Tracy–Widom distribution

The Tracy–Widom distribution is defined as the limit:[8]

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

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:[8]

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

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 slide 56 in Edelman (2003) and 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).

Shen & Serkh (2022) developed a spectral algorithm for the eigendecomposition of the integral operator ${\displaystyle A_{s}}$, which can be used to rapidly evaluate Tracy–Widom distributions, or, more generally, the distributions of the kth largest level at the soft edge scaling limit of Gaussian ensembles, to machine accuracy.

• Wigner semicircle distribution
• Marchenko–Pastur distribution

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