||This article may be too technical for most readers to understand. (November 2014)|
In practical terms, Tracy-Widom is the crossover function between the two phases of weakly versus strongly coupled components in a system. It also appears in the distribution of the length of the longest increasing subsequence of random permutations (Baik, Deift & Johansson 1999), in current fluctuations of the asymmetric simple exclusion process (ASEP) with step initial condition (Johansson 2000, Tracy & Widom 2009), and in simplified mathematical models of the behavior of the longest common subsequence problem on random inputs (Majumdar & Nechaev 2005). 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 (or ) as predicted by (Prähofer & Spohn 2000).
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).
The Tracy-Widom distribution is defined as the limit:
The shift by is used to keep the distributions centered at 0. The multiplication by is used because the standard deviation of the distributions scales as .
of the operator As on square integrable functions on the half line (s, ∞) with kernel given in terms of Airy functions Ai by
It can also be given as an integral
in terms of a solution of a Painlevé equation of type II
where q, called the Hastings–McLeod solution, satisfies the boundary condition
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 (Tracy & Widom 1996):
For an extension of the definition of the Tracy–Widom distributions Fβ 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 (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.
For a simple approximation based on a shifted gamma distribution see Chiani (2012).
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