CircularSymplecticMatrixDistribution

CircularSymplecticMatrixDistribution[n]

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

Details

Background & Context

Examples

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Basic Examples  (2)

Generate a pseudorandom matrix from unitary symplectic group:

The random matrix is unitary:

It also verifies the symplectic self-duality condition:

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

Scope  (3)

Generate a single pseudorandom matrix:

Generate a set of pseudorandom matrices:

Compute statistical properties numerically:

Applications  (1)

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):

Define the distribution of the value of the Hamiltonian on random CSE 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, taking into account that they come in pairs:

Compare the histogram of sample level spacings with the closed form, also known as Wigner surmise for Dyson index 4:

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

Compare the histogram with PDF of ChiSquareDistribution:

Possible Issues  (1)

A matrix from CircularSymplecticMatrixDistribution need not be symplectic:

Use CircularQuaternionMatrixDistribution to randomly generate a unitary symplectic matrix:

Introduced in 2015
 (10.3)