Med Library . org

Open Source Encyclopedia

Tracy–Widom distribution

Welcome to MedLibrary.org. For best results, we recommend beginning with the navigation links at the top of the page, which can guide you through our collection of over 14,000 medication labels and package inserts. For additional information on other topics which are not covered by our database of medications, just enter your topic in the search box below:

The Tracy–Widom distribution, introduced by Craig Tracy and Harold Widom (1993, 1994), is the probability distribution of the largest eigenvalue of a random hermitian matrix in the edge scaling limit. It also appears in the distribution of the length of the longest increasing subsequence of random permutations (Baik, Deift & Johansson 1999) and in current fluctuations of the asymmetric simple exclusion process (ASEP) with step initial condition (Johansson 2000, Tracy & Widom 2009). See (Takeuchi & Sano 2010, 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 F_2 (or F_1) as predicted by (Prähofer & Spohn 2000).

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

F_2(s) = \det(I - A_s)\,

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

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

It can also be given as an integral

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

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

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

\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 (Tracy & Widom 1996):

F_1(s)=\exp\left(-\frac{1}{2}\int_s^\infty q(x)\,dx\right)\, \left(F_2(s)\right)^{1/2}

and

F_4(s/\sqrt{2})=\cosh\left(\frac{1}{2}\int_s^\infty q(x)\, dx\right)\, \left(F_2(s)\right)^{1/2}.

The distribution F1 is of particular interest in multivariate statistics (Johnstone 2007, 2008, 2009). 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).

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 (2009) 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 Johnstone et al. (2009) and MATLAB package 'RMLab' by Dieng (2006).

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

References

Additional reading

External links