Random Matrix Laws & Jacobi Operators

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. 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. At a Glance

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

5. JacobiMatrix

6. Gil Strang’s Favorite Matrixencoded in Cupcakes

7. Computing the Jacobi encodingFrom the moments[Golub,Welsch 1969] • Form Hankel matrix of moments • R=Cholesky(H)

8. 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

9. Example: Normal Distribution Moments  Hermite Recurrence

10. Example ChebfunLanczos Run [Verbatim from Pedro Gonnet’s November 2011 Run] Thanks to Bernie Wang

11. Too Small

12. RMT Big laws: Toeplitz + Boundary [Anshelevich,2010] (Free Meixner) [E, Dubbs, 2014] That’s pretty special! Corresponds to 2nd order differences with boundary

13. 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

14. Semicircle Law

15. Marcenko-Pastur Law

16. McKay Law

17. Wachter Law

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

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

20. Jacobi Matrix J =

21. Jacobi Matrix J =

22. 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 Toeplitzform

23. Algorithm • Compute truncated Jacobi from a few initial moments 1a. (or run a few steps of Lanczos on the density) • Compute g(x)= • Approximate density = 5x5 example

24. Algorithm • Compute truncated Jacobi from a few initial moments 1a. (or run a few steps of Lanczos on the density) • Compute g(x)= • Approximate density = k x k example 5x5 example “It’s like replacing .1666… with 1/6 and not .16” “No need to move off real axis” Replaces infinitely equal α’s and β’s

25. Mathematica

26. Fast convergence! Theory g[2] approx

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

28. Moments

29. Free Cumulants

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

31. At a Glance

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

33. Classical Orthogonal Polynomials • Triangular Sparsity structure of monomial expansion: • Hermite: even/odd: • Generally Pngoes from 0 to n • Tridiagonalsparsity of 3-term recurrence

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

35. Hermite Polynomials becomeMultivariate Hermite Polynomials Orthogonal with respect to 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. Monomials becomeJack 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. Multivariate Hermite Polynomials (β=1) [Chikuse, 1992] X … matrix Polynomial evaluated at eigenvalues of X

38. (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. β=2 Special case • 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. Classical Orthogonal Polynomials • Triangular Sparsity structure of monomial expansion: • Hermite: even/odd: • Generally Pngoes from 0 to n • Tridiagonalsparsity of 3-term recurrence Extensions to multivariate case?? Before extending, a few slides about these multivariate polynomials and their applications.

41. 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. 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

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

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

45. At a Glance

46. Hermite Jacobi Matrix

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

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

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