# Tracy–Widom distribution

Tracy–Widom distributions for β=1,2,4

The Tracy–Widom distribution, introduced by Template:Harvs, is the probability distribution of the normalized largest eigenvalue of a random Hermitian matrix.[1]

In practical terms, Tracy-Widom is the crossover function between the two phases of weakly versus strongly coupled components in a system.[2] It also appears in the distribution of the length of the longest increasing subsequence of random permutations Template:Harv, in current fluctuations of the asymmetric simple exclusion process (ASEP) with step initial condition (Template:Harvnb, Template:Harvnb), and in simplified mathematical models of the behavior of the longest common subsequence problem on random inputs Template:Harv. See (Template:Harvnb, Template:Harvnb) 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 (Template:Harvnb).

The distribution F1 is of particular interest in multivariate statistics Template:Harv. For a discussion of the universality of Fβ, β = 1, 2, and 4, see Template:Harvtxt. For an application of F1 to inferring population structure from genetic data see Template:Harvtxt.

## Definition

The Tracy-Widom distribution is defined as the limit:[3]

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

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 Template:Harv:

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

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 Template:Harvtxt.

## 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 Template:Harvtxt using MATLAB. These approximation techniques were further analytically justified in Template:Harvtxt 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 Template:Harvtxt 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. Template:Harvtxt 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 kurtosis of the distributions Fβ.

β Mean Variance Skewness 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 Template:Harvtxt and MATLAB package 'RMLab' by Template:Harvtxt.

For a simple approximation based on a shifted gamma distribution see Template:Harvtxt.

## Footnotes

1. Dominici, D. (2008) Special Functions and Orthogonal Polynomials American Math. Soc.
2. Mysterious Statistical Law May Finally Have an Explanation, wired.com 2014-10-27
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