represents a Gaussian unitary matrix distribution with matrix dimensions {n,n} and scale parameter σ.


represents a Gaussian unitary matrix distribution with unit scale parameter.


Background & Context


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Basic Examples  (4)

Generate a pseudorandom GUE matrix:

Check that it is Hermitian:

Independent real components of a matrix from GaussianUnitaryMatrixDistribution are jointly Gaussian, uncorrelated, with entries off the diagonal having half the variance of entries on the diagonal:

Use MatrixPropertyDistribution to sample eigenvalues of GUE matrices:

Mean and variance:

Scope  (4)

Generate a single pseudorandom matrix:

Generate a set of pseudorandom matrices:

Compute statistical properties numerically:

Distribution parameters estimation:

Estimate the distribution parameters from sample data:

Compare LogLikelihood of the distributions:

Applications  (3)

Sample eigenvalue spacing distribution in a 2by2 GUE matrix:

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

Sample the joint distribution of eigenvalues of 2-by-2 GUE matrix:

Use RandomSample to randomly permute eigenvalues to compensate for algorithmspecific ordering:

Visualize estimated density:

Compare the estimated density to the known closed form result:

Evaluate the density for the case of 2-by-2 GUE matrices:

Compare the density to the histogram density estimate from the sample:

Confirm the agreement with a goodness-of-fit test:

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  (4)

MatrixExp applied to with sampled from GaussianUnitaryMatrixDistribution is unitary:

Estimate the parameters of the distribution of elements of matrices drawn from GaussianUnitaryMatrixDistribution, assuming they are independent and normally distributed:

The spectral density of large GUE matrix converges to WignerSemicircleDistribution:

Compare the histogram with the PDF:

The distribution of scaled largest eigenvalue of large GUE matrices converges to TracyWidomDistribution:

Compare sample histogram with the PDF of TracyWidomDistribution[2]:

Wolfram Research (2015), GaussianUnitaryMatrixDistribution, Wolfram Language function, (updated 2017).


Wolfram Research (2015), GaussianUnitaryMatrixDistribution, Wolfram Language function, (updated 2017).


Wolfram Language. 2015. "GaussianUnitaryMatrixDistribution." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2017.


Wolfram Language. (2015). GaussianUnitaryMatrixDistribution. Wolfram Language & System Documentation Center. Retrieved from


@misc{reference.wolfram_2024_gaussianunitarymatrixdistribution, author="Wolfram Research", title="{GaussianUnitaryMatrixDistribution}", year="2017", howpublished="\url{}", note=[Accessed: 21-July-2024 ]}


@online{reference.wolfram_2024_gaussianunitarymatrixdistribution, organization={Wolfram Research}, title={GaussianUnitaryMatrixDistribution}, year={2017}, url={}, note=[Accessed: 21-July-2024 ]}