The Tracy–Widom distribution is a probability distribution from random matrix theory introduced by . It is the distribution of the normalized largest eigenvalue of a random Hermitian matrix. The distribution is defined as a Fredholm determinant.

In practical terms, Tracy–Widom is the crossover function between the two phases of weakly versus strongly coupled components in a system.
It also appears in the distribution of the length of the longest increasing subsequence of random permutations, as large-scale statistics in the Kardar-Parisi-Zhang equation, in current fluctuations of the asymmetric simple exclusion process (ASEP) with step initial condition, and in simplified mathematical models of the behavior of the longest common subsequence problem on random inputs. See  and  for experimental testing (and verifying) that the interface fluctuations of a growing droplet (or substrate) are described by the TW distribution  (or ) as predicted by .

The distribution  is of particular interest in multivariate statistics. For a discussion of the universality of , , see . For an application of  to inferring population structure from genetic data see .
In 2017 it was proved that the distribution F is not infinitely divisible.

Definition as a law of large numbers
Let  denote the cumulative distribution function of the Tracy–Widom distribution with given . It can be defined as a law of large numbers, similar to the central limit theorem.

There are typically three Tracy–Widom distributions, , with . They correspond to the three gaussian ensembles: orthogonal (), unitary (), and symplectic ().

In general, consider a gaussian ensemble with beta value , with its diagonal entries having variance 1, and off-diagonal entries having variance , and let  be probability that an  matrix sampled from the ensemble have maximal eigenvalue , then definewhere  denotes the largest eigenvalue of the random matrix. The shift by  centers the distribution, since at the limit, the eigenvalue distribution converges to the semicircular distribution with radius . The multiplication by  is used because the standard deviation of the distribution scales as  (first derived in ).

For example:

where the matrix is sampled from the gaussian unitary ensemble with off-diagonal variance . 

The definition of the Tracy–Widom distributions  may be extended to all  (Slide 56 in , ). 

One may naturally ask for the limit distribution of second-largest eigenvalues, third-largest eigenvalues, etc. They are known.

Functional forms

Fredholm determinant 
 can be given as the Fredholm determinant

of the kernel  ("Airy kernel") on square integrable functions on the half line , given in terms of Airy functions Ai by

Painlevé transcendents 
 can also be given as an integral

in terms of a solution of a Painlevé equation of type II

with boundary condition  This function  is a Painlevé transcendent.

Other distributions are also expressible in terms of the same :

Functional equations 
Define then

Occurrences 
Other than in random matrix theory, the Tracy–Widom distributions occur in many other probability problems. 

Let  be the length of the longest increasing subsequence in a random permutation sampled uniformly from , the permutation group on n elements. Then the cumulative distribution function of  converges to .

Asymptotics

Probability density function 
Let  be the probability density function for the distribution, thenIn particular, we see that it is severely skewed to the right: it is much more likely for  to be much larger than  than to be much smaller. This could be intuited by seeing that the limit distribution is the semicircle law, so there is "repulsion" from the bulk of the distribution, forcing  to be not much smaller than .

At the  limit, a more precise expression is (equation 49 )for some positive number  that depends on .

Cumulative distribution function 
At the  limit,and at the  limit,where  is the Riemann zeta function, and .

This allows derivation of  behavior of . For example,

Painlevé transcendent 

The Painlevé transcendent has asymptotic expansion at  (equation 4.1 of )This is necessary for numerical computations, as the  solution is unstable: any deviation from it tends to drop it to the  branch instead.

Numerics

Numerical techniques for obtaining numerical solutions to the Painlevé equations of the types II and V, and numerically evaluating eigenvalue distributions of random matrices in the beta-ensembles were first presented by  using MATLAB. These approximation techniques were further analytically justified in  and used to provide numerical evaluation of Painlevé II and Tracy–Widom distributions (for ) in S-PLUS. These distributions have been tabulated in  to four significant digits for values of the argument in increments of 0.01; a statistical table for p-values was also given in this work.  gave accurate and fast algorithms for the numerical evaluation of  and the density functions 
 for .  These algorithms can be used to compute numerically the mean, variance, skewness and excess kurtosis of the distributions . 
   

Functions for working with the Tracy–Widom laws are also presented in the R package 'RMTstat' by  and MATLAB package 'RMLab' by .

For a simple approximation based on a shifted gamma distribution see .

 developed a spectral algorithm for the eigendecomposition of the integral operator , which can be used to rapidly evaluate Tracy–Widom distributions, or, more generally, the distributions of the th largest level at the soft edge scaling limit of Gaussian ensembles, to machine accuracy.

See also 
 Wigner semicircle distribution
 Marchenko–Pastur distribution

Footnotes

References
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Further reading
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External links
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At the Far Ends of a New Universal Law, Quanta Magazine

Continuous distributions
Random matrices
Special functions