### Random Matrix Laws & Jacobi Operators

```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
Random Matrix Idea
Jacobi Operator Idea
Key Point
Moment Matching
Algorithm
1. Probability
Densities as
Jacobi Operators
• Key Limit Density
Laws
• Other Limit Density
Laws
• Toeplitz + Boundary
• Asymptotically
Toeplitz
2. Multivariate
Orthogonal
Polynomials
• Multivariate weights
for β-Ensembles
• Generalization of
triangular and
tridiagonal
structure
Young Diagrams
• Genus Expansion
• q-Hermite Jacobi
operator
• Application of
Algorithm in 1
Explicit Generalized
Harer-Zagier
Formula
3. Natural q-GUE
integrals
(q-theory)
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
BetaRand(3/2,3/2) (then x 4x-2)
(Bessel Function) [Wigner]
Cauchy Transform
R-Transform
Orthogonal Polynomials
(Cheybshev of 2nd 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
RMT Law
Formula
0.4
Hermite
Semicircle Law
Wigner 1955
Free CLT
0.3
0.2
0.1
0
Laguerre
MarcenkoPastur Law
1967
−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
Jacobi
Wachter Law
1980
0
0.5
1
1.5
2
Too Small
Gegenbauer
random
regular
graphs
Mckay Law
1981
(a=b=v/2)
11/55
RMT Big laws: Toeplitz + Boundary
Hermite
Law
Jacobi
Encoding
Semicircle Law
1955
x=a
y=b
Free CLT
Laguerre
MarcenkoPastur Law
1967
x=parameter
y=b
Free Poisson
[Anshelevich, 2010] (Free Meixner)
[E, Dubbs, 2014]
Jacobi
That’s pretty special!
Corresponds to
2nd order differences with boundary
Wachter Law
1980
x=parameter
y=parameter
Free Binomial
Gengenbauer
Mackay Law
1981
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)=
k x kexample
example
5x5
“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
Narayana
Photo
Unavailable
Wigner and Narayana
[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
Random Matrix Idea
Jacobi Operator Idea
Key Point
Moment Matching
Algorithm
1. Probability
Densities as
Jacobi Operators
• Key Limit Density
Laws
• Other Limit Density
Laws
• Toeplitz + Boundary
• Asymptotically
Toeplitz
2. Multivariate
Orthogonal
Polynomials
• Multivariate weights
for β-Ensembles
• Generalization of
triangular and
tridiagonal
structure
Young Diagrams
• Genus Expansion
• q-Hermite Jacobi
operator
• Application of
Algorithm in 1
Explicit Generalized
Harer-Zagier
Formula
3. Natural q-GUE
integrals
(q-theory)
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 Pn 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 Pn 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
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),…
35/55
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),…
36/55
Multivariate Hermite Polynomials
(β=1)
[Chikuse, 1992]
X … matrix
Polynomial evaluated at eigenvalues of37/55
X
(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 Pn 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
β=2
General β
Hermite, Laguerre
YOUNG
(Baker,Forrester)
YOUNG
(Baker,Forrester)
Jacobi
????
????
General Weight
Functions
YOUNG
(Venkataramana, E)
Probably NOT YOUNG
?????
(Venkataramana, E)
42/55
The second question
• What Is the sparsity pattern of the analog of
=
=
?
43/55
children in the Young
Diagram
44/55
At a Glance
Random Matrix Idea
Jacobi Operator Idea
Key Point
Moment Matching
Algorithm
1. Probability
Densities as
Jacobi Operators
• Key Limit Density
Laws
• Other Limit Density
Laws
• Toeplitz + Boundary
• Asymptotically
Toeplitz
2. Multivariate
Orthogonal
Polynomials
• Multivariate weights
for β-Ensembles
• Generalization of
triangular and
tridiagonal
structure
Young Diagrams
• Genus Expansion
• q-Hermite Jacobi
operator
• Application of
Algorithm in 1
Explicit Generalized
Harer-Zagier
Formula
3. Natural q-GUE
integrals
(q-theory)
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: MurnaghanNakayama 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
```