", note=[Accessed: 10-September-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_circularsymplecticmatrixdistribution, organization={Wolfram Research}, title={CircularSymplecticMatrixDistribution}, year={2015}, url={https://reference.wolfram.com/language/ref/CircularSymplecticMatrixDistribution.html}, note=[Accessed: 10-September-2024 ]}