# CircularOrthogonalMatrixDistribution

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

# Details # Background & Context

• , also referred to as the circular orthogonal ensemble (COE), represents a statistical distribution over the unitary and symmetric complex matrices, namely complex square matrices satisfying both and , where denotes the transpose of , the conjugate transpose of and is the identity matrix. Here, the parameter n is called the dimension parameter of the distribution and may be any positive integer. Despite the name "circular orthogonal matrix distribution", while matrices belonging to this distribution are unitary ( ), they are not necessarily orthogonal ( ).
• Along with the circular symplectic and circular unitary matrix distributions (CircularSymplecticMatrixDistribution and CircularUnitaryMatrixDistribution, respectively), the circular orthogonal matrix distribution was one of three circle matrix ensembles originally devised by Freeman Dyson in 1962 as a tool to study quantum mechanics. Probabilistically, the circular orthogonal matrix distribution represents a uniform distribution over the collection of symmetric unitary square matrices, while mathematically it is a so-called Haar measure on the subset of all symmetric matrices within the unitary group . Matrix ensembles like the circular orthogonal 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 orthogonal 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,CircularOrthogonalMatrixDistribution[n]], written more concisely as ACircularOrthogonalMatrixDistribution[n], can be used to assert that a random matrix A is distributed according to a circular orthogonal matrix distribution. Such an assertion can be used in functions such as MatrixPropertyDistribution.
• The trace, eigenvalues and norm of variates distributed according to circular orthogonal matrix distribution may be computed using Tr, Eigenvalues and Norm, respectively. Such variates may also be examined with MatrixFunction, MatrixPower, and related real quantities such as the real part (Re), imaginary part (Im) and complex argument (Arg) can be plotted using MatrixPlot.
• CircularOrthogonalMatrixDistribution is related to a number of other distributions. As discussed above, it is qualitatively similar to other circular matrix distributions such as CircularQuaternionMatrixDistribution, CircularRealMatrixDistribution, CircularSymplecticMatrixDistribution and CircularUnitaryMatrixDistribution. Originally, the circular matrix ensembles were derived as generalizations of the so-called Gaussian ensembles, and so CircularOrthogonalMatrixDistribution is related to GaussianOrthogonalMatrixDistribution, GaussianSymplecticMatrixDistribution and GaussianUnitaryMatrixDistribution. CircularOrthogonalMatrixDistribution is also related to MatrixNormalDistribution, MatrixTDistribution, WishartMatrixDistribution, InverseWishartMatrixDistribution, TracyWidomDistribution and WignerSemicircleDistribution.

# 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: