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


  • CircularSymplecticMatrixDistribution is also known as circular symplectic ensemble, or CSE.
  • CircularSymplecticMatrixDistribution represents a uniform distribution over the self-dual unitary quaternionic square matrices of dimension n.
  • The dimension parameter n can be any positive integer.
  • CircularSymplecticMatrixDistribution can be used with such functions as MatrixPropertyDistribution and RandomVariate.

Background & Context

  • CircularSymplecticMatrixDistribution[n], also referred to as the circular symplectic ensemble (CSE), represents a statistical distribution over the unitary and self-dual complex matrices, namely complex square matrices of even dimension satisfying both and , where denotes the conjugate transpose of , the identity matrix, the transpose of and is a symplectic matrix of the form with the Kronecker product. The parameter n is called the dimension parameter of the distribution and may be any positive integer. Despite the name "circular symplectic matrix distribution", matrices belonging to this distribution need not be symplectic.
  • Along with the circular orthogonal and circular unitary matrix distributions (CircularOrthogonalMatrixDistribution and CircularUnitaryMatrixDistribution, respectively), the circular symplectic matrix distribution was one of three so-called circle matrix ensembles originally devised by Freeman Dyson in 1962 as a tool to study quantum mechanics. Probabilistically, the circular symplectic matrix distribution represents a uniform distribution over the self-dual unitary quaternionic square matrices. Matrix ensembles like the circular symplectic matrix distribution are of considerable importance in the study of random matrix theory, as well as in various branches of physics and mathematics.
  • RandomVariate can be used to give one or more machine- or arbitrary-precision (the latter via the WorkingPrecision option) pseudorandom variates from a circular symplectic matrix distribution, and the mean, median, variance, raw moments and central moments of a collection of such variates may then be computed using Mean, Median, Variance, Moment and CentralMoment, respectively. Distributed[A,CircularSymplecticMatrixDistribution[n]], written more concisely as ACircularSymplecticMatrixDistribution[n] , can be used to assert that a random matrix A is distributed according to a circular symplectic matrix distribution. Such an assertion can then be used in functions such as MatrixPropertyDistribution.
  • The trace, eigenvalues and norm of variates distributed according to circular symplectic matrix distribution may be computed using Tr, Eigenvalues and Norm, respectively. Such variates may also be examined with MatrixFunction, MatrixPower, and real quantities related thereto, such as the real part (Re), imaginary part (Im) and complex argument (Arg), can be plotted using MatrixPlot.
  • CircularSymplecticMatrixDistribution is related to a number of other distributions. As discussed above, it is qualitatively similar to other circular matrix distributions such as CircularQuaternionMatrixDistribution, CircularRealMatrixDistribution, CircularOrthogonalMatrixDistribution and CircularUnitaryMatrixDistribution. Originally, the circular matrix ensembles were derived as generalizations of the so-called Gaussian ensembles, and so CircularSymplecticMatrixDistribution is related to GaussianOrthogonalMatrixDistribution, GaussianSymplecticMatrixDistribution and GaussianUnitaryMatrixDistribution. CircularSymplecticMatrixDistribution is also related to MatrixNormalDistribution, MatrixTDistribution, WishartMatrixDistribution, InverseWishartMatrixDistribution, TracyWidomDistribution and WignerSemicircleDistribution.


open allclose all

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_2024_circularsymplecticmatrixdistribution, author="Wolfram Research", title="{CircularSymplecticMatrixDistribution}", year="2015", howpublished="\url{}", note=[Accessed: 20-May-2024 ]}


@online{reference.wolfram_2024_circularsymplecticmatrixdistribution, organization={Wolfram Research}, title={CircularSymplecticMatrixDistribution}, year={2015}, url={}, note=[Accessed: 20-May-2024 ]}