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

- 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.
Examples
open allclose allBasic Examples (2)Summary of the most common use cases
Generate a pseudorandom matrix from unitary symplectic group:

https://wolfram.com/xid/0g0re02oodlh03c1my2a0c6-lcf25m

https://wolfram.com/xid/0g0re02oodlh03c1my2a0c6-kp68hx

It also verifies the symplectic self-duality condition:

https://wolfram.com/xid/0g0re02oodlh03c1my2a0c6-jsy9ui

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

https://wolfram.com/xid/0g0re02oodlh03c1my2a0c6-eq63i5

Scope (3)Survey of the scope of standard use cases
Generate a single pseudorandom matrix:

https://wolfram.com/xid/0g0re02oodlh03c1my2a0c6-d13sc9

Generate a set of pseudorandom matrices:

https://wolfram.com/xid/0g0re02oodlh03c1my2a0c6-y6wkjj

Compute statistical properties numerically:

https://wolfram.com/xid/0g0re02oodlh03c1my2a0c6-6gedk9

https://wolfram.com/xid/0g0re02oodlh03c1my2a0c6-3m1a2c

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):

https://wolfram.com/xid/0g0re02oodlh03c1my2a0c6-eodd25
Define the distribution of the value of the Hamiltonian on random CSE matrix:

https://wolfram.com/xid/0g0re02oodlh03c1my2a0c6-lnb1ly
Compute the sample mean of the Hamiltonian for systems of different size:

https://wolfram.com/xid/0g0re02oodlh03c1my2a0c6-l5kr5a

Plot the sample means and compare them with thermodynamic limit:

https://wolfram.com/xid/0g0re02oodlh03c1my2a0c6-vau38

Properties & Relations (2)Properties of the function, and connections to other functions
Distribution of phase angle of the eigenvalues:

https://wolfram.com/xid/0g0re02oodlh03c1my2a0c6-po5mg

https://wolfram.com/xid/0g0re02oodlh03c1my2a0c6-c5tff8

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

https://wolfram.com/xid/0g0re02oodlh03c1my2a0c6-k7htx4
Compare the histogram of sample level spacings with the closed form, also known as Wigner surmise for Dyson index 4:

https://wolfram.com/xid/0g0re02oodlh03c1my2a0c6-i7msh9

https://wolfram.com/xid/0g0re02oodlh03c1my2a0c6-ccvw5k

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

https://wolfram.com/xid/0g0re02oodlh03c1my2a0c6-dnlvx8

https://wolfram.com/xid/0g0re02oodlh03c1my2a0c6-et9c22
Compare the histogram with PDF of ChiSquareDistribution:

https://wolfram.com/xid/0g0re02oodlh03c1my2a0c6-rbum

Possible Issues (1)Common pitfalls and unexpected behavior
A matrix from CircularSymplecticMatrixDistribution need not be symplectic:

https://wolfram.com/xid/0g0re02oodlh03c1my2a0c6-bq6rxq

https://wolfram.com/xid/0g0re02oodlh03c1my2a0c6-dnpjex

https://wolfram.com/xid/0g0re02oodlh03c1my2a0c6-dsnn3u

Use CircularQuaternionMatrixDistribution to randomly generate a unitary symplectic matrix:

https://wolfram.com/xid/0g0re02oodlh03c1my2a0c6-e1fw6s

https://wolfram.com/xid/0g0re02oodlh03c1my2a0c6-bpwc97

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
]}
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
]}