random matrix theory numerical computation and remarkable applications alan edelman mathematics...
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Random Matrix TheoryNumerical Computation
and Remarkable Applications
Alan EdelmanMathematics
Computer Science & AI Labs
Computer Science & AI Laboratories
AMS Short CourseJanuary 8, 2013San Diego, CA
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A Personal Theme
• A Computational Trick can also be a Theoretical Trick
– A View: Math stands on its own.
– My View: Rigors of coding, modern numerical linear algebra, and the quest for efficiency has revealed deep mathematics.
• Tridiagonal/Bidiagonal Models• Stochastic Operators• Sturm Sequences/Ricatti Diffusion• Method of Ghosts and Shadows
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Outline
• Random Matrix Headlines
• Crash Course in Theory
• Crash Course on being a Random Matrix Theory user
• How I Got Into This Business: Random Condition Numbers
• Good Computations Leads to Good Mathematics
• (If Time) Ghosts and Shadows
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Outline
• Random Matrix Headlines
• Crash Course in Theory
• Crash Course on being a Random Matrix Theory user
• How I Got Into This Business: Random Condition Numbers
• Good Computations Leads to Good Mathematics
• (If Time) Ghosts and Shadows
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Early View of RMT
Heavy atoms too hard. Let’s throwup our hands and pretend energy levels come from a random matrix
Our viewRandomness is a structure! A NICE STRUCTURE!!!!
Think sampling elections, central limit theorems, self-organizing systems, randomized algorithms,…
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Random matrix theory in the natural progression of mathematics
• Scalar statistics
• Vector statistics
• Matrix statistics
Established Statistics
Newer Mathematics
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Outline
• Random Matrix Headlines
• Crash Course in Theory
• Crash Course on being a Random Matrix Theory user
• How I Got Into This Business: Random Condition Numbers
• Good Computations Leads to Good Mathematics
• (If Time) Ghosts and Shadows
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Crash course to introduce the Theory
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Class Notes from 18.338
Normal Distribution1733
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Semicircle Distribution 1955
Semicircle 1955
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Tracy-Widom Distribution 1993
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n random ±1’s
eig(A+Q’BQ)
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Free Probability
• Gives the distribution of the eigenvalues of A+Q’BQ given that of A and B
• (as n∞ theoretically, works well for finite n in practice)
• Can be explained with simple calculus to engineers usually in under 30 minutes
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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
Crash Course on White Noise and Brownian Motion
x=[0:h:1]; % h=.001
dW=randn(length(x),1)*sqrt(h); % white noise
W=cumsum(dW); %Brownian motion
plot(x,W)
Free Brownian Motion isthe limit of W where each elementof dW is a GOE *sqrt(h)
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W = anything + cumsum(dW)Interpolates anything to gaussians
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Outline
• Random Matrix Headlines
• Crash Course in Theory
• Crash Course on being a Random Matrix Theory user
• How I Got Into This Business: Random Condition Numbers
• Good Computations Leads to Good Mathematics
• (If Time) Ghosts and Shadows
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The GUE (Gaussian Unitary Ensemble)
• A=randn(n)+i*randn(n); S=(A+A’)/sqrt(4n)
• Eigenvalues follow semicircle law
• Eigenvalue repel! Spacings follow a known law:
http://matematiku.wordpress.com/2011/05/04/nontrivial-zeros-and-the-eigenvalues-of-random-matrices/
SPACINGS!
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Applications
• Parked Cars in London
• Zeros of the Riemann Zeta Function
• Busses in Cuernevaca, Mexico
• …..
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The Marcenko-Pastur Law
The density of the singular values of a normalized rectangular random matrix with aspect ratio r and iid elements (in the infinite limit, etc.)
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Covariance Matrix Estimation:
Source: http://www.math.nyu.edu/fellows_fin_math/gatheral/RandomMatrixCovariance2008.pdf Page 27
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RM Tool – Raj (U Michigan)
Free probability toolMathematics: The Polynomial Method
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Outline
• Random Matrix Headlines
• Crash Course in Theory
• Crash Course on being a Random Matrix Theory user
• How I Got Into This Business: Random Condition Numbers
• Good Computations Leads to Good Mathematics
• (If Time) Ghosts and Shadows
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Numerical Analysis: Condition Numbers
• (A) = “condition number of A”
• If A=UV’ is the svd, then (A) = max/min .
• One number that measures digits lost in finite precision and general matrix “badness”
– Small=good
– Large=bad
• The condition of a random matrix???
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Von Neumann & co.
• Solve Ax=b via x= (A’A) -1A’ b
M A-1
• Matrix Residual: ||AM-I||2
• ||AM-I||2< 2002 n
• How should we estimate ?
• Assume, as a model, that the elements of A are independent standard normals!
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Von Neumann & co. estimates (1947-1951)
• “For a ‘random matrix’ of order n the expectation value has been shown to be about n”
Goldstine, von Neumann
• “… we choose two different values of , namely n and 10n”Bargmann, Montgomery, vN
• “With a probability ~1 … < 10n”Goldstine, von Neumann
X
P(<n) 0.02P(< 10n)0.44
P(<10n) 0.80
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Random cond numbers, n
2/2/23
42 xxex
xy
Distribution of /n
Experiment with n=200Page 33
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Finite n
n=10 n=25
n=50 n=100
Convergence proved by Tao and VuOpen question: why so fast Page 34
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Tao-Vu ('09) “the rigorous proof”!
• Basic idea (NLA reformulation)...Consider a 2x2 block QR decomposition of M:
1. The smallest singular value of R22
, scaled by √n/s, is a good estimate for σ
n!
2. R22
(viewed as the product Q2
T M2) is roughly s x s Gaussian
M = (M1 M
2) = QR = (Q
1 Q
2)( )
Note: Q2T M
2 = R
22
R11
R12
n-s
R22
s
n-s s n-s s
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Sanity Checks on the smallest singular value
Gaussians +/- 1 (note many singulars)
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Bounds from the proof
• “C is a sufficiently large const (104 suffices)”
• Implied constants in O(...) depend on E|ξ|C
– For ξ = Gaussian, this is 9999!!• s = n500/C
– To get s = 10, n ≈ 1020?• Various tail bounds go as n-1/C
– To get 1% chance of failure, n ≈ 1020000??
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Good Computation Good Mathematics
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Outline
• Random Matrix Headlines
• Crash Course in Theory
• Crash Course on being a Random Matrix Theory user
• How I Got Into This Business: Random Condition Numbers
• Good Computations Leads to Good Mathematics
• (If Time) Ghosts and Shadows
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Outline
• Random Matrix Headlines
• Crash Course in Theory
• Crash Course on being a Random Matrix Theory user
• How I Got Into This Business: Random Condition Numbers
• Good Computations Leads to Good Mathematics
• (If Time) Ghosts and Shadows
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Eigenvalues of GOE (β=1)• Naïve Way:
MATLAB®: A=randn(n); S=(A+A’)/sqrt(2*n);eig(S)
R:
A=matrix(rnorm(n*n),ncol=n);S=(a+t(a))/sqrt(2*n);eigen(S,symmetric=T,only.values=T)$values;
Mathematica: A=RandomArray[NormalDistribution[],{n,n}];S=(A+Transpose[A])/Sqrt[n];Eigenvalues[s]
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Tridiagonal Model More Efficient
(Silverstein, Trotter, etc)
Beta Hermite ensemblegi ~N(0,2)
LAPACK’s DSTEQRStorage: O(n) (vs O(n2))Time: O(n2) (vs O(n3))Real Matrices
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Histogram without Histogramming:Sturm Sequences
• Count #eigs < 0.5: Count sign changes in
Det( (A-0.5*I)[1:k,1:k] )
• Count #eigs in [x,x+h]
Take difference in number of sign changes at x+h and x
Mentioned in Dumitriu and E 2006, Used theoretically in Albrecht, Chan, and E 2008
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A good computational trick is a good theoretical trick!
Finite Semi-Circle Laws for Any Beta!
Finite Tracy-Widom Laws for Any Beta!
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Efficient Tracy Widom Simulation
• Naïve Way:
A=randn(n); S=(A+A’)/sqrt(2*n);max(eig(S))
• Better Way:
• Only create the 10n1/3 initial segment of the diagonal and off-diagonal as the “Airy” function tells us that the max eig hardly depends on the rest
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Stochastic Operator – the best way
,dW β
2 x
dxd
2
2
+-
converges to
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Obervation
• Distributions you have seen are asymptotic limits!
• The matrices were left behind.
• Now we have stochastic operators whose distributions themselves can be studied.
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Tracy Widom Best Way
,dW β
2 x
dxd
2
2
+-
MATLAB:Diagonal =(-2/h^2)*ones(1,N) – x +(2/sqrt(beta))*randn(1,N)/sqrt(h)Off Diagonal = (1/h^2)*ones(1,N-1)
See applications by Alex Bloemendal, Balint Virag etc.
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Outline
• Random Matrix Headlines
• Crash Course in Theory
• Crash Course on being a Random Matrix Theory user
• How I Got Into This Business: Random Condition Numbers
• Good Computations Leads to Good Mathematics
• (If Time) Ghosts and Shadows
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The method of Ghosts and Shadows
for Beta Ensembles
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Introduction to Ghosts• G1 is a standard normal N(0,1)
• G2 is a complex normal (G1 +iG1)
• G4 is a quaternion normal (G1 +iG1+jG1+kG1)
• Gβ (β>0) seems to often work just fine
“Ghost Gaussian”
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Chi-squared
• Defn: χβ is the sum of β iid squares of standard normals if β=1,2,…
• Generalizes for non-integer β as the “gamma” function interpolates factorial
• χ β is the sqrt of the sum of squares (which generalizes) (wikipedia chi-distriubtion)
• |G1| is χ 1 , |G2| is χ 2, |G4| is χ 4
• So why not |G β | is χ β ?
• I call χ β the shadow of G β
2
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Scary Ideas in Mathematics
• Zero• Negative• Radical• Irrational• Imaginary• Ghosts: Something like a sometimes commutative algebra of
random variables that generalizes random Reals, Complexes, and Quaternions and inspires theoretical results and numerical computation
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Did you say “commutative”??
• Quaternions don’t commute.
• Yes but random quaternions do!
• If x and y are G4 then x*y and y*x are identically distributed!
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RMT Densities
• Hermite:
c ∏|λi-λj|β e-∑λi
2/2 (Gaussian Ensemble)
• Laguerre:
c ∏|λi-λj|β ∏λim e-∑λi (Wishart Matrices)
• Jacobi:
c ∏|λi-λj|β ∏λim1 ∏(1-λi)m2 (Manova Matrices)
• Fourier:
c ∏|λi-λj|β (on the complex unit circle) (Circular Ensembles)
(orthogonalized by Jack Polynomials)Page 58
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Wishart Matrices (arbitrary covariance)
• G=mxn matrix of Gaussians
• Σ=mxn semidefinite matrix
• G’G Σ is similar to A=Σ½G’GΣ-½
• For β=1,2,4, the joint eigenvalue density of A has a formula:
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Joint Eigenvalue density of G’G Σ
The “0F0” function is a hypergeometric function of two matrix arguments that depends only on the eigenvalues of the matrices. Formulas and software exist.
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Generalization of Laguerre
• Laguerre:
• Versus Wishart:
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General β?
The joint density:
is a probability density for all β>0.
Goals:• Algorithm for sampling from this density• Get a feel for the density’s “ghost” meaning
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Main Result
• An algorithm derived from ghosts that samples eigenvalues
• A MATLAB implementation that is consistent with other beta-ized formulas
– Largest Eigenvalue
– Smallest Eigenvalue
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Working with Ghosts
Real quantity
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More practice with Ghosts
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Bidiagonalizing Σ=I
• Z’Z has the Σ=I density giving a special case of
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The Algorithm for Z=GΣ½
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The Algorithm for Z=GΣ½
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Removing U and V
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Algorithm cont.
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Completion of Recursion
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Numerical Experiments – Largest Eigenvalue
• Analytic Formula for largest eigenvalue dist
• E and Koev: software to compute
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m5n4beta0.750M120.1234.a.fig
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Smallest Eigenvalue as Well
The cdf of the smallest eigenvalue,
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Cdf’s of smallest eigenvalue
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Goals
• Continuum of Haar Measures generalizing orthogonal, unitary, symplectic
• Place finite random matrix theory “β”into same framework as infinite random matrix theory: specifically β as a knob to turn down the randomness, e.g. Airy Kernel
–d2/dx2+x+(2/β½)dW White Noise
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Formally• Let Sn=2π/Γ(n/2)=“surface area of sphere”
• Defined at any n= β>0.• A β-ghost x is formally defined by a function fx(r) such that ∫∞ fx(r)
rβ-1Sβ-1dr=1.• Note: For β integer, the x can be realized as a random spherically
symmetric variable in β dimensions• Example: A β-normal ghost is defined by
• f(r)=(2π)-β/2e-r2/2
• Example: Zero is defined with constant*δ(r).• Can we do algebra? Can we do linear algebra?• Can we add? Can we multiply?
r=0
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Understanding ∏|λi-λj|β
• Define volume element (dx)^ by
(r dx)^=rβ(dx)^ (β-dim volume, like fractals, but don’t really see any fractal theory here)
• Jacobians: A=QΛQ’ (Sym Eigendecomposition)
Q’dAQ=dΛ+(Q’dQ)Λ- Λ(Q’dQ)
(dA)^=(Q’dAQ)^= diagonal ^ strictly-upper
diagonal = ∏dλi =(dΛ)^
off-diag = ∏((Q’dQ)ij(λi-λj))^=(Q’dQ)^ ∏|λi-λj|β
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Conclusion
• Random Matrices are Really Useful!
• The totality of the subject is huge
– Try to get to know it from all corners!
• Most Problems still unsolved!
• A good computational trick is a good theoretical trick!
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Numerical Tools
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Entertainment
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Random Triangles, Random Matrices, and Lewis Carroll
Alan EdelmanMathematics
Computer Science & AI Labs
Gilbert StrangMathematics
Computer Science & AI LaboratoriesPresentation Author, 2003Page 90
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What do triangles look like?
Popular triangles (Google!) are all acute
Textbook (generic) triangles are always acute
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What is the probability that a random triangle is acute?
January 20, 1884
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Depends on your definition of random: One easy case!
Uniform on the space(Angle 1)+(Angle 2)+(Angle 3)=180o
(0,180,0)
(0,0,180) (180,0,0)(90,0, 90)
(90,90,0)(0,90, 90) (45,90,45)
(45,45,90) (90,45,45)
(120,30,30)
Acute
Obtuse
ObtuseObtuse
Right Right
(60.60.60)
(30,120,30)
(30,30,120)
Right
Prob(Acute)=¼
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Another case/same answer: normals! P(acute)=¼
3 vertices x 2 coordinates = 6 independent Standard Normals
Experiment: A=randn(2,3)
=triangle vertices
Not the same probability measure!
Open problem:give a satisfactory explanation of why both measures should give the same answer
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An interesting experiment
Compute side lengths normalized to a2+b2+c2=1Plot (a2,b2,c2) in the plane x+y+z=1
Black=Obtuse Blue=Acute Dot density largest near the perimeter
Dot density = uniform on hemisphere as it appears to the eye from above
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Kendall and others, “Shape Space”
Kendall “Father” of modern probability theory in Britain.
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Connection to Linear Algebra
The problem is equivalent to knowing the condition number distribution of a random 2x2 matrix of normals normalized to Frobenius norm 1.
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Connection to Shape Theory
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In Terms of Singular Values
A=(2x2 Orthogonal)(Diagonal)(Rotation(θ))
Longitude on hemisphere = 2θz-coordinate on hemisphere = determinant
Condition Number density (Edelman 89) =
Or the normalized determinant is uniform:
Also ellipticity statistic in multivariate statistics!Page 99
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What are the Eigenvalues of a Sum of (Non-Commuting) Random Symmetric Matrices? : A "Quantum Information" Inspired Answer.
Alan Edelman
Ramis Movassagh
Presentation Author, 2003Page 100
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Example Resultp=1 classical probabilityp=0 isotropic convolution (finite free probability)
We call this “isotropic entanglement”
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Simple Question
The eigenvalues of
where the diagonals are random, and randomly ordered. Too easy?Page 102
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Another Question
where Q is orthogonal with Haar measure. (Infinite limit = Free probability)
The eigenvalues of
T
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Quantum Information Question
where Q is somewhat complicated. (This is the general sum of two symmetric matrices)
The eigenvalues of
T
I like to think of the two extremes as localized eigenvectors and delocalizedeigenvectors!
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Moments?
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Wishart
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Stochastic Differential Operators
• Eigenvalues may be as important as stochastic differential equations
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Everyone’s Favorite Tridiagonal
-2 11 -2 1
11 -2
… …
…
…
…1n2
d2
dx2
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Everyone’s Favorite Tridiagonal
-2 11 -2 1
11 -2
… …
…
…
…1n2
d2
dx2
1(βn)1/2+
G
G
G
dWβ1/2+
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Conclusion
• Random Matrix Theory is rich, exciting, and ripe for applications
• Go out there and use a random matrix result in your area
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Equilibrium Measures (kind of a maximum likelihood distribution)Riemann-Hilbert Problems
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Multivariate Orthogonal Polynomials&Hypergeometrics of Matrix Argument
• The important special functions of the 21st century
• Begin with w(x) on I–∫ pκ(x)pλ(x) Δ(x)β ∏i w(xi)dxi = δκλ
– Jack Polynomials orthogonal for w=1 on the unit circle. Analogs of xm
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Multivariate Hypergeometric Functions
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Multivariate Hypergeometric Functions
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Hypergeometric Functions of Matrix Argument, Zonal
Polynomials, Jack Polynomials
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Exact computationof “finite” Tracy Widomlaws
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Mops (Dumitriu etc. 2004) Symbolic
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A=randn(n); S=(A+A’)/2; trace(S^4)
det(S^3)
Symbolic MOPS applications
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Symbolic MOPS applications
β=3; hist(eig(S))
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Smallest eigenvalue statistics
A=randn(m,n); hist(min(svd(A).^2))
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