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


Background & Context


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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,


Wolfram Research (2015), CircularSymplecticMatrixDistribution, Wolfram Language function,


Wolfram Language. 2015. "CircularSymplecticMatrixDistribution." Wolfram Language & System Documentation Center. Wolfram Research.


Wolfram Language. (2015). CircularSymplecticMatrixDistribution. Wolfram Language & System Documentation Center. Retrieved from


@misc{reference.wolfram_2023_circularsymplecticmatrixdistribution, author="Wolfram Research", title="{CircularSymplecticMatrixDistribution}", year="2015", howpublished="\url{}", note=[Accessed: 03-December-2023 ]}


@online{reference.wolfram_2023_circularsymplecticmatrixdistribution, organization={Wolfram Research}, title={CircularSymplecticMatrixDistribution}, year={2015}, url={}, note=[Accessed: 03-December-2023 ]}