Tracy–Widom distribution

Densities of Tracy–Widom distributions for β = 1, 2, 4

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.

In practical terms, Tracy–Widom is the crossover function between the two phases of weakly versus strongly coupled components in a system.^[1] It also appears in the distribution of the length of the longest increasing subsequence of random permutations,^[2] in current fluctuations of the asymmetric simple exclusion process (ASEP) with step initial condition,^[3] and in simplified mathematical models of the behavior of the longest common subsequence problem on random inputs.^[4] See Takeuchi & Sano (2010) and 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.^[5] For a discussion of the universality of F_β, 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). In 2017 it was proved that the distribution F is not infinitely divisible.^[6]

Definition

The Tracy–Widom distribution is defined as the limit:^[7]

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

where $\lambda_{\max}$ denotes the largest eigenvalue of the random matrix. 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

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 A_s 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

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:^[7]

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 F_β to all β > 0 see Ramírez, Rider & Virág (2006).

Numerical approximations

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.

β	Mean	Variance	Skewness	Excess 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 (2014).

Footnotes

↑ Mysterious Statistical Law May Finally Have an Explanation, wired.com 2014-10-27
↑ Baik, Deift & Johansson (1999).
↑ Johansson (2000); Tracy & Widom (2009)).
↑ Majumdar & Nechaev (2005).
↑ Johnstone (2007, 2008, 2009).
↑ Domínguez-Molina (2017).
1 2 Tracy & Widom (1996).

References

Baik, J.; Deift, P.; Johansson, K. (1999), "On the distribution of the length of the longest increasing subsequence of random permutations", Journal of the American Mathematical Society, 12 (4): 1119–1178, doi:10.1090/S0894-0347-99-00307-0, JSTOR 2646100, MR 1682248 .
Bornemann, F. (2010), "On the numerical evaluation of distributions in random matrix theory: A review with an invitation to experimental mathematics", Markov Processes and Related Fields, 16 (4): 803–866, arXiv:0904.1581, Bibcode:2009arXiv0904.1581B .
Chiani, M. (2014), "Distribution of the largest eigenvalue for real Wishart and Gaussian random matrices and a simple approximation for the Tracy–Widom distribution", Journal of Multivariate Analysis, 129: 69–81, arXiv:1209.3394, doi:10.1016/j.jmva.2014.04.002 .
Deift, P. (2007), "Universality for mathematical and physical systems" (PDF), International Congress of Mathematicians (Madrid, 2006), European Mathematical Society, pp. 125–152, arXiv:math-ph/0603038, doi:10.4171/022-1/7, MR 2334189 .
Dieng, Momar (2006), RMLab, a MATLAB package for computing Tracy-Widom distributions and simulating random matrices .
Domínguez-Molina, J.Armando (2017), "The Tracy-Widom distribution is not infinitely divisible", Statistics & Probability Letters, 213 (1): 56–60 .
Johansson, K. (2000), "Shape fluctuations and random matrices", Communications in Mathematical Physics, 209 (2): 437–476, arXiv:math/9903134, Bibcode:2000CMaPh.209..437J, doi:10.1007/s002200050027 .
Johansson, K. (2002), "Toeplitz determinants, random growth and determinantal processes" (PDF), Proc. International Congress of Mathematicians (Beijing, 2002), 3, Beijing: Higher Ed. Press, pp. 53–62, MR 1957518 .
Johnstone, I. M. (2007), "High dimensional statistical inference and random matrices" (PDF), International Congress of Mathematicians (Madrid, 2006), European Mathematical Society, pp. 307–333, arXiv:math/0611589, doi:10.4171/022-1/13, MR 2334195 .
Johnstone, I. M. (2008), "Multivariate analysis and Jacobi ensembles: largest eigenvalue, Tracy–Widom limits and rates of convergence", Annals of Statistics, 36 (6): 2638–2716, arXiv:0803.3408, doi:10.1214/08-AOS605, PMC 2821031, PMID 20157626 .
Johnstone, I. M. (2009), "Approximate null distribution of the largest root in multivariate analysis", Annals of Applied Statistics, 3 (4): 1616–1633, arXiv:1009.5854, doi:10.1214/08-AOAS220, PMC 2880335, PMID 20526465 .
Majumdar, Satya N.; Nechaev, Sergei (2005), "Exact asymptotic results for the Bernoulli matching model of sequence alignment", Physical Review E, 72 (2): 020901, 4, arXiv:q-bio/0410012, Bibcode:2005PhRvE..72b0901M, doi:10.1103/PhysRevE.72.020901, MR 2177365 .
Patterson, N.; Price, A. L.; Reich, D. (2006), "Population structure and eigenanalysis", PLoS Genetics, 2 (12): e190, doi:10.1371/journal.pgen.0020190, PMC 1713260, PMID 17194218 .
Prähofer, M.; Spohn, H. (2000), "Universal distributions for growing processes in 1+1 dimensions and random matrices", Physical Review Letters, 84 (21): 4882–4885, arXiv:cond-mat/9912264, Bibcode:2000PhRvL..84.4882P, doi:10.1103/PhysRevLett.84.4882, PMID 10990822 .
Takeuchi, K. A.; Sano, M. (2010), "Universal fluctuations of growing interfaces: Evidence in turbulent liquid crystals", Physical Review Letters, 104 (23): 230601, arXiv:1001.5121, Bibcode:2010PhRvL.104w0601T, doi:10.1103/PhysRevLett.104.230601, PMID 20867221
Takeuchi, K. A.; Sano, M.; Sasamoto, T.; Spohn, H. (2011), "Growing interfaces uncover universal fluctuations behind scale invariance", Scientific Reports, 1: 34, arXiv:1108.2118, Bibcode:2011NatSR...1E..34T, doi:10.1038/srep00034
Tracy, C. A.; Widom, H. (1993), "Level-spacing distributions and the Airy kernel", Physics Letters B, 305 (1–2): 115–118, arXiv:hep-th/9210074, Bibcode:1993PhLB..305..115T, doi:10.1016/0370-2693(93)91114-3 .
Tracy, C. A.; Widom, H. (1994), "Level-spacing distributions and the Airy kernel", Communications in Mathematical Physics, 159 (1): 151–174, arXiv:hep-th/9211141, Bibcode:1994CMaPh.159..151T, doi:10.1007/BF02100489, MR 1257246 .
Tracy, C. A.; Widom, H. (1996), "On orthogonal and symplectic matrix ensembles", Communications in Mathematical Physics, 177 (3): 727–754, arXiv:solv-int/9509007, Bibcode:1996CMaPh.177..727T, doi:10.1007/BF02099545, MR 1385083
Tracy, C. A.; Widom, H. (2002), "Distribution functions for largest eigenvalues and their applications" (PDF), Proc. International Congress of Mathematicians (Beijing, 2002), 1, Beijing: Higher Ed. Press, pp. 587–596, MR 1989209 .
Tracy, C. A.; Widom, H. (2009), "Asymptotics in ASEP with step initial condition", Communications in Mathematical Physics, 290 (1): 129–154, arXiv:0807.1713, Bibcode:2009CMaPh.290..129T, doi:10.1007/s00220-009-0761-0 .

External links

Kuijlaars, Universality of distribution functions in random matrix theory (PDF) .
Tracy, C. A.; Widom, H., The distributions of random matrix theory and their applications (PDF) .
Johnstone, Iain; Ma, Zongming; Perry, Patrick; Shahram, Morteza (2009), Package 'RMTstat' (PDF) .
Quanta Magazine: At the Far Ends of a New Universal Law

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[1] Mysterious Statistical Law May Finally Have an Explanation, wired.com 2014-10-27

[FOOTNOTEBaikDeiftJohansson1999-2] Baik, Deift & Johansson (1999).

[3] Johansson (2000); Tracy & Widom (2009)).

[FOOTNOTEMajumdarNechaev2005-4] Majumdar & Nechaev (2005).

[5] Johnstone (2007, 2008, 2009).

[FOOTNOTEDomínguez-Molina2017-6] Domínguez-Molina (2017).

[FOOTNOTETracyWidom1996-7] 1 2 Tracy & Widom (1996).

Probability distributions
List
Discrete univariate with finite support	Benford Bernoulli beta-binomial binomial categorical hypergeometric Poisson binomial Rademacher discrete uniform Zipf Zipf–Mandelbrot
Discrete univariate with infinite support	beta negative binomial Borel Conway–Maxwell–Poisson discrete phase-type Delaporte extended negative binomial Gauss–Kuzmin geometric logarithmic negative binomial parabolic fractal Poisson Skellam Yule–Simon zeta
Continuous univariate supported on a bounded interval	arcsine ARGUS Balding–Nichols Bates beta beta rectangular Irwin–Hall Kumaraswamy logit-normal noncentral beta raised cosine reciprocal triangular U-quadratic uniform Wigner semicircle
Continuous univariate supported on a semi-infinite interval	Benini Benktander 1st kind Benktander 2nd kind beta prime Burr chi-squared chi Dagum Davis exponential-logarithmic Erlang exponential F folded normal Flory–Schulz Fréchet gamma gamma/Gompertz generalized inverse Gaussian Gompertz half-logistic half-normal Hotelling's T-squared hyper-Erlang hyperexponential hypoexponential inverse chi-squared scaled inverse chi-squared inverse Gaussian inverse gamma Kolmogorov Lévy log-Cauchy log-Laplace log-logistic log-normal Lomax matrix-exponential Maxwell–Boltzmann Maxwell–Jüttner Mittag-Leffler Nakagami noncentral chi-squared Pareto phase-type poly-Weibull Rayleigh relativistic Breit–Wigner Rice shifted Gompertz truncated normal type-2 Gumbel Weibull Discrete Weibull Wilks's lambda
Continuous univariate supported on the whole real line	Cauchy exponential power Fisher's z Gaussian q generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Johnson's S_U Landau Laplace asymmetric Laplace logistic noncentral t normal (Gaussian) normal-inverse Gaussian skew normal slash stable Student's t type-1 Gumbel Tracy–Widom variance-gamma Voigt
Continuous univariate with support whose type varies	generalized extreme value generalized Pareto Marchenko–Pastur q-exponential q-Gaussian q-Weibull shifted log-logistic Tukey lambda
Mixed continuous-discrete univariate	rectified Gaussian
Multivariate (joint)	Discrete Ewens multinomial Dirichlet-multinomial negative multinomial Continuous Dirichlet generalized Dirichlet multivariate Laplace multivariate normal multivariate stable multivariate t normal-inverse-gamma normal-gamma Matrix-valued inverse matrix gamma inverse-Wishart matrix normal matrix t matrix gamma normal-inverse-Wishart normal-Wishart Wishart
Directional	Univariate (circular) directional Circular uniform univariate von Mises wrapped normal wrapped Cauchy wrapped exponential wrapped asymmetric Laplace wrapped Lévy Bivariate (spherical) Kent Bivariate (toroidal) bivariate von Mises Multivariate von Mises–Fisher Bingham
Degenerate and singular	Degenerate Dirac delta function Singular Cantor
Families	Circular compound Poisson elliptical exponential natural exponential location–scale maximum entropy mixture Pearson Tweedie wrapped