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# Tracy–Widom distribution

Densities of Tracy–Widom distributions for β = 1, 2, 4

The Tracy–Widom distribution, introduced by Craig Tracy and Harold Widom (1993, 1994), is the probability distribution of the normalized largest eigenvalue of a random Hermitian matrix.

In practical terms, Tracy–Widom is the crossover function between the two phases of weakly versus strongly coupled components in a system.[1] It also appears in the distribution of the length of the longest increasing subsequence of random permutations,[2] in current fluctuations of the asymmetric simple exclusion process (ASEP) with step initial condition,[3] and in simplified mathematical models of the behavior of the longest common subsequence problem on random inputs.[4] See Takeuchi & Sano (2010) and Takeuchi et al. (2011) for experimental testing (and verifying) that the interface fluctuations of a growing droplet (or substrate) are described by the TW distribution ${\displaystyle F_{2}}$ (or ${\displaystyle F_{1}}$) as predicted by Prähofer & Spohn (2000).

The distribution F1 is of particular interest in multivariate statistics.[5] For a discussion of the universality of Fβ, β = 1, 2, and 4, see Deift (2007). For an application of F1 to inferring population structure from genetic data see Patterson, Price & Reich (2006). In 2017 it was proved that the distribution F is not infinitely divisible.[6]

## Definition

The Tracy–Widom distribution is defined as the limit:[7]

${\displaystyle F_{2}(s)=\lim _{n\rightarrow \infty }\operatorname {Prob} \left((\lambda _{\max }-{\sqrt {2n}})({\sqrt {2}})n^{1/6}\leq s\right),}$

where ${\displaystyle \lambda _{\max }}$ denotes the largest eigenvalue of the random matrix. The shift by ${\displaystyle {\sqrt {2n}}}$ is used to keep the distributions centered at 0. The multiplication by ${\displaystyle ({\sqrt {2}})n^{1/6}}$ is used because the standard deviation of the distributions scales as ${\displaystyle n^{-1/6}}$.

## Equivalent formulations

The cumulative distribution function of the Tracy–Widom distribution can be given as the Fredholm determinant

${\displaystyle F_{2}(s)=\det(I-A_{s})\,}$

of the operator As on square integrable functions on the half line (s, ∞) with kernel given in terms of Airy functions Ai by

${\displaystyle {\frac {\mathrm {Ai} (x)\mathrm {Ai} '(y)-\mathrm {Ai} '(x)\mathrm {Ai} (y)}{x-y}}.\,}$

It can also be given as an integral

${\displaystyle F_{2}(s)=\exp \left(-\int _{s}^{\infty }(x-s)q^{2}(x)\,dx\right)}$

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

${\displaystyle q^{\prime \prime }(s)=sq(s)+2q(s)^{3}\,}$

where q, called the Hastings–McLeod solution, satisfies the boundary condition

${\displaystyle \displaystyle q(s)\sim {\textrm {Ai}}(s),s\rightarrow \infty .}$

## Other Tracy–Widom distributions

The distribution F2 is associated to unitary ensembles in random matrix theory. There are analogous Tracy–Widom distributions F1 and F4 for orthogonal (β = 1) and symplectic ensembles (β = 4) that are also expressible in terms of the same Painlevé transcendent q:[7]

${\displaystyle F_{1}(s)=\exp \left(-{\frac {1}{2}}\int _{s}^{\infty }q(x)\,dx\right)\,\left(F_{2}(s)\right)^{1/2}}$

and

${\displaystyle F_{4}(s/{\sqrt {2}})=\cosh \left({\frac {1}{2}}\int _{s}^{\infty }q(x)\,dx\right)\,\left(F_{2}(s)\right)^{1/2}.}$

For an extension of the definition of the Tracy–Widom distributions Fβ to all β > 0 see Ramírez, Rider & Virág (2006).

## Numerical approximations

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 Edelman & Persson (2005) using MATLAB. These approximation techniques were further analytically justified in Bejan (2005) and used to provide numerical evaluation of Painlevé II and Tracy–Widom distributions (for β = 1, 2, and 4) in S-PLUS. These distributions have been tabulated in Bejan (2005) 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. Bornemann (2010) gave accurate and fast algorithms for the numerical evaluation of Fβ and the density functions fβ(s) = dFβ/ds for β = 1, 2, and 4. These algorithms can be used to compute numerically the mean, variance, skewness and excess kurtosis of the distributions Fβ.

β Mean Variance Skewness Excess kurtosis
1 −1.2065335745820 1.607781034581 0.29346452408 0.1652429384
2 −1.771086807411 0.8131947928329 0.224084203610 0.0934480876
4 −2.306884893241 0.5177237207726 0.16550949435 0.0491951565

Functions for working with the Tracy–Widom laws are also presented in the R package 'RMTstat' by Johnstone et al. (2009) and MATLAB package 'RMLab' by Dieng (2006).

For a simple approximation based on a shifted gamma distribution see Chiani (2014).