1. 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)

2. 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)

3. 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

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

5. 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

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

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

8. 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

9. 9/55 Example: Normal Distribution Moments  Hermite Recurrence

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

11. 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

12. 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

13. 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]

14. 14/55 Semicircle Law

15. 15/55 Marcenko-Pastur Law

16. 16/55 McKay Law

17. 17/55 Wachter Law

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

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

20. 20/55 Jacobi Matrix J =

21. 21/55 Jacobi Matrix J =

22. 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

23. 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

24. 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”

25. 25/55 Mathematica

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

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

28. 28/55 Moments

29. 29/55 Free Cumulants

30. 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)

31. 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

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

33. 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

34. 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.

35. 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),…

36. 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),…

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

38. 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.

39. 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

40. 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.

41. 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

42. 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)

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

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

45. 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

46. 46/55 Hermite Jacobi Matrix

47. 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

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

49. 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

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

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

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

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

54. 54/55 Extension to general q

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