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, in current fluctuations of the asymmetric simple exclusion process (ASEP) with step initial condition, and in simplified mathematical models of the behavior of the longest common subsequence problem on random inputs. 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 (or ) as predicted by Prähofer & Spohn (2000).
The distribution F1 is of particular interest in multivariate statistics. 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). In 2017 it was proved that the distribution F is not infinitely divisible.
where denotes the largest eigenvalue of the random matrix. 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 .
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:
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 (2010) 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 excess kurtosis of the distributions Fβ.
For a simple approximation based on a shifted gamma distribution see Chiani (2014).
- 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.
- Bejan, Andrei Iu. (2005), Largest eigenvalues and sample covariance matrices. Tracy–Widom and Painleve II: Computational aspects and realization in S-Plus with applications (PDF), M.Sc. dissertation, Department of Statistics, The University of Warwick.
- 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, doi:10.1090/S0894-0347-2011-00703-0.
- 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