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

The Tracy–Widom distribution is a probability distribution from random matrix theory introduced by Craig Tracy and Harold Widom (1993, 1994). It is the distribution of the normalized largest eigenvalue of a random Hermitian matrix. The distribution is defined as a Fredholm determinant.

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] as large-scale statistics in the Kardar-Parisi-Zhang equation,^[3] in current fluctuations of the asymmetric simple exclusion process (ASEP) with step initial condition,^[4] and in simplified mathematical models of the behavior of the longest common subsequence problem on random inputs.^[5] 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 ${\displaystyle F_{2}}$ (or ${\displaystyle F_{1}}$) as predicted by Prähofer & Spohn (2000).

The distribution ${\displaystyle F_{1}}$ is of particular interest in multivariate statistics.^[6] For a discussion of the universality of ${\displaystyle F_{\beta }}$, ${\displaystyle \beta =1,2,4}$, see Deift (2007). For an application of ${\displaystyle F_{1}}$ 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.^[7]

Definition as a law of large numbers

Let ${\displaystyle F_{\beta }}$ denote the cumulative distribution function of the Tracy–Widom distribution with given ${\displaystyle \beta }$. It can be defined as a law of large numbers, similar to the central limit theorem.

There are typically three Tracy–Widom distributions, ${\displaystyle F_{\beta }}$, with ${\displaystyle \beta \in \{1,2,4\}}$. They correspond to the three gaussian ensembles: orthogonal (${\displaystyle \beta =1}$), unitary (${\displaystyle \beta =2}$), and symplectic (${\displaystyle \beta =4}$).

In general, consider a gaussian ensemble with beta value ${\displaystyle \beta }$, with its diagonal entries having variance 1, and off-diagonal entries having variance ${\displaystyle \sigma ^{2}}$, and let ${\displaystyle F_{N,\beta }(s)}$ be probability that an ${\displaystyle N\times N}$ matrix sampled from the ensemble have maximal eigenvalue ${\displaystyle \leq s}$, then define^[8]

$${\displaystyle F_{\beta }(x)=\lim _{N\to \infty }F_{N,\beta }(\sigma (2N^{1/2}+N^{-1/6}x))=\lim _{N\to \infty }Pr(N^{1/6}(\lambda _{max}/\sigma -2N^{1/2})\leq x)}$$

where ${\displaystyle \lambda _{\max }}$ denotes the largest eigenvalue of the random matrix. The shift by ${\displaystyle 2\sigma N^{1/2}}$ centers the distribution, since at the limit, the eigenvalue distribution converges to the semicircular distribution with radius ${\displaystyle 2\sigma N^{1/2}}$. The multiplication by ${\displaystyle N^{1/6}}$ is used because the standard deviation of the distribution scales as ${\displaystyle N^{-1/6}}$ (first derived in ^[9]).

For example:^[10]

${\displaystyle F_{2}(x)=\lim _{N\to \infty }\operatorname {Prob} \left((\lambda _{\max }-{\sqrt {4N}})N^{1/6}\leq x\right),}$

where the matrix is sampled from the gaussian unitary ensemble with off-diagonal variance ${\displaystyle 1}$.

The definition of the Tracy–Widom distributions ${\displaystyle F_{\beta }}$ may be extended to all ${\displaystyle \beta >0}$ (Slide 56 in Edelman (2003), Ramírez, Rider & Virág (2006)).

One may naturally ask for the limit distribution of second-largest eigenvalues, third-largest eigenvalues, etc. They are known.^[11][8]

Functional forms

Functional equations

Define

$${\displaystyle {\begin{aligned}F(x)&=\exp \left(-{\frac {1}{2}}\int _{x}^{\infty }(y-x)q(y)^{2}\,dy\right)\\E(x)&=\exp \left(-{\frac {1}{2}}\int _{x}^{\infty }q(y)\,dy\right)\end{aligned}}}$$

then^[8]

$${\displaystyle F_{1}(x)=E(x)F(x),\quad F_{2}(x)=F(x)^{2},\quad \quad F_{4}\left({\frac {x}{\sqrt {2}}}\right)={\frac {1}{2}}\left(E(x)+{\frac {1}{E(x)}}\right)F(x)}$$

Occurrences

Other than in random matrix theory, the Tracy–Widom distributions occur in many other probability problems.^[12]

Let ${\displaystyle l_{n}}$ be the length of the longest increasing subsequence in a random permutation sampled uniformly from ${\displaystyle S_{n}}$, the permutation group on n elements. Then the cumulative distribution function of ${\displaystyle {\frac {l_{n}-2N^{1/2}}{N^{1/6}}}}$ converges to ${\displaystyle F_{2}}$.^[13]

Asymptotics

Probability density function

Let ${\displaystyle f_{\beta }(x)=F_{\beta }'(x)}$ be the probability density function for the distribution, then^[12]

$${\displaystyle f_{\beta }(x)\sim {\begin{cases}e^{-{\frac {\beta }{24}}|x|^{3}},\quad x\to -\infty \\e^{-{\frac {2\beta }{3}}|x|^{3/2}},\quad x\to +\infty \end{cases}}}$$

In particular, we see that it is severely skewed to the right: it is much more likely for ${\displaystyle \lambda _{max}}$ to be much larger than ${\displaystyle 2\sigma {\sqrt {N}}}$ than to be much smaller. This could be intuited by seeing that the limit distribution is the semicircle law, so there is "repulsion" from the bulk of the distribution, forcing ${\displaystyle \lambda _{max}}$ to be not much smaller than ${\displaystyle 2\sigma {\sqrt {N}}}$. At the ${\displaystyle x\to -\infty }$ limit, a more precise expression is (equation 49 ^[12])

$${\displaystyle f_{\beta }(x)\sim \tau _{\beta }|x|^{(\beta ^{2}+4-6\beta )/16\beta }\exp \left[-\beta {\frac {|x|^{3}}{24}}+{\sqrt {2}}{\frac {\beta -2}{6}}|x|^{3/2}\right]}$$

for some positive number ${\displaystyle \tau _{\beta }}$ that depends on ${\displaystyle \beta }$.

Cumulative distribution function

At the ${\displaystyle x\to +\infty }$ limit,^[14]

$${\displaystyle {\begin{aligned}F(x)&=1-{\frac {e^{-{\frac {4}{3}}x^{3/2}}}{32\pi x^{3/2}}}{\biggl (}1-{\frac {35}{24x^{3/2}}}+{\cal {O}}(x^{-3}){\biggr )},\\E(x)&=1-{\frac {e^{-{\frac {2}{3}}x^{3/2}}}{4{\sqrt {\pi }}x^{3/2}}}{\biggl (}1-{\frac {41}{48x^{3/2}}}+{\cal {O}}(x^{-3}){\biggr )}\end{aligned}}}$$

and at the ${\displaystyle x\to -\infty }$ limit,

$${\displaystyle {\begin{aligned}F(x)&=2^{1/48}e^{{\frac {1}{2}}\zeta ^{\prime }(-1)}{\frac {e^{-{\frac {1}{24}}|x|^{3}}}{|x|^{1/16}}}\left(1+{\frac {3}{2^{7}|x|^{3}}}+O(|x|^{-6})\right)\\E(x)&={\frac {1}{2^{1/4}}}e^{-{\frac {1}{3{\sqrt {2}}}}|x|^{3/2}}{\Biggl (}1-{\frac {1}{24{\sqrt {2}}|x|^{3/2}}}+{\cal {O}}(|x|^{-3}){\Biggr )}.\end{aligned}}}$$

where ${\displaystyle \zeta }$ is the Riemann zeta function, and ${\displaystyle \zeta '(-1)=-0.1654211437}$. This allows derivation of ${\displaystyle x\to \pm \infty }$ behavior of ${\displaystyle F_{\beta }}$. For example,

$${\displaystyle {\begin{aligned}1-F_{2}(x)&={\frac {1}{32\pi x^{3/2}}}e^{-4x^{3/2}/3}(1+O(x^{-3/2})),\\F_{2}(-x)&={\frac {2^{1/24}e^{\zeta ^{\prime }(-1)}}{x^{1/8}}}e^{-x^{3}/12}{\biggl (}1+{\frac {3}{2^{6}x^{3}}}+O(x^{-6}){\biggr )}.\end{aligned}}}$$

Painlevé transcendent

The Painlevé transcendent has asymptotic expansion at ${\displaystyle x\to -\infty }$ (equation 4.1 of ^[15])

$${\displaystyle q(x)={\sqrt {-{\frac {x}{2}}}}\left(1+{\frac {1}{8}}x^{-3}-{\frac {73}{128}}x^{-6}+{\frac {10657}{1024}}x^{-9}+O(x^{-12})\right)}$$

This is necessary for numerical computations, as the ${\displaystyle q\sim {\sqrt {-x/2}}}$ solution is unstable: any deviation from it tends to drop it to the ${\displaystyle q\sim -{\sqrt {-x/2}}}$ branch instead.^[16]

Numerics

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 ${\displaystyle \beta =1,2,4}$) 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 ${\displaystyle F_{\beta }}$ and the density functions ${\displaystyle f_{\beta }(s)=dF_{\beta }/ds}$ for ${\displaystyle \beta =1,2,4}$. These algorithms can be used to compute numerically the mean, variance, skewness and excess kurtosis of the distributions ${\displaystyle F_{\beta }}$.^[17]

${\displaystyle \beta }$	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).

Shen & Serkh (2022) developed a spectral algorithm for the eigendecomposition of the integral operator ${\displaystyle A_{s}}$, which can be used to rapidly evaluate Tracy–Widom distributions, or, more generally, the distributions of the ${\displaystyle k}$th largest level at the soft edge scaling limit of Gaussian ensembles, to machine accuracy.

Tracy-Widom and KPZ universality

The Tracy-Widom distribution appears as a limit distribution in the universality class of the KPZ equation. For example it appears under ${\displaystyle t^{1/3}}$ scaling of the one-dimensional KPZ equation with fixed time.^[18]

See also

• Wigner semicircle distribution
• Marchenko–Pastur distribution

Footnotes

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, arXiv:math/9810105, 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, S2CID 15889291.
• Sasamoto, Tomohiro; Spohn, Herbert (2010), "One-Dimensional Kardar-Parisi-Zhang Equation: An Exact Solution and its Universality", Physical Review Letters, 104 (23): 230602, arXiv:1002.1883, Bibcode:2010PhRvL.104w0602S, doi:10.1103/PhysRevLett.104.230602, PMID 20867222, S2CID 34945972
• 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, S2CID 14133017.
• 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, arXiv:1601.02898, doi:10.1016/j.spl.2016.11.029, S2CID 119676736.
• 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, S2CID 16291076.
• Johansson, K. (2002), "Toeplitz determinants, random growth and determinantal processes" (PDF), Proc. International Congress of Mathematicians (Beijing, 2002), vol. 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, S2CID 88524958.
• 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, PMID 16196539, S2CID 11390762.
• 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, S2CID 20814566.
• Shen, Z.; Serkh, K. (2022), "On the evaluation of the eigendecomposition of the Airy integral operator", Applied and Computational Harmonic Analysis, 57: 105–150, arXiv:2104.12958, doi:10.1016/j.acha.2021.11.003, S2CID 233407802.
• 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, S2CID 19315093
• 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, PMC 3216521, PMID 22355553
• 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, S2CID 119690132.
• 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, S2CID 13912236.
• 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, S2CID 17398688
• Tracy, C. A.; Widom, H. (2002), "Distribution functions for largest eigenvalues and their applications" (PDF), Proc. International Congress of Mathematicians (Beijing, 2002), vol. 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, S2CID 14730756.

Further reading

• 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.
• Edelman, A. (2003), Stochastic Differential Equations and Random Matrices, SIAM Applied Linear Algebra.
• Ramírez, J. A.; Rider, B.; Virág, B. (2006), "Beta ensembles, stochastic Airy spectrum, and a diffusion", Journal of the American Mathematical Society, 24 (4): 919–944, arXiv:math/0607331, Bibcode:2006math......7331R, doi:10.1090/S0894-0347-2011-00703-0, S2CID 10226881.

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).
• At the Far Ends of a New Universal Law, Quanta Magazine