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A p p e a r a n c e
F r o m W i k i p e d i a , t h e f r e e e n c y c l o p e d i a
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 , 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. 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
{\displaystyle F_{2}}
(or
F
1
{\displaystyle F_{1}}
) as predicted by Prähofer & Spohn (2000) .
The distribution
F
1
{\displaystyle F_{1}}
is of particular interest in multivariate statistics .[6] For a discussion of the universality of
F
β
{\displaystyle F_{\beta }}
,
β
=
1
,
2
,
4
{\displaystyle \beta =1,2,4}
, see Deift (2007) . For an application of
F
1
{\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.
Definition as a law of large numbers [ edit ]
Let
F
β
{\displaystyle F_{\beta }}
denote the cumulative distribution function of the Tracy–Widom distribution with given
β
{\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,
F
β
{\displaystyle F_{\beta }}
, with
β
∈
{
1
,
2
,
4
}
{\displaystyle \beta \in \{1,2,4\}}
. They correspond to the three gaussian ensembles : orthogonal (
β
=
1
{\displaystyle \beta =1}
), unitary (
β
=
2
{\displaystyle \beta =2}
), and symplectic (
β
=
4
{\displaystyle \beta =4}
).
In general, consider a gaussian ensemble with beta value
β
{\displaystyle \beta }
, with its diagonal entries having variance 1, and off-diagonal entries having variance
σ
2
{\displaystyle \sigma ^{2}}
, and let
F
N
,
β
(
s
)
{\displaystyle F_{N,\beta }(s )}
be probability that an
N
×
N
{\displaystyle N\times N}
matrix sampled from the ensemble have maximal eigenvalue
≤
s
{\displaystyle \leq s}
, then define[8]
F
β
(
x
)
=
lim
N
→
∞
F
N
,
β
(
σ
(
2
N
1
/
2
+
N
−
1
/
6
x
)
)
=
lim
N
→
∞
P
r
(
N
1
/
6
(
λ
m
a
x
/
σ
−
2
N
1
/
2
)
≤
x
)
{\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
λ
max
{\displaystyle \lambda _{\max }}
denotes the largest eigenvalue of the random matrix. The shift by
2
σ
N
1
/
2
{\displaystyle 2\sigma N^{1/2}}
centers the distribution, since at the limit, the eigenvalue distribution converges to the semicircular distribution with radius
2
σ
N
1
/
2
{\displaystyle 2\sigma N^{1/2}}
. The multiplication by
N
1
/
6
{\displaystyle N^{1/6}}
is used because the standard deviation of the distribution scales as
N
−
1
/
6
{\displaystyle N^{-1/6}}
(first derived in [9] ).
For example:
F
2
(
x
)
=
lim
N
→
∞
Prob
(
(
λ
max
−
4
N
)
N
1
/
6
≤
x
)
,
{\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
1
{\displaystyle 1}
.
The definition of the Tracy–Widom distributions
F
β
{\displaystyle F_{\beta }}
may be extended to all
β
>
0
{\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 [ edit ]
Fredholm determinant [ edit ]
F
2
{\displaystyle F_{2}}
can be given as the Fredholm determinant
F
2
(
s
)
=
det
(
I
−
A
s
)
=
1
+
∑
n
=
1
∞
(
−
1
)
n
n
!
∫
(
s
,
∞
)
n
det
i
,
j
=
1
,
.
.
.
,
n
[
A
s
(
x
i
,
x
j
)
]
d
x
1
⋯
d
x
n
{\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}}
of the kernel
A
s
{\displaystyle A_{s}}
("Airy kernel") on square integrable functions on the half line
(
s
,
∞
)
{\displaystyle (s,\infty )}
, given in terms of Airy functions Ai by
A
s
(
x
,
y
)
=
{
A
i
(
x
)
A
i
′
(
y
)
−
A
i
′
(
x
)
A
i
(
y
)
x
−
y
if
x
≠
y
A
i
′
(
x
)
2
−
x
(
A
i
(
x
)
)
2
if
x
=
y
{\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}}}
Painlevé transcendents [ edit ]
F
2
{\displaystyle F_{2}}
can also be given as an integral
F
2
(
s
)
=
exp
(
−
∫
s
∞
(
x
−
s
)
q
2
(
x
)
d
x
)
{\displaystyle F_{2}(s )=\exp \left(-\int _{s}^{\infty }(x-s)q^{2}(x )\,dx\right)}
in terms of a solution[note 1] of a Painlevé equation of type II
q
′
′
(
s
)
=
s
q
(
s
)
+
2
q
(
s
)
3
{\displaystyle q^{\prime \prime }(s )=sq(s )+2q(s )^{3}\,}
with boundary condition
q
(
s
)
∼
Ai
(
s
)
,
s
→
∞
.
{\textstyle \displaystyle q(s )\sim {\textrm {Ai}}(s ),s\to \infty .}
This function
q
{\displaystyle q}
is a Painlevé transcendent .
Other distributions are also expressible in terms of the same
q
{\displaystyle q}
:
F
1
(
s
)
=
exp
(
−
1
2
∫
s
∞
q
(
x
)
d
x
)
(
F
2
(
s
)
)
1
/
2
F
4
(
s
/
2
)
=
cosh
(
1
2
∫
s
∞
q
(
x
)
d
x
)
(
F
2
(
s
)
)
1
/
2
.
{\displaystyle {\begin{aligned}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{aligned}}}
Functional equations [ edit ]
Define
F
(
x
)
=
exp
(
−
1
2
∫
x
∞
(
y
−
x
)
q
(
y
)
2
d
y
)
E
(
x
)
=
exp
(
−
1
2
∫
x
∞
q
(
y
)
d
y
)
{\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]
F
1
(
x
)
=
E
(
x
)
F
(
x
)
,
F
2
(
x
)
=
F
(
x
)
2
,
F
4
(
x
2
)
=
1
2
(
E
(
x
)
+
1
E
(
x
)
)
F
(
x
)
{\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 [ edit ]
Other than in random matrix theory, the Tracy–Widom distributions occur in many other probability problems.[12]
Let
l
n
{\displaystyle l_{n}}
be the length of the longest increasing subsequence in a random permutation sampled uniformly from
S
n
{\displaystyle S_{n}}
, the permutation group on n elements. Then the cumulative distribution function of
l
n
−
2
N
1
/
2
N
1
/
6
{\displaystyle {\frac {l_{n}-2N^{1/2}}{N^{1/6}}}}
converges to
F
2
{\displaystyle F_{2}}
.[13]
Asymptotics [ edit ]
Probability density function [ edit ]
Let
f
β
(
x
)
=
F
β
′
(
x
)
{\displaystyle f_{\beta }(x )=F_{\beta }'(x )}
be the probability density function for the distribution, then[12]
f
β
(
x
)
∼
{
e
−
β
24
|
x
|
3
,
x
→
−
∞
e
−
2
β
3
|
x
|
3
/
2
,
x
→
+
∞
{\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
λ
m
a
x
{\displaystyle \lambda _{max}}
to be much larger than
2
σ
N
{\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
λ
m
a
x
{\displaystyle \lambda _{max}}
to be not much smaller than
2
σ
N
{\displaystyle 2\sigma {\sqrt {N}}}
.
At the
x
→
−
∞
{\displaystyle x\to -\infty }
limit, a more precise expression is (equation 49 [12] )
f
β
(
x
)
∼
τ
β
|
x
|
(
β
2
+
4
−
6
β
)
/
16
β
exp
[
−
β
|
x
|
3
24
+
2
β
−
2
6
|
x
|
3
/
2
]
{\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
τ
β
{\displaystyle \tau _{\beta }}
that depends on
β
{\displaystyle \beta }
.
Cumulative distribution function [ edit ]
At the
x
→
+
∞
{\displaystyle x\to +\infty }
limit,[14]
F
(
x
)
=
1
−
e
−
4
3
x
3
/
2
32
π
x
3
/
2
(
1
−
35
24
x
3
/
2
+
O
(
x
−
3
)
)
,
E
(
x
)
=
1
−
e
−
2
3
x
3
/
2
4
π
x
3
/
2
(
1
−
41
48
x
3
/
2
+
O
(
x
−
3
)
)
{\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
x
→
−
∞
{\displaystyle x\to -\infty }
limit,
F
(
x
)
=
2
1
/
48
e
1
2
ζ
′
(
−
1
)
e
−
1
24
|
x
|
3
|
x
|
1
/
16
(
1
+
3
2
7
|
x
|
3
+
O
(
|
x
|
−
6
)
)
E
(
x
)
=
1
2
1
/
4
e
−
1
3
2
|
x
|
3
/
2
(
1
−
1
24
2
|
x
|
3
/
2
+
O
(
|
x
|
−
3
)
)
.
{\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
ζ
{\displaystyle \zeta }
is the Riemann zeta function , and
ζ
′
(
−
1
)
=
−
0.1654211437
{\displaystyle \zeta '(-1)=-0.1654211437}
.
This allows derivation of
x
→
±
∞
{\displaystyle x\to \pm \infty }
behavior of
F
β
{\displaystyle F_{\beta }}
. For example,
1
−
F
2
(
x
)
=
1
32
π
x
3
/
2
e
−
4
x
3
/
2
/
3
(
1
+
O
(
x
−
3
/
2
)
)
,
F
2
(
−
x
)
=
2
1
/
24
e
ζ
′
(
−
1
)
x
1
/
8
e
−
x
3
/
12
(
1
+
3
2
6
x
3
+
O
(
x
−
6
)
)
.
{\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 [ edit ]
The Painlevé transcendent has asymptotic expansion at
x
→
−
∞
{\displaystyle x\to -\infty }
(equation 4.1 of [15] )
q
(
x
)
=
−
x
2
(
1
+
1
8
x
−
3
−
73
128
x
−
6
+
10657
1024
x
−
9
+
O
(
x
−
12
)
)
{\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
q
∼
−
x
/
2
{\displaystyle q\sim {\sqrt {-x/2}}}
solution is unstable: any deviation from it tends to drop it to the
q
∼
−
−
x
/
2
{\displaystyle q\sim -{\sqrt {-x/2}}}
branch instead.[16]
Numerics [ edit ]
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
,
4
{\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
F
β
{\displaystyle F_{\beta }}
and the density functions
f
β
(
s
)
=
d
F
β
/
d
s
{\displaystyle f_{\beta }(s )=dF_{\beta }/ds}
for
β
=
1
,
2
,
4
{\displaystyle \beta =1,2,4}
. These algorithms can be used to compute numerically the mean , variance , skewness and excess kurtosis of the distributions
F
β
{\displaystyle F_{\beta }}
.[17]
β
{\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
A
s
{\displaystyle A_{s}}
, which can be used to rapidly evaluate Tracy–Widom distributions, or, more generally, the distributions of the
k
{\displaystyle k}
th largest level at the soft edge scaling limit of Gaussian ensembles, to machine accuracy.
Tracy-Widom and KPZ universality [ edit ]
The Tracy-Widom distribution appears as a limit distribution in the universality class of the KPZ equation . For example it appears under
t
1
/
3
{\displaystyle t^{1/3}}
scaling of the one-dimensional KPZ equation with fixed time.[18]
See also [ edit ]
^ Sasamoto & Spohn (2010)
^ Johansson (2000) ; Tracy & Widom (2009) ).
^ Johnstone (2007 , 2008 , 2009 ).
^ a b c Tracy, Craig A.; Widom, Harold (2009b). "The Distributions of Random Matrix Theory and their Applications" . In Sidoravičius, Vladas (ed.). New Trends in Mathematical Physics . Dordrecht: Springer Netherlands. pp. 753–765. doi :10.1007/978-90-481-2810-5_48 . ISBN 978-90-481-2810-5 .
^ Forrester, P. J. (1993-08-09). "The spectrum edge of random matrix ensembles" . Nuclear Physics B . 402 (3 ): 709–728. Bibcode :1993NuPhB.402..709F . doi :10.1016/0550-3213(93 )90126-A . ISSN 0550-3213 .
^ Dieng, Momar (2005). "Distribution functions for edge eigenvalues in orthogonal and symplectic ensembles: Painlevé representations" . International Mathematics Research Notices . 2005 (37 ): 2263–2287. doi :10.1155/IMRN.2005.2263 . ISSN 1687-0247 .
^ a b c Majumdar, Satya N; Schehr, Grégory (2014-01-31). "Top eigenvalue of a random matrix: large deviations and third order phase transition" . Journal of Statistical Mechanics: Theory and Experiment . 2014 (1 ): P01012. arXiv :1311.0580 . Bibcode :2014JSMTE..01..012M . doi :10.1088/1742-5468/2014/01/p01012 . ISSN 1742-5468 . S2CID 119122520 .
^ Baik, Deift & Johansson 1999
^ Baik, Jinho; Buckingham, Robert; DiFranco, Jeffery (2008-02-26). "Asymptotics of Tracy-Widom Distributions and the Total Integral of a Painlevé II Function" . Communications in Mathematical Physics . 280 (2 ): 463–497. arXiv :0704.3636 . Bibcode :2008CMaPh.280..463B . doi :10.1007/s00220-008-0433-5 . ISSN 0010-3616 . S2CID 16324715 .
^ Tracy, Craig A.; Widom, Harold (May 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 . ISSN 0370-2693 . S2CID 13912236 .
^ Bender, Carl M.; Orszag, Steven A. (1999-10-29). Advanced Mathematical Methods for Scientists and Engineers I: Asymptotic Methods and Perturbation Theory . Springer Science & Business Media. pp. 163–165. ISBN 978-0-387-98931-0 .
^ Su, Zhong-gen; Lei, Yu-huan; Shen, Tian (2021-03-01). "Tracy-Widom distribution, Airy2 process and its sample path properties" . Applied Mathematics-A Journal of Chinese Universities . 36 (1 ): 128–158. doi :10.1007/s11766-021-4251-2 . ISSN 1993-0445 . S2CID 237903590 .
^ Amir, Gideon; Corwin, Ivan; Quastel, Jeremy (2010). "Probability distribution of the free energy of the continuum directed random polymer in 1 + 1 dimensions". Communications on Pure and Applied Mathematics . 64 (4 ). Wiley: 466--537. arXiv :1003.0443 . doi :10.1002/cpa.20347 .
^ called "Hastings–McLeod solution". Published by
Hastings, S.P., McLeod, J.B.: A boundary value problem associated with the second Painlevé transcendent and the Korteweg-de Vries equation. Arch. Ration. Mech. Anal. 73 , 31–51 (1980)
References [ edit ]
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 [ edit ]
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. (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 [ edit ]
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
R e t r i e v e d f r o m " https://en.wikipedia.org/w/index.php?title=Tracy–Widom_distribution&oldid=1195832418 "
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