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Random Matrix Laws & Jacobi Operators Alan Edelman MIT May 19, 2014 joint with Alex Dubbs and Praveen Venkataramana (acknowledging gratefully the help from Bernie Wang)

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2/55 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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3/55 At a Glance Random Matrix IdeaJacobi Operator IdeaKey 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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4/55 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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5/55 Encoding Probability Densities Density 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 2 nd kind) Jacobi Matrix

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6/55 Gil Strang’s Favorite Matrix encoded in Cupcakes

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7/55 Computing the Jacobi encoding From the moments [Golub,Welsch 1969] 1.Form Hankel matrix of moments 2.R=Cholesky(H) 3.

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8/55 Computing the Jacobi encoding From 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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9/55 Example: Normal Distribution Moments Hermite Recurrence

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10/55 Example Chebfun Lanczos Run [Verbatim from Pedro Gonnet’s November 2011 Run] Thanks to Bernie Wang

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11/55 RMT LawFormula HermiteSemicircle Law Wigner 1955 Free CLT LaguerreMarcenko- Pastur Law 1967 JacobiWachter Law 1980 Gegenbauer random regular graphs Mckay Law 1981 (a=b=v/2) Too Small

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12/55 RMT Big laws: Toeplitz + Boundary That’s pretty special! Corresponds to 2 nd order differences with boundary [Anshelevich, 2010] (Free Meixner) [E, Dubbs, 2014] LawJacobi Encoding HermiteSemicircle Law 1955 Free CLT x=a y=b LaguerreMarcenko- Pastur Law 1967 Free Poisson x=parameter y=b JacobiWachter Law 1980 Free Binomial x=parameter y=parameter GengenbauerMackay Law 1981 x=a y=parameter

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13/55 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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14/55 Semicircle Law

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15/55 Marcenko-Pastur Law

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16/55 McKay Law

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17/55 Wachter Law

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18/55 What RM are these other three? [Anshelevich, 2010]

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19/55 Another interesting Random Matrix Law The singular values (squared) of Density: Moments:

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20/55 Jacobi Matrix J =

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21/55 Jacobi Matrix J =

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22/55 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 Idea A moment algorithm that “looks for” an eventually Toeplitz form Idea A moment algorithm that “looks for” an eventually Toeplitz form

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23/55 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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24/55 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 k x k example Replaces infinitely equal 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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25/55 Mathematica

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26/55 Fast convergence! Theory g[2] approx

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27/55 Even the normal distribution (not particularly well approximated by Toeplitz) It’s not a random matrix law!

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28/55 Moments

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29/55 Free Cumulants

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30/55 Wigner and Narayana Marcenko-Pastur = Limiting Density for Laguerre Moments are Narayana Polynomials! Narayana probably would not have known [Wigner, 1957] Narayana Photo Unavailable (Narayana was 27)

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31/55 At a Glance Random Matrix IdeaJacobi Operator IdeaKey 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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32/55 Multivariate Orthogonal Polynomials In random matrix theory and elsewhere The orthogonal polynomials associated with the weight of general beta distributions

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33/55 Classical Orthogonal Polynomials Triangular Sparsity structure of monomial expansion: Hermite: even/odd: Generally P n goes from 0 to n Tridiagonal sparsity of 3-termrecurrence

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34/55 Classical Orthogonal Polynomials Triangular Sparsity structure of monomial expansion: Hermite: even/odd: Generally P n goes from 0 to n Tridiagonal sparsity of 3-termrecurrence Extensions to multivariate case?? Before extending, a few slides about these multivariate polynomials and their applications.

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35/55 Hermite Polynomials become Multivariate Hermite Polynomials Orthogonal with respect to Indexed by degree k=0,1,2,3,… Symmetric scalar valued polynomials Indexed by partitions (multivariate degree): (),(1),(2),(1,1),(3),(2,1),(1,1,1),…

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36/55 Monomials become Jack Polynomials Orthogonal on the unit circle Orthogonal on copies of the unit circle with respect to circular ensemble measure Symmetric scalar valued polynomials Indexed by partitions (multivariate degree): (),(1),(2),(1,1),(3),(2,1),(1,1,1),…

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37/55 Multivariate Hermite Polynomials (β=1) X … matrix Polynomial evaluated at eigenvalues of X [Chikuse, 1992]

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38/55 (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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39/55 Special case β=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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40/55 Classical Orthogonal Polynomials Triangular Sparsity structure of monomial expansion: Hermite: even/odd: Generally P n goes from 0 to n Tridiagonal sparsity of 3-termrecurrence Extensions to multivariate case?? Before extending, a few slides about these multivariate polynomials and their applications.

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41/55 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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42/55 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=2General β Hermite, LaguerreYOUNG (Baker,Forrester) YOUNG (Baker,Forrester) Jacobi???? General Weight Functions YOUNG (Venkataramana, E) Probably NOT YOUNG ????? (Venkataramana, E)

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43/55 The second question What Is the sparsity pattern of the analog of = = ?

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44/55 Answer You, your parents and children in the Young Diagram

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45/55 At a Glance Random Matrix IdeaJacobi Operator IdeaKey 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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46/55 Hermite Jacobi Matrix

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47/55 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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48/55 Theorem: This is true for any weight function for which you have the Jacobi matrix Proof: (Venkataramana, E 2014)

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49/55 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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50/55 q-Hermite Jacobi Matrix q 1 recovers classical Hermite

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51/55 Genus expansion formula (β=2) Harer-Zagier formula

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52/55 When β=2: Murnaghan- Nakayama Rule Power function can be expanded in schur functions For example

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53/55 q-Harer Zagier formula [Venkataramana, E 2014]

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54/55 Extension to general q

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55/55 Conclusion This conference theme is fantastic Jacobi Operators Random Matrices Multivarite Jacobi: Much to Explore

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