The Random Matrix Technique of Ghosts and Shadows

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The Random Matrix Technique of Ghosts and Shadows

• Alan Edelman
• Dept of Mathematics
• Computer Science and AI Laboratories
• Massachusetts Institute of Technology
• (with thanks to Plamen Koev)

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Short followed by the Movie

• Some interesting computational techniques
• Random Matrix Theory Theorems, Applications,
  and Software
• A new application can be more valuable than a
  theorem!
• A well crafted experiment or package is not a
  theorem but it can be as important or even more
  to the field!
• The Main Show The Method of Ghosts and Shadows
  in Random Matrix Theory
• Yet another nail in the threefold way coffin

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Semi-Circle Law

• Naïve Way
• MATLAB Arandn(n) S(AA)/sqrt(2n)eig
  (S)
• R
• Amatrix(rnorm(nn),ncoln)S(at(a))/sqrt(
  2n)eigen(S,symmetricT,only.valuesT)values
• Mathematica ARandomArrayNormalDistribution,n
  ,nS(ATransposeA)/SqrtnEigenvaluess

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Compute All the Eigenvalues

• Sym Tridiagonal ß1real, ß2complex,
  ß4quaternion, ß2½?

Diagonals N(0,2), Off-diagonals chi
random-variables N2000 12 seconds vs. 0.2
seconds (factor of 60!!) (Dumitriu E 2002)
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Histogram without HistogrammingSturm Sequences

• Count eigs lt 0.5 Count sign changes in
• Det (A-0.5I)1k,1k
• Count eigs in x,xh
• Take difference in number of sign changes at
  xh 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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Stochastic Differential Eigen-Equations

• Tridiagonal Models Suggest SDEs with Brownian
  motion as infinite limit
• E 2002
• E and Sutton 2005, 2007
• Brian Rider and (Ramirez, Cambronero, Virag,
  etc.)
• (Lots of beautiful results!)
• (Not todays talk)

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Tracy-Widom ComputationsEigenvalues without
the whole matrix!
Never construct the entire tridiagonal
matrix! Just say the upper 10n1/3 by 10n1/3
Compute largest eigenvalue of that, perhaps
using Lanczos with shift and invert strategy! Can
compute for n amazingly large!!!!! And any beta.
E 2003 Persson and E 2005
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Other Computational Results

• Infinite Random Matrix Theory
• The Free Probability Calculator (Raj)
• Finite Random Matrix Theory
• MOPS (Ioana Dumitriu) (ß orthogonal polynomials)
• Hypergeometrics of Matrix Argument (Plamen Koev)
  (ß distribution functions for finite stats such
  as the finite Tracy-Widom laws for Laguerre)
• Good stuff, but not today.

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Ghosts and Shadows
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Scary Ideas in Mathematics

• Zero
• Negative
• Radical
• Irrational
• Imaginary
• Ghosts Something like a commutative algebra of
  random variables that generalizes Reals,
  Complexes, and Quaternions and inspires
  theoretical results and numerical computation

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RMT Densities

• Hermite
• c ??i-?jß e-??i2/2 (Gaussian Ensemble)
• Laguerre
• c ??i-?jß ??im e-??i (Wishart Matrices)
• Jacobi
• c ??i-?jß ??im1 ?(1-?i)m2 (Manova Matrices)
• Fourier
• c ??i-?jß (on the complex unit circle) Jack
  Polynomials

Traditional Story Count the real parameters
ß1,2,4 for real, complex, quaternion Application
s ß1 All of Multivariate Statistics ß2parked
cars in London, wireless networks
ß4 There and almost nobody cares Dyson 1962
Threefold Way Three Division Rings
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ß-Ghosts

• ß1 has one real Gaussian (G)
• ß2 has two real Gaussians (GiG)
• ß4 has four real Gaussians (GiGjGkG)
• ß1 has one real part and (ß-1) Ghost parts

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Introductory Theory

• There is an advanced theory emerging (some other
  day)
• Informally
• A ß-ghost is a spherically symmetric random
  variable defined on Rß
• A shadow is a derived real or complex quantity

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Goals

• Continuum of Haar Measureas generalizing
  orthogonal, unitary, symplectic
• New Definition of Jack Polynomials generalizing
  the zonals
• Computations! E.g. Moments of the Haar Measures
• Place finite random matrix theory ßinto same
  framework as infinite random matrix theory
  specifically ß as a knob to turn down the
  randomness, e.g. Airy Kernel
• d2/dx2x(2/ß½)dW ?White Noise

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Formally

• Let Sn2p/G(n/2)suface area of sphere
• Defined at any n ßgt0.
• A ß-ghost x is formally defined by a function
  fx(r) such that ?8 fx(r) rß-1Sß-1dr1.
• Note For ß integer, the x can be realized as a
  random spherically symmetric variable in ß
  dimensions
• Example A ß-normal ghost is defined by
  f(r)(2p)-ß/2e-r2/2
• Example Zero is defined with constantd(r).
• Can we do algebra? Can we do linear algebra?
• Can we add? Can we multiply?

r0
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A few more operations

• ??x?? is a real random variable whose density is
  given by fx(r)
• (xx)/2 is real random variable given by
  multiplying ??x?? by a beta distributed random
  variable representing a coordinate on the sphere

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Representations

• Ive tried a few on for size. My favorite right
  now is
• A complex number z with ??z??, the radius and
  Re(z), the real part.

Addition of Independent Ghosts

• Addition returns a spherically symmetric object
• Have an integral formula
• Prefer Add the real part, imaginary part
  completed to keep spherical symmetry

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Multiplication of Independent Ghosts

• Just multiply ??z??s and plug in spherical
  symmetry
• Multiplication is commutative
• (Important Example Quaternions dont commute,
  but spherically symmetric random variables do!)

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Shadow Example

• Given a ghost Gaussian, Gß, the length is a real
  chi-beta variable ? ß variable.

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Linear Algebra Example

• Given a ghost Gaussian, Gß, the length is a real
  chi-beta variable ? ß variable.
• Gram-Schmidt
• ? ? ?
• or
• Q

Gß Gß Gß
Gß Gß Gß
Gß Gß Gß
?3ß Gß Gß
Gß Gß
Gß Gß
?3ß Gß Gß
?2ß Gß
Gß
?3ß Gß Gß
?2ß Gß
?ß
H3
H2
H1
Gß Gß Gß
Gß Gß Gß
Gß Gß Gß
?3ß Gß Gß
?2ß Gß
?ß
Q has ß -Haar Measure! We have computed
moments! (more later)
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Tridiagonalizing Example
Gß Gß Gß Gß
Gß Gß Gß Gß
Gß Gß Gß Gß

Gß Gß Gß Gß Gß
Symmetric Part of
?
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Understanding ??i-?jß

• Define volume element (dx) by
• (r dx)rß(dx) (ß-dim volume, like fractals,
  but dont really see any fractal theory here)
• Jacobians AQ?Q (Sym Eigendecomposition)
• QdAQd?(QdQ)?- ?(QdQ)
• (dA)(QdAQ) diagonal strictly-upper
• diagonal ?d?i (d?)
• off-diag ?((QdQ)ij(?i-?j))(QdQ)
  ??i-?jß

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Haar Measure

• ß1 EQ(trace(AQBQ)k)?C?(A)C?(B)/C?(I)
• Forward Method Suppose you know C?s a-priori.
  (Jack Polynomials!)
• Let A and B be diagonal indeterminants (Think
  Generating Functions)
• Then can formally obtain moments of Q
• Example E(q112q222) (na-1)/(n(n-1)(na))
• a2/ ß
• Can Gram-Schmidt the ghosts. Same answers coming
  up!

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Further Uses of Ghosts

• Multivariate Othogonal Polynomials
• Largest Eigenvalues/Smallest Eigenvalues
• Noncentral Distributions
• Expect Lots of Uses to be discovered