represents a Tracy–Widom distribution with Dyson index β.
- The Tracy–Widom distribution is the limiting probability distribution of the normalized largest eigenvalue of a random matrix that belongs to a Gaussian ensemble.
- TracyWidomDistribution allows the Dyson index β to be 1, 2, or 4 according to the underlying matrix distribution:
GaussianOrthogonalMatrixDistribution β2 GaussianUnitaryMatrixDistribution β4 GaussianSymplecticMatrixDistribution
- TracyWidomDistribution can be used with such functions as Mean, CDF, and RandomVariate.
Examplesopen allclose all
Basic Examples (4)
Compare its histogram to the PDF:
Compute the Mean of Tracy–Widom distribution to 50‐digit precision:
Use MatrixPropertyDistribution to represent the normalized largest eigenvalue of a matrix from Gaussian unitary ensemble:
Sample from MatrixPropertyDistribution:
When n and p (the dimension of the covariance matrix Σ) are both large, the scaled largest eigenvalue of a matrix from Wishart ensemble with identity covariance is approximately distributed as Tracy–Widom distribution of :
Define a function to find the length of LongestOrderedSequence in the given sequence:
Properties & Relations (2)
Tracy–Widom distributions can be well approximated by GammaDistribution in the central region:
Match GammaDistribution with Tracy–Widom distribution of with the first three moments:
Wolfram Research (2015), TracyWidomDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/TracyWidomDistribution.html.
Wolfram Language. 2015. "TracyWidomDistribution." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/TracyWidomDistribution.html.
Wolfram Language. (2015). TracyWidomDistribution. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/TracyWidomDistribution.html