Tracy–Widom distribution

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The Tracy–Widom distribution, introduced by Craig Tracy and Harold Widom (1993, 1994), is the probability distribution of the normalized largest eigenvalue of a random Hermitian matrix.[1]

In practical terms, Tracy-Widom is the crossover function between the two phases of weakly versus strongly coupled components in a system.[2] 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  F_2 (or F_1) 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).

Definition[edit]

The Tracy-Widom distribution is defined as the limit:[3]

F_2(s) = \lim\limits_{n\rightarrow \infty}{\rm Prob}\left((\lambda_{\rm max}-\sqrt{2n})(\sqrt{2})n^{1/6}\leq s\right),

The shift by \sqrt{2n} is used to keep the distributions centered at 0. The multiplication by (\sqrt{2})n^{1/6} is used because the standard deviation of the distributions scales as n^{-1/6}.

Equivalent formulations[edit]

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 functions 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.

Other Tracy-Widom Distributions[edit]

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}.

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

Numerical approximations[edit]

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

Footnotes[edit]

  1. ^ Dominici, D. (2008) Special Functions and Orthogonal Polynomials American Math. Soc.
  2. ^ Mysterious Statistical Law May Finally Have an Explanation, wired.com 2014-10-27
  3. ^ Tracy, C. A.; Widom, H. (1996), "On orthogonal and symplectic matrix ensembles", Communications in Mathematical Physics 177 (3): 727–754, Bibcode:1996CMaPh.177..727T, doi:10.1007/BF02099545, MR 1385083 

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