Random Matrix Laws&

Jacobi Operators

Alan EdelmanMIT

May 19, 2014

joint with Alex Dubbs and Praveen Venkataramana(acknowledging gratefully the help from Bernie Wang)

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Conference Blurb

• Recent years have seen significant progress in the understanding of asymptotic spectral properties of random matrices and related systems.

• One particularly interesting aspect is the multifaceted connection with properties of orthogonal polynomial systems, encoded in Jacobi matrices (and their analogs)

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At a GlanceRandom Matrix Idea Jacobi Operator Idea Key Point

1. Probability Densities as

Jacobi Operators

• Key Limit Density Laws

• Other Limit Density Laws

• Toeplitz + Boundary• Asymptotically

Toeplitz

Moment Matching Algorithm

2. Multivariate Orthogonal Polynomials

• Multivariate weights for β-Ensembles

• Generalization of triangular and tridiagonal structure

Young Diagrams

3. Natural q-GUE integrals (q-theory)

• Genus Expansion • q-Hermite Jacobi operator

• Application of Algorithm in 1

Explicit Generalized Harer-Zagier

Formula

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Jacobi Operators (Symmetric Tridiagonal Format)

Three term recurrence coefficients for orthogonal polynomials displayed as a Jacobi matrix

Classically derived through Gram-Schmidt…

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Encoding Probability DensitiesDensity

Moments

Random Number Generator BetaRand(3/2,3/2) (then x 4x-2)

Fourier Transform (Bessel Function) [Wigner]

Cauchy Transform

R-Transform

Orthogonal Polynomials (Cheybshev of 2nd kind)

Jacobi Matrix

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Gil Strang’s Favorite Matrixencoded in Cupcakes

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Computing the Jacobi encodingFrom the moments [Golub,Welsch 1969]

1. Form Hankel matrix of moments

2. R=Cholesky(H)3.

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Computing the Jacobi encodingFrom the weight (Continuous Lanczos)

• Inner product:• Computes Jacobi Parameters and orthogonal polynomials• Discrete version very successful for eigenvalues of sparse

symmetric matrices• May be computed with Chebfun

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Example: Normal Distribution Moments Hermite Recurrence

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Example Chebfun Lanczos Run[Verbatim from Pedro Gonnet’s November 2011 Run]

Thanks toBernie Wang

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RMT Law Formula

Hermite Semicircle LawWigner 1955

Free CLT

Laguerre Marcenko-Pastur Law

1967

Jacobi Wachter Law 1980

Gegenbauerrandom regular graphs

Mckay Law1981

(a=b=v/2)

Too Small

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RMT Big laws: Toeplitz + Boundary

That’s pretty special!Corresponds to 2nd order differences with boundary

[Anshelevich, 2010] (Free Meixner)[E, Dubbs, 2014]

Law JacobiEncoding

Hermite Semicircle Law 1955

Free CLT

x=ay=b

Laguerre Marcenko-Pastur Law

1967

Free Poisson

x=parametery=b

Jacobi Wachter Law 1980

Free Binomial

x=parametery=parameter

Gengenbauer Mackay Law1981

x=ay=parameter

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Anshelevich Theory• Describe all weight Functions whose Jacobi encoding

is Toeplitz off the first row and column• This is a terrific result, which directly lets us

characterize

• McKay often thrown in with Wachter, but seems worth distinguishing as special

• Known as “free Meixner,” but I prefer to emphasize the Toeplitz plus boundary aspect

[Anshelevich, 2010]

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Semicircle Law

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Marcenko-Pastur Law

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McKay Law

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Wachter Law

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What RM are these other three?

[Anshelevich, 2010]

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Another interesting Random Matrix Law• The singular values (squared) of

• Density:

• Moments:

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Jacobi Matrix

J =

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Jacobi Matrix

J =

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Implication?

• The four big laws are Toeplitz + size 1 border• The svd law seems to be heading towards Toeplitz• Enough laws “want” to be Toeplitz

IdeaA moment algorithm that “looks for” an eventually Toeplitz form

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Algorithm

1. Compute truncated Jacobi from a few initial moments 1a. (or run a few steps of Lanczos on the density)

2. Compute g(x)=

3. Approximate density =

5x5 example

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Algorithm

1. Compute truncated Jacobi from a few initial moments 1a. (or run a few steps of Lanczos on the density)

2. Compute g(x)=

3. Approximate density =

5x5 examplek x k example

Replaces infinitelyequal α’s and β’s

“It’s like replacing .1666… with 1/6 and not .16”“No need to move off real axis”

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Mathematica

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Fast convergence!

Theoryg[2] approx

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Even the normal distribution• (not particularly well approximated by Toeplitz)• It’s not a random matrix law!

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Moments

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Free Cumulants

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Wigner and Narayana

• Marcenko-Pastur = Limiting Density for Laguerre• Moments are Narayana Polynomials!• Narayana probably would not have known

[Wigner, 1957]

NarayanaPhoto

Unavailable

(Narayana was 27)

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At a GlanceRandom Matrix Idea Jacobi Operator Idea Key Point

1. Probability Densities as

Jacobi Operators

• Key Limit Density Laws

• Other Limit Density Laws

• Toeplitz + Boundary• Asymptotically

Toeplitz

Moment Matching Algorithm

2. Multivariate Orthogonal Polynomials

• Multivariate weights for β-Ensembles

• Generalization of triangular and tridiagonal structure

Young Diagrams

3. Natural q-GUE integrals (q-theory)

• Genus Expansion • q-Hermite Jacobi operator

• Application of Algorithm in 1

Explicit Generalized Harer-Zagier

Formula

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Multivariate Orthogonal Polynomials

• In random matrix theory and elsewhere• The orthogonal polynomials associated with the

weight of general beta distributions

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Classical Orthogonal Polynomials• Triangular Sparsity structure of monomial

expansion:• Hermite: even/odd:• Generally Pn goes from 0 to n

• Tridiagonal sparsity of 3-term recurrence

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Classical Orthogonal Polynomials• Triangular Sparsity structure of monomial

expansion:• Hermite: even/odd:• Generally Pn goes from 0 to n

• Tridiagonal sparsity of 3-term recurrence

Extensions to multivariate case?? Before extending, a few slidesabout these multivariate polynomials and their applications.

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Hermite Polynomials becomeMultivariate Hermite Polynomials

Orthogonal with respect to

Orthogonal with respect to

Indexed by degree k=0,1,2,3,…

Symmetric scalar valued polynomialsIndexed by partitions (multivariate degree):(),(1),(2),(1,1),(3),(2,1),(1,1,1),…

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Monomials becomeJack Polynomials

Orthogonal on the unit circle

Orthogonal on copiesof the unit circle with respect tocircular ensemble measure

Symmetric scalar valued polynomialsIndexed by partitions (multivariate degree):(),(1),(2),(1,1),(3),(2,1),(1,1,1),…

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Multivariate Hermite Polynomials (β=1)

X … matrix Polynomial evaluated at eigenvalues of X

[Chikuse, 1992]

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(Selberg Integrals and)Combinatorics of mult polynomials:

Graphs on Surfaces(Thanks to Mike LaCroix)

• Hermite: Maps with one Vertex Coloring

• Laguerre: Bipartite Maps with multiple Vertex Colorings

• Jacobi: We know it’s there, but don’t have it quite yet.

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Special caseβ=2

• Balderrama, Graczyk and Urbina (original proof)• β=2 (only!): explicit formula for multivariate

orthogonal polynomials in terms of univariate orthogonal polynomials.

• Generalizes Schur Polynomial construction in an important way

• New proof reduces to orthogonality of Schur’s

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Classical Orthogonal Polynomials• Triangular Sparsity structure of monomial

expansion:• Hermite: even/odd:• Generally Pn goes from 0 to n

• Tridiagonal sparsity of 3-term recurrence

Extensions to multivariate case?? Before extending, a few slidesabout these multivariate polynomials and their applications.

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What we know about the first question

• Sometimes follows the Young Diagram• Hermite always follows Young diagram for all β

• Laguerre always follows Young diagram for all β

• (Baker and Forrester 1998)

Young Diagram

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What we know• Young Diagram for Hermite, Laguerre for all β• Young Diagram for all weight functions for β=2 (can

be derived from schur polynomials)• Numerical evidence suggests answer does not

follow Young diagram for all weight functions for all beta

• Open Questions remain

β=2 General β

Hermite, Laguerre YOUNG(Baker,Forrester)

YOUNG(Baker,Forrester)

Jacobi ???? ????

General Weight Functions

YOUNG(Venkataramana, E)

Probably NOT YOUNG ?????(Venkataramana, E)

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The second question • What Is the sparsity pattern of the analog of

=

= ?

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Answer

You, your parents andchildren in the YoungDiagram

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At a GlanceRandom Matrix Idea Jacobi Operator Idea Key Point

1. Probability Densities as

Jacobi Operators

• Key Limit Density Laws

• Other Limit Density Laws

• Toeplitz + Boundary• Asymptotically

Toeplitz

Moment Matching Algorithm

2. Multivariate Orthogonal Polynomials

• Multivariate weights for β-Ensembles

• Generalization of triangular and tridiagonal structure

Young Diagrams

3. Natural q-GUE integrals (q-theory)

• Genus Expansion • q-Hermite Jacobi operator

• Application of Algorithm in 1

Explicit Generalized Harer-Zagier

Formula

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Hermite Jacobi Matrix

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The Jacobi matrix Defines the moments of the normal

Similarly there is a recipe for

that does not require knowledge of the multivariate β=2 Hermite weight

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Theorem: This is true for any weight function for which you have the Jacobi matrix

• Proof: (Venkataramana, E 2014)

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Proof Idea

• We can use the wonderful formula

• To compute integrals of any symmetric polynomial against

• without needing to know w(x) explicitly

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q-Hermite Jacobi Matrix

q1 recovers classical Hermite

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Genus expansion formula (β=2)

Harer-Zagier formula

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When β=2: Murnaghan-Nakayama Rule • Power function can be expanded in schur functions• For example

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q-Harer Zagier formula

[Venkataramana, E 2014]

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Extension to general q

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Conclusion

• This conference theme is fantastic

Jacobi Operators

Random Matrices

• Multivarite Jacobi: Much to Explore