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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 (or ) as predicted by (Prähofer & Spohn 2000).
of the operator As on square integrable function 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
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):
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.
For an extension of the definition of the Tracy–Widom distributions Fβ to all β>0 see Ramírez, Rider & Virág (2006).
- 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.
- Deift, P. (2007), "Universality for mathematical and physical systems", International Congress of Mathematicians (Madrid, 2006), European Mathematical Society, pp. 125–152, MR 2334189.
- Johansson, K. (2000), "Shape fluctuations and random matrices", Communications in Mathematical Physics 209 (2): 151–174, arXiv:math/9903134, Bibcode:2000CMaPh.209..437J, doi:10.1007/s002200050027.
- Johansson, K. (2002), "Toeplitz determinants, random growth and determinantal processes", 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", International Congress of Mathematicians (Madrid, 2006), European Mathematical Society, pp. 307–333, 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.
- 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, 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, Bibcode:1996CMaPh.177..727T, doi:10.1007/BF02099545, MR 1385083.
- Tracy, C. A.; Widom, H. (2002), "Distribution functions for largest eigenvalues and their applications", 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.
- Bejan, Andrei Iu. (2005), Largest eigenvalues and sample covariance matrices. Tracy-Widom and Painleve II: Computational aspects and realization in S-Plus with applications, M.Sc. dissertation, Department of Statistics, The University of Warwick.
- 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.
- Edelman, A.; Persson, P.-O. (2005), Numerical Methods for Eigenvalue Distributions of Random Matrices, arXiv:math-ph/0501068, Bibcode:2005math.ph...1068E.
- Ramírez, J. A.; Rider, B.; Virág, B. (2006), Beta ensembles, stochastic Airy spectrum, and a diffusion, arXiv:math/0607331, Bibcode:2006math......7331R.
- Kuijlaars, Universality of distribution functions in random matrix theory.
- Tracy, C. A.; Widom, H., The distributions of random matrix theory and their applications.
- Johnstone, Iain; Ma, Zongming; Perry, Patrick; Shahram, Morteza (2009), Package 'RMTstat'.
- Dieng, Momar (2006), Package ``RMLab.