# CircularUnitaryMatrixDistribution

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

# Examples

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## Basic Examples(2)

Generate a random CUE matrix:

Verify that the matrix is unitary:

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

## Scope(3)

Generate a random unitary matrix:

Generate a set of random unitary matrices:

Compute statistical properties numerically:

## Applications(4)

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 number of permutations of elements in which the longest increasing subsequence is at most of length can computed by averaging over , where are drawn from :

Compare with direct count:

The joint distribution of the eigenvalues for CircularUnitaryMatrixDistribution 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 CUE matrix:

Compute the sample mean of the Hamiltonian for systems of different size:

Plot the sample means and compare them with thermodynamic limit:

Construct Brownian motion on CUE by using matrices from GaussianUnitaryMatrixDistribution as infinitesimal generators:

Generate a Brownian path with initial matrix sampled from CircularUnitaryMatrixDistribution:

Compute the phase of the eigenvalues and compare them with the PDF of the eigenvalues of matrices from CircularUnitaryMatrixDistribution:

## 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 2:

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

Compare the histogram with PDF of ChiSquareDistribution:

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

#### Text

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

#### CMS

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

#### APA

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

#### BibTeX

@misc{reference.wolfram_2024_circularunitarymatrixdistribution, author="Wolfram Research", title="{CircularUnitaryMatrixDistribution}", year="2015", howpublished="\url{https://reference.wolfram.com/language/ref/CircularUnitaryMatrixDistribution.html}", note=[Accessed: 02-August-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_circularunitarymatrixdistribution, organization={Wolfram Research}, title={CircularUnitaryMatrixDistribution}, year={2015}, url={https://reference.wolfram.com/language/ref/CircularUnitaryMatrixDistribution.html}, note=[Accessed: 02-August-2024 ]}