Tracy–Widom distribution

From Wikipedia, the free encyclopedia

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).

For a simple approximation based on a shifted gamma distribution see Chiani (2012).

References

Additional reading

External links

 
Continuous univariate supported on the whole real line (−∞, ∞)
This article is issued from Wikipedia. The text is available under the Creative Commons Attribution/Share Alike; additional terms may apply for the media files.