Tracy–Widom distribution

Short description: Probability distribution

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. It also appears in the distribution of the length of the longest increasing subsequence of random permutations, as large-scale statistics in the Kardar-Parisi-Zhang equation, 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) and (Takeuchi Sano) for experimental testing (and verifying) that the interface fluctuations of a growing droplet (or substrate) are described by the TW distribution [math]\displaystyle{ F_2 }[/math] (or [math]\displaystyle{ F_1 }[/math]) as predicted by (Prähofer Spohn).

The distribution [math]\displaystyle{ F_1 }[/math] is of particular interest in multivariate statistics. For a discussion of the universality of [math]\displaystyle{ F_\beta }[/math], [math]\displaystyle{ \beta=1,2,4 }[/math], see (Deift 2007). For an application of [math]\displaystyle{ F_1 }[/math] to inferring population structure from genetic data see (Patterson Price). In 2017 it was proved that the distribution F is not infinitely divisible.

Definition as a law of large numbers

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

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

In general, consider a gaussian ensemble with beta value [math]\displaystyle{ \beta }[/math], with its diagonal entries having variance 1, and off-diagonal entries having variance [math]\displaystyle{ \sigma^2 }[/math], and let [math]\displaystyle{ F_{N, \beta}(s) }[/math] be probability that an [math]\displaystyle{ N\times N }[/math] matrix sampled from the ensemble have maximal eigenvalue [math]\displaystyle{ \leq s }[/math], then define[math]\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) }[/math]where [math]\displaystyle{ \lambda_{\max} }[/math] denotes the largest eigenvalue of the random matrix. The shift by [math]\displaystyle{ 2\sigma N^{1/2} }[/math] centers the distribution, since at the limit, the eigenvalue distribution converges to the semicircular distribution with radius [math]\displaystyle{ 2\sigma N^{1/2} }[/math]. The multiplication by [math]\displaystyle{ N^{1/6} }[/math] is used because the standard deviation of the distribution scales as [math]\displaystyle{ N^{-1/6} }[/math] (first derived in ).

For example:

[math]\displaystyle{ F_2(x) = \lim_{N\to \infty} \operatorname{Prob}\left( (\lambda_{\max}-\sqrt{4N})N^{1/6}\leq x\right), }[/math]

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

The definition of the Tracy–Widom distributions [math]\displaystyle{ F_\beta }[/math] may be extended to all [math]\displaystyle{ \beta \gt 0 }[/math] (Slide 56 in (Edelman 2003), (Ramírez Rider)).

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

Functional forms

Fredholm determinant

[math]\displaystyle{ F_2 }[/math] can be given as the Fredholm determinant

[math]\displaystyle{ F_2(s) = \det(I - A_s) = 1 + \sum_{n=1}^\infty \frac{(-1)^n}{n!} \int_{(s, \infty)^n} \det_{i, j = 1, ..., n}[A_s(x_i, x_j)]dx_1\cdots dx_n }[/math]

of the kernel [math]\displaystyle{ A_s }[/math] ("Airy kernel") on square integrable functions on the half line [math]\displaystyle{ (s,\infty) }[/math], given in terms of Airy functions Ai by

[math]\displaystyle{ A_s(x, y) = \begin{cases} \frac{\mathrm{Ai}(x)\mathrm{Ai}'(y) - \mathrm{Ai}'(x)\mathrm{Ai}(y)}{x-y} \quad \text{if }x\neq y \\ Ai' (x)^2- x (Ai(x))^2 \quad \text{if }x=y \end{cases} }[/math]

Painlevé transcendents

[math]\displaystyle{ F_2 }[/math] can also be given as an integral

[math]\displaystyle{ F_2(s) = \exp\left(-\int_s^\infty (x-s)q^2(x)\,dx\right) }[/math]

in terms of a solution of a Painlevé equation of type II

[math]\displaystyle{ q^{\prime\prime}(s) = sq(s)+2q(s)^3\, }[/math]

with boundary condition [math]\displaystyle{ \displaystyle q(s) \sim \textrm{Ai}(s), s\to\infty. }[/math] This function [math]\displaystyle{ q }[/math] is a Painlevé transcendent.

Other distributions are also expressible in terms of the same [math]\displaystyle{ q }[/math]:

[math]\displaystyle{ \begin{align} F_1(s) &=\exp\left(-\frac{1}{2}\int_s^\infty q(x)\,dx\right)\, \left(F_2(s)\right)^{1/2} \\ F_4(s/\sqrt{2}) &=\cosh\left(\frac{1}{2}\int_s^\infty q(x)\, dx\right)\, \left(F_2(s)\right)^{1/2}. \end{align} }[/math]

Functional equations

Define [math]\displaystyle{ \begin{align} F(x) &= \exp\left(-\frac{1}{2}\int_{x}^{\infty}(y-x)q(y)^{2}\,d y\right) \\ E(x) &= \exp\left(-\frac{1}{2}\int_{x}^{\infty}q(y)\,d y\right) \end{align} }[/math]then[math]\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) }[/math]

Occurrences

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

Let [math]\displaystyle{ l_n }[/math] be the length of the longest increasing subsequence in a random permutation sampled uniformly from [math]\displaystyle{ S_n }[/math], the permutation group on n elements. Then the cumulative distribution function of [math]\displaystyle{ \frac{l_n - 2N^{1/2}}{N^{1/6}} }[/math] converges to [math]\displaystyle{ F_2 }[/math].

Asymptotics

Probability density function

Let [math]\displaystyle{ f_\beta(x) = F_\beta'(x) }[/math] be the probability density function for the distribution, then[math]\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} }[/math]In particular, we see that it is severely skewed to the right: it is much more likely for [math]\displaystyle{ \lambda_{max} }[/math] to be much larger than [math]\displaystyle{ 2\sigma\sqrt{N} }[/math] 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 [math]\displaystyle{ \lambda_{max} }[/math] to be not much smaller than [math]\displaystyle{ 2\sigma\sqrt{N} }[/math].

At the [math]\displaystyle{ x\to -\infty }[/math] limit, a more precise expression is (equation 49 )[math]\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] }[/math]for some positive number [math]\displaystyle{ \tau_\beta }[/math] that depends on [math]\displaystyle{ \beta }[/math].

Cumulative distribution function

At the [math]\displaystyle{ x\to +\infty }[/math] limit,[math]\displaystyle{ \begin{align} 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{align} }[/math]and at the [math]\displaystyle{ x\to -\infty }[/math] limit,[math]\displaystyle{ \begin{align} 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{align} }[/math]where [math]\displaystyle{ \zeta }[/math] is the Riemann zeta function, and [math]\displaystyle{ \zeta' (-1) = -0.1654211437 }[/math].

This allows derivation of [math]\displaystyle{ x\to \pm\infty }[/math] behavior of [math]\displaystyle{ F_\beta }[/math]. For example,[math]\displaystyle{ \begin{align} 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{align} }[/math]

Painlevé transcendent

The Painlevé transcendent has asymptotic expansion at [math]\displaystyle{ x \to -\infty }[/math] (equation 4.1 of )[math]\displaystyle{ q(x) = \sqrt{-\frac{x}{2}} \left(1 + \frac 18 x^{-3} - \frac{73}{128} x^{-6} + \frac{10657}{1024}x^{-9} + O(x^{-12})\right) }[/math]This is necessary for numerical computations, as the [math]\displaystyle{ q\sim \sqrt{-x/2} }[/math] solution is unstable: any deviation from it tends to drop it to the [math]\displaystyle{ q \sim -\sqrt{-x/2} }[/math] branch instead.

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

[math]\displaystyle{ \beta }[/math]	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 Ma) and MATLAB package 'RMLab' by (Dieng 2006).

For a simple approximation based on a shifted gamma distribution see (Chiani 2014).

(Shen Serkh) developed a spectral algorithm for the eigendecomposition of the integral operator [math]\displaystyle{ A_s }[/math], which can be used to rapidly evaluate Tracy–Widom distributions, or, more generally, the distributions of the [math]\displaystyle{ k }[/math]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 [math]\displaystyle{ t^{1/3} }[/math] scaling of the one-dimensional KPZ equation with fixed time.

See also

• Wigner semicircle distribution
• Marchenko–Pastur distribution

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 .
• 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, 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, doi:10.1016/j.jmva.2014.04.002 .
• Sasamoto, Tomohiro; Spohn, Herbert (2010), "One-Dimensional Kardar-Parisi-Zhang Equation: An Exact Solution and its Universality", Physical Review Letters 104 (23): 230602, doi:10.1103/PhysRevLett.104.230602, PMID 20867222, Bibcode: 2010PhRvL.104w0602S 
• Deift, P. (2007), "Universality for mathematical and physical systems", International Congress of Mathematicians (Madrid, 2006), European Mathematical Society, pp. 125–152, doi:10.4171/022-1/7, http://icm2006.mathunion.org/proceedings/Vol_I/11.pdf .
• 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, doi:10.1016/j.spl.2016.11.029 .
• Johansson, K. (2000), "Shape fluctuations and random matrices", Communications in Mathematical Physics 209 (2): 437–476, doi:10.1007/s002200050027, Bibcode: 2000CMaPh.209..437J .
• 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, http://www.mathunion.org/ICM/ICM2002.3/Main/icm2002.3.0053.0062.ocr.pdf .
• Johnstone, I. M. (2007), "High dimensional statistical inference and random matrices", International Congress of Mathematicians (Madrid, 2006), European Mathematical Society, pp. 307–333, doi:10.4171/022-1/13, http://www.icm2006.org/proceedings/Vol_I/17.pdf .
• 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, doi:10.1214/08-AOS605, PMID 20157626 .
• Johnstone, I. M. (2009), "Approximate null distribution of the largest root in multivariate analysis", Annals of Applied Statistics 3 (4): 1616–1633, doi:10.1214/08-AOAS220, 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, doi:10.1103/PhysRevE.72.020901, PMID 16196539, Bibcode: 2005PhRvE..72b0901M .
• Patterson, N.; Price, A. L. (2006), "Population structure and eigenanalysis", PLOS Genetics 2 (12): e190, doi:10.1371/journal.pgen.0020190, 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, doi:10.1103/PhysRevLett.84.4882, PMID 10990822, Bibcode: 2000PhRvL..84.4882P .
• Shen, Z.; Serkh, K. (2022), "On the evaluation of the eigendecomposition of the Airy integral operator", Applied and Computational Harmonic Analysis 57: 105–150, doi:10.1016/j.acha.2021.11.003 .
• Takeuchi, K. A.; Sano, M. (2010), "Universal fluctuations of growing interfaces: Evidence in turbulent liquid crystals", Physical Review Letters 104 (23): 230601, doi:10.1103/PhysRevLett.104.230601, PMID 20867221, Bibcode: 2010PhRvL.104w0601T 
• 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, PMID 22355553, Bibcode: 2011NatSR...1E..34T 
• "Level-spacing distributions and the Airy kernel", Physics Letters B 305 (1–2): 115–118, 1993, doi:10.1016/0370-2693(93)91114-3, Bibcode: 1993PhLB..305..115T .
• "Level-spacing distributions and the Airy kernel", Communications in Mathematical Physics 159 (1): 151–174, 1994, doi:10.1007/BF02100489, Bibcode: 1994CMaPh.159..151T .
• "On orthogonal and symplectic matrix ensembles", Communications in Mathematical Physics 177 (3): 727–754, 1996, doi:10.1007/BF02099545, Bibcode: 1996CMaPh.177..727T 
• "Distribution functions for largest eigenvalues and their applications", Proc. International Congress of Mathematicians (Beijing, 2002), 1, Beijing: Higher Ed. Press, 2002, pp. 587–596, http://www.mathunion.org/ICM/ICM2002.1/Main/icm2002.1.0587.0596.ocr.pdf .
• "Asymptotics in ASEP with step initial condition", Communications in Mathematical Physics 290 (1): 129–154, 2009, doi:10.1007/s00220-009-0761-0, Bibcode: 2009CMaPh.290..129T .

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, M.Sc. dissertation, Department of Statistics, The University of Warwick, http://www.cl.cam.ac.uk/~aib29/TWinSplus.pdf .
• Edelman, A.; Persson, P.-O. (2005), Numerical Methods for Eigenvalue Distributions of Random Matrices, Bibcode: 2005math.ph...1068E .
• Edelman, A. (2003), Stochastic Differential Equations and Random Matrices, SIAM Applied Linear Algebra, https://math.mit.edu/~edelman/talks/2003/siam2003.ppt .
• 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, doi:10.1090/S0894-0347-2011-00703-0, Bibcode: 2006math......7331R .

External links

• Kuijlaars, Universality of distribution functions in random matrix theory, http://web.mit.edu/sea06/agenda/talks/Kuijlaars.pdf .
• The distributions of random matrix theory and their applications, http://www.math.ucdavis.edu/~tracy/talks/SITE7.pdf .
• Johnstone, Iain; Ma, Zongming; Perry, Patrick; Shahram, Morteza (2009), Package 'RMTstat', https://cran.r-project.org/web/packages/RMTstat/RMTstat.pdf .
• At the Far Ends of a New Universal Law, Quanta Magazine