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)

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) 2/55

At a Glance 1. Probability Densities as Jacobi Operators 2. Multivariate Orthogonal Polynomials 3. Natural q-GUE integrals (q-theory) •

Random Matrix Idea

• • • Key Limit Density Laws Other Limit Density Laws Multivariate weights for β-Ensembles Genus Expansion • •

Jacobi Operator Idea

Toeplitz + Boundary Asymptotically Toeplitz • • • Generalization of triangular and tridiagonal structure q-Hermite Jacobi operator Application of Algorithm in 1

Key Point

Moment Matching Algorithm Young Diagrams Explicit Generalized Harer-Zagier Formula 3/55

Jacobi Operators (Symmetric Tridiagonal Format) Three term recurrence coefficients for orthogonal polynomials displayed as a Jacobi matrix Classically derived through Gram-Schmidt… 4/55

Encoding Probability Densities

Density Moments Random Number Generator Fourier Transform Cauchy Transform R-Transform Orthogonal Polynomials BetaRand(3/2,3/2) (then x  4x-2) (Bessel Function) [Wigner] (Cheybshev of 2 nd kind) Jacobi Matrix 5/55

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

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

7/55

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 8/55

Example: Normal Distribution Moments  Hermite Recurrence 9/55

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

Hermite

RMT Law

Semicircle Law Wigner 1955 Free CLT Laguerre Marcenko Pastur Law 1967 Jacobi Wachter Law 1980 Gegenbauer random regular graphs Mckay Law 1981 (a=b=v/2)

Formula

0.4

0.3

0.2

0.1

0 −0.1

−2.5

1.5

−2 −1.5

−1 −0.5

0 0.5

1 1.5

2 2.5

1 0.5

0 0 0.5

1 1.5

2 Too Small 11/55

RMT Big laws: Toeplitz + Boundary

Law

Hermite Semicircle Law 1955

Jacobi Encoding

x=a y=b [Anshelevich, 2010] (Free Meixner) [E, Dubbs, 2014] That’s pretty special!

Corresponds to 2 nd order differences with boundary Laguerre Jacobi Free CLT Marcenko Pastur Law 1967 Free Poisson Wachter Law 1980 Gengenbauer Free Binomial Mackay Law 1981 x=parameter y=b x=parameter y=parameter x=a y=parameter 12/55

Anshelevich Theory [Anshelevich, 2010] • • 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 13/55

Semicircle Law 14/55

Marcenko-Pastur Law 15/55

McKay Law 16/55

Wachter Law 17/55

What RM are these other three?

[Anshelevich, 2010] 18/55

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

Jacobi Matrix J = 20/55

Jacobi Matrix J = 21/55

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 22/55

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)= 5x5 example 3. Approximate density = 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)= “It’s like replacing .1666… with 1/6 and not .16” “No need to move off real axis” 3. Approximate density = Replaces infinitely equal α’s and β’s 24/55

Mathematica 25/55

Fast convergence!

Theory g[2] approx 26/55

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

27/55

Moments 28/55

Free Cumulants 29/55

Wigner and Narayana Narayana Photo Unavailable [Wigner, 1957] (Narayana was 27) • • • Marcenko-Pastur = Limiting Density for Laguerre Moments are Narayana Polynomials!

Narayana probably would not have known 30/55

At a Glance 1. Probability Densities as Jacobi Operators 2. Multivariate Orthogonal Polynomials 3. Natural q-GUE integrals (q-theory) •

Random Matrix Idea

• • • Key Limit Density Laws Other Limit Density Laws Multivariate weights for β-Ensembles Genus Expansion • •

Jacobi Operator Idea

Toeplitz + Boundary Asymptotically Toeplitz • • • Generalization of triangular and tridiagonal structure q-Hermite Jacobi operator Application of Algorithm in 1

Key Point

Moment Matching Algorithm Young Diagrams Explicit Generalized Harer-Zagier Formula 31/55

Multivariate Orthogonal Polynomials • In random matrix theory and elsewhere • The orthogonal polynomials associated with the weight of general beta distributions 32/55

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

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) ,… Orthogonal with respect to 35/55

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

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

(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.

38/55

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 39/55

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

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 41/55

• 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 Hermite, Laguerre Jacobi General Weight Functions

β=2

YOUNG (Baker,Forrester) ????

YOUNG (Venkataramana, E)

General β

YOUNG (Baker,Forrester) ????

Probably NOT YOUNG ?????

(Venkataramana, E) 42/55

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

?

43/55

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

At a Glance 1. Probability Densities as Jacobi Operators 2. Multivariate Orthogonal Polynomials 3. Natural q-GUE integrals (q-theory) •

Random Matrix Idea

• • • Key Limit Density Laws Other Limit Density Laws Multivariate weights for β-Ensembles Genus Expansion • •

Jacobi Operator Idea

Toeplitz + Boundary Asymptotically Toeplitz • • • Generalization of triangular and tridiagonal structure q-Hermite Jacobi operator Application of Algorithm in 1

Key Point

Moment Matching Algorithm Young Diagrams Explicit Generalized Harer-Zagier Formula 45/55

Hermite Jacobi Matrix 46/55

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 47/55

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

Proof Idea • We can use the wonderful formula • To compute integrals of any symmetric polynomial against • without needing to know w(x) explicitly 49/55

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

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

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

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

Extension to general q 54/55

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