CircularOrthogonalMatrixDistribution

represents a circular orthogonal matrix distribution with matrix dimensions {n,n}.

Examples

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Basic Examples(2)

Generate a pseudorandom COE matrix:

It is both unitary and symmetric:

Sample a random point on a sphere using MatrixPropertyDistribution:

The distribution is visibly clustered around the axes:

Scope(3)

Generate a single pseudorandom matrix:

Generate a set of pseudorandom matrices:

Compute statistical properties numerically:

Applications(2)

Define distribution of complex arguments of random matrix eigenvalues:

Sample the phases of eigenvalues followed by random permutations:

Visualize joint phase distribution together with the closed-form PDF:

The joint distribution of the eigenvalues for CircularOrthogonalMatrixDistribution 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 Hamiltonian on random COE 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:

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

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

Compare the histogram with PDF of ChiSquareDistribution:

Possible Issues(2)

A matrix from CircularOrthogonalMatrixDistribution need not be orthogonal:

Use CircularRealMatrixDistribution to sample a random orthogonal real-valued matrix:

Matrix from CircularOrthogonalMatrixDistribution can be represented as matrix , where matrix follows CircularUnitaryMatrixDistribution:

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

Text

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

CMS

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

APA

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

BibTeX

@misc{reference.wolfram_2024_circularorthogonalmatrixdistribution, author="Wolfram Research", title="{CircularOrthogonalMatrixDistribution}", year="2015", howpublished="\url{https://reference.wolfram.com/language/ref/CircularOrthogonalMatrixDistribution.html}", note=[Accessed: 17-June-2024 ]}

BibLaTeX

@online{reference.wolfram_2024_circularorthogonalmatrixdistribution, organization={Wolfram Research}, title={CircularOrthogonalMatrixDistribution}, year={2015}, url={https://reference.wolfram.com/language/ref/CircularOrthogonalMatrixDistribution.html}, note=[Accessed: 17-June-2024 ]}