WOLFRAM

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

Details

Background & Context

Examples

open allclose all

Basic Examples  (2)Summary of the most common use cases

Generate a pseudorandom matrix from unitary symplectic group:

The random matrix is unitary:

Out[2]=2

It also verifies the symplectic self-duality condition:

Out[6]=6

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

Out[1]=1

Scope  (3)Survey of the scope of standard use cases

Generate a single pseudorandom matrix:

Out[10]=10

Generate a set of pseudorandom matrices:

Out[1]=1

Compute statistical properties numerically:

Out[2]=2

Applications  (1)Sample problems that can be solved with this function

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:

Out[3]=3

Plot the sample means and compare them with thermodynamic limit:

Out[4]=4

Properties & Relations  (2)Properties of the function, and connections to other functions

Distribution of phase angle of the eigenvalues:

Out[3]=3

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:

Out[12]=12

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

Compare the histogram with PDF of ChiSquareDistribution:

Out[3]=3

Possible Issues  (1)Common pitfalls and unexpected behavior

A matrix from CircularSymplecticMatrixDistribution need not be symplectic:

Out[4]=4

Use CircularQuaternionMatrixDistribution to randomly generate a unitary symplectic matrix:

Out[7]=7
Wolfram Research (2015), CircularSymplecticMatrixDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/CircularSymplecticMatrixDistribution.html.
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.

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.

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

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

BibTeX

@misc{reference.wolfram_2025_circularsymplecticmatrixdistribution, author="Wolfram Research", title="{CircularSymplecticMatrixDistribution}", year="2015", howpublished="\url{https://reference.wolfram.com/language/ref/CircularSymplecticMatrixDistribution.html}", note=[Accessed: 08-July-2025 ]}

@misc{reference.wolfram_2025_circularsymplecticmatrixdistribution, author="Wolfram Research", title="{CircularSymplecticMatrixDistribution}", year="2015", howpublished="\url{https://reference.wolfram.com/language/ref/CircularSymplecticMatrixDistribution.html}", note=[Accessed: 08-July-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_circularsymplecticmatrixdistribution, organization={Wolfram Research}, title={CircularSymplecticMatrixDistribution}, year={2015}, url={https://reference.wolfram.com/language/ref/CircularSymplecticMatrixDistribution.html}, note=[Accessed: 08-July-2025 ]}

@online{reference.wolfram_2025_circularsymplecticmatrixdistribution, organization={Wolfram Research}, title={CircularSymplecticMatrixDistribution}, year={2015}, url={https://reference.wolfram.com/language/ref/CircularSymplecticMatrixDistribution.html}, note=[Accessed: 08-July-2025 ]}