Tracy–Widom distribution

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 ${\displaystyle n^{-1/6}}$.

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 A_s 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

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

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 F_β to all β > 0 see Ramírez, Rider & Virág (2006).

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 F_β and the density functions f_β(s) = dF_β/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 F_β.

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).

• Wigner semicircle distribution
• Marchenko–Pastur distribution

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• Quanta Magazine: At the Far Ends of a New Universal Law