TracyWidomDistribution
✖
TracyWidomDistribution
Details

- 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.
Examples
open allclose allBasic Examples (4)Summary of the most common use cases

https://wolfram.com/xid/0me9vohvc7yf3sr80a-ux2bof

Cumulative distribution function:

https://wolfram.com/xid/0me9vohvc7yf3sr80a-fglibo


https://wolfram.com/xid/0me9vohvc7yf3sr80a-vbfs2x


https://wolfram.com/xid/0me9vohvc7yf3sr80a-jecqsf


https://wolfram.com/xid/0me9vohvc7yf3sr80a-vkfoux

Scope (4)Survey of the scope of standard use cases
Generate a sample of pseudorandom numbers from a Tracy–Widom distribution:

https://wolfram.com/xid/0me9vohvc7yf3sr80a-qhtk5j
Compare its histogram to the PDF:

https://wolfram.com/xid/0me9vohvc7yf3sr80a-03mwaz

Moments of Tracy–Widom distributions are not available in closed form:

https://wolfram.com/xid/0me9vohvc7yf3sr80a-d8vxs7


https://wolfram.com/xid/0me9vohvc7yf3sr80a-hkhusa

Find their machine‐precision approximation:

https://wolfram.com/xid/0me9vohvc7yf3sr80a-coq6u


https://wolfram.com/xid/0me9vohvc7yf3sr80a-kwqqo

Compute the Mean of Tracy–Widom distribution to 50‐digit precision:

https://wolfram.com/xid/0me9vohvc7yf3sr80a-h4cizk


https://wolfram.com/xid/0me9vohvc7yf3sr80a-w9ogck


https://wolfram.com/xid/0me9vohvc7yf3sr80a-qccu4o

Applications (4)Sample problems that can be solved with this function
Use MatrixPropertyDistribution to represent the normalized largest eigenvalue of a matrix from Gaussian unitary ensemble:

https://wolfram.com/xid/0me9vohvc7yf3sr80a-rxoefr
Sample from MatrixPropertyDistribution:

https://wolfram.com/xid/0me9vohvc7yf3sr80a-b2536j
Compare the histogram with the PDF:

https://wolfram.com/xid/0me9vohvc7yf3sr80a-f5b58t

Perform similar computations with Gaussian orthogonal ensemble and Gaussian symplectic ensemble:

https://wolfram.com/xid/0me9vohvc7yf3sr80a-nf9db

https://wolfram.com/xid/0me9vohvc7yf3sr80a-sydlem

https://wolfram.com/xid/0me9vohvc7yf3sr80a-ctbngy
Compare the histograms with the PDFs:

https://wolfram.com/xid/0me9vohvc7yf3sr80a-duu5j0


https://wolfram.com/xid/0me9vohvc7yf3sr80a-bbrjl9

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 :

https://wolfram.com/xid/0me9vohvc7yf3sr80a-hkqwkr
Sample the scaled largest eigenvalue:

https://wolfram.com/xid/0me9vohvc7yf3sr80a-ez2ac6
Compare the histogram of scaled largest eigenvalues with the PDF:

https://wolfram.com/xid/0me9vohvc7yf3sr80a-deemk4

Compare the CDFs of Tracy–Widom distributions with the leading order asymptotic expansion on the left tail in log scale:

https://wolfram.com/xid/0me9vohvc7yf3sr80a-g3n4zc

https://wolfram.com/xid/0me9vohvc7yf3sr80a-e00606

https://wolfram.com/xid/0me9vohvc7yf3sr80a-gsodfu

https://wolfram.com/xid/0me9vohvc7yf3sr80a-drh1ps

https://wolfram.com/xid/0me9vohvc7yf3sr80a-icg43

Define a function to find the length of LongestOrderedSequence in the given sequence:

https://wolfram.com/xid/0me9vohvc7yf3sr80a-i0ic14
Find lengths of the longest increasing subsequence in random permutations of list :

https://wolfram.com/xid/0me9vohvc7yf3sr80a-g3ts8h
Compare the scaled length of the longest increasing subsequence with Tracy–Widom distribution of :

https://wolfram.com/xid/0me9vohvc7yf3sr80a-tc3yc

https://wolfram.com/xid/0me9vohvc7yf3sr80a-dkd2ef

Properties & Relations (2)Properties of the function, and connections to other functions
CDFs of Tracy–Widom distributions for different values of β are related to each other:

https://wolfram.com/xid/0me9vohvc7yf3sr80a-chs8am

https://wolfram.com/xid/0me9vohvc7yf3sr80a-jmh9o8

Tracy–Widom distributions can be well approximated by GammaDistribution in the central region:

https://wolfram.com/xid/0me9vohvc7yf3sr80a-by9f82

Match GammaDistribution with Tracy–Widom distribution of with the first three moments:

https://wolfram.com/xid/0me9vohvc7yf3sr80a-iqtpua


https://wolfram.com/xid/0me9vohvc7yf3sr80a-qib61c

Compare the CDFs between in log scale:

https://wolfram.com/xid/0me9vohvc7yf3sr80a-dekcmk

Wolfram Research (2015), TracyWidomDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/TracyWidomDistribution.html.
Text
Wolfram Research (2015), TracyWidomDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/TracyWidomDistribution.html.
Wolfram Research (2015), TracyWidomDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/TracyWidomDistribution.html.
CMS
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. Wolfram Research. https://reference.wolfram.com/language/ref/TracyWidomDistribution.html.
APA
Wolfram Language. (2015). TracyWidomDistribution. Wolfram Language & System Documentation Center. Retrieved from 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
BibTeX
@misc{reference.wolfram_2025_tracywidomdistribution, author="Wolfram Research", title="{TracyWidomDistribution}", year="2015", howpublished="\url{https://reference.wolfram.com/language/ref/TracyWidomDistribution.html}", note=[Accessed: 29-March-2025
]}
BibLaTeX
@online{reference.wolfram_2025_tracywidomdistribution, organization={Wolfram Research}, title={TracyWidomDistribution}, year={2015}, url={https://reference.wolfram.com/language/ref/TracyWidomDistribution.html}, note=[Accessed: 29-March-2025
]}