Generate a random CUE matrix:

Verify that the matrix is unitary:

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

The eigenvalues lie on a unit circle:

Generate a random unitary matrix:

Generate a set of random unitary matrices:

Compute statistical properties numerically:

Expectation:

Simulate from the distribution:

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 CircularUnitaryMatrixDistribution[n]:

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:

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: