Combinatorial Models for Random Matrices with
Gaussian Entries
Michael La Croix
Massachusetts Institute of Technology
January 27, 2015
1
Random Matrices and Maps
What are Random Matrices?
Moments of random matrices count maps
What are Maps?
2
An Algebraic Perspective
Encoding Non-oriented Maps
Encoding Oriented Maps
Generalizing to Hypermaps
Generating Series
3
Combining Continuous and Discrete Perspectives
A Integration Formula for zonal polynomials
4
Generalizing β
Jack symmetric functions
A Recurrence
Michael La Croix
Combinatorial Models for Random Matrices
January 27, 2015
1 / 34
Collaborators
David M. Jackson
Alan Edelman
University of Waterloo
Waterloo, ON, Canada
MIT
Cambridge, MA, U.S.A.
Michael La Croix
Combinatorial Models for Random Matrices
January 27, 2015
1 / 34
Outline
1
Random Matrices and Maps
What are Random Matrices?
Moments of random matrices count maps
What are Maps?
2
An Algebraic Perspective
Encoding Non-oriented Maps
Encoding Oriented Maps
Generalizing to Hypermaps
Generating Series
3
Combining Continuous and Discrete Perspectives
A Integration Formula for zonal polynomials
4
Generalizing β
Jack symmetric functions
A Recurrence
Michael La Croix
Combinatorial Models for Random Matrices
January 27, 2015
2 / 34
Random Matrices
A Random matrix is a matrix with random elements.

Often:
A11 A12
questions involve eigenvalues,
 A21 A22

A= .
..
entries are mostly independent,
 ..
.
entries are real or complex.
A
A
n1
n2

. . . A1m
. . . A2m 

.. 
..
.
. 
. . . Anm
Some Combinatorial Random Matrices
Ginibre
A
Hermite (Gaussian)
H = 21 (A + A∗ )
Laguerre (Wishart)
W = AA∗
Less Combinatorial Random Matrices
Jacobi / MANOVA
Michael La Croix
Unitary
Orthogonal
Combinatorial Models for Random Matrices
Circular
January 27, 2015
2 / 34
Universality
A typical theorem about random matrices:
1
Shows dependence on entry distribution is weak.
2
Establishes the result when entries are i.i.d. Gaussian.
We’ll focus on step 2.
Example (Wigner’s Semicircle Law)
2k
E(tr(H )) = n
Michael La Croix
k−1
1
2k
+ O(nk−2 )
k+1 k
Combinatorial Models for Random Matrices
January 27, 2015
3 / 34
Gaussian Random Variables Have Combinatorial Moments
Real Gaussians, X, with mean 0 and variance 1
Z
2
1
m!! if n = 2m
n
n − x2
E X =√
x e
dx =
0
if n is odd
2π R
Complex Gaussians, Z, with mean 0 and variance 1
Z
1
k! if k = l
k l
k l −|z|2
z z e
dz =
E Z Z =
0 if k 6= l
π C
Michael La Croix
Combinatorial Models for Random Matrices
January 27, 2015
4 / 34
Gaussian Ensembles
For β ∈ {1, 2, 4} an element of the β-Gaussian ensemble is constructed as
A = G + G∗
where G is n × n with i.i.d. Guassian entries selected from {R, C, H}.
Motivating Question
What is the value of E(f (A)), when f is a symmetric function of the
eigenvalues of its argument?
Example

 510n + 720n2 + 360n3 + 90n4 β = 1
5
3
60n2 + 45n4
β=2
E(tr(A ) tr(A )) =
 255
45 4
2
3
− 16 n + 45n − 45n + 2 n
β=4
Michael La Croix
Combinatorial Models for Random Matrices
January 27, 2015
5 / 34
Gaussian Ensembles
For β ∈ {1, 2, 4} an element of the β-Gaussian ensemble is constructed as
A = G + G∗
where G is n × n with i.i.d. Guassian entries selected from {R, C, H}.
Motivating Question
What is the value of E(f (A)), when f is a symmetric function of the
eigenvalues of its argument?
Example

 5n + 5n2 + 2n3
4
n + 2n3
E(tr(A )) =
 5
5 2
3
4 n − 2 n + 2n
Michael La Croix
Combinatorial Models for Random Matrices
β=1
β=2
β=4
January 27, 2015
5 / 34
Gaussian Ensembles
For β ∈ {1, 2, 4} an element of the β-Gaussian ensemble is constructed as
A = G + G∗
where G is n × n with i.i.d. Guassian entries selected from {R, C, H}.
Motivating Question
What is the value of E(f (A)), when f is a symmetric function of the
eigenvalues of its argument?
Example

 5n + 5n2 + 2n3
4
n + 2n3
E(tr(A )) =

(1 + b + 3b2 )n + 5bn2 + 2n3
Michael La Croix
Combinatorial Models for Random Matrices
β=1
β=2
2
β = 1+b
January 27, 2015
5 / 34
3
2
n
n
2
3
n
2
n
2
n
n
n
n
n
n
n
n
2
Michael La Croix
Combinatorial Models for Random Matrices
January 27, 2015
6 / 34
What’s being counted?
Example (for β ∈ {1, 2, 4})
i
tr(A4 ) =
X
Aij Ajk Akl Ali
A ij
A li
j
A jk
i,j,k,l
l
A kl
k
E tr(A4 ) = (n)1 E(A11 A11 A11 A11 ) + (n)2 E(2A11 A12 A22 A21 )
+ (n)2 E(4A11 A11 A12 A21 ) + (n)4 E(A12 A23 A34 A41 )
+ (n)2 E(A12 A21 A12 A21 ) + (n)3 E(4A11 A12 A23 A31 )
+ (n)3 E(2A12 A21 A13 A31 )
Michael La Croix
Combinatorial Models for Random Matrices
January 27, 2015
7 / 34
What’s being counted?
Example (for β ∈ {1, 2, 4})
i
tr(A4 ) =
X
Aij Ajk Akl Ali
A ij
A li
1
1
1
1
j
A jk
i,j,k,l
l
A kl
k
E tr(A4 ) = (n)1 E(A11 A11 A11 A11 ) + (n)2 E(2A11 A12 A22 A21 )
+ (n)2 E(4A11 A11 A12 A21 ) + (n)4 E(A12 A23 A34 A41 )
+ (n)2 E(A12 A21 A12 A21 ) + (n)3 E(4A11 A12 A23 A31 )
+ (n)3 E(2A12 A21 A13 A31 )
Michael La Croix
Combinatorial Models for Random Matrices
January 27, 2015
7 / 34
What’s being counted?
Example (for β ∈ {1, 2, 4})
i
tr(A4 ) =
X
Aij Ajk Akl Ali
A ij
A li
j
A jk
i,j,k,l
l
A kl
k
E tr(A4 ) = (n)1 E(A11 A11 A11 A11 ) + (n)2 E(2A11 A12 A22 A21 )
1
1
1
2
1
1
1
1
1
2
1
1
1
1
1
1
2
1
2
1
+ (n)2 E(4A11 A11 A12 A21 ) + (n)4 E(A12 A23 A34 A41 )
+ (n)2 E(A12 A21 A12 A21 ) + (n)3 E(4A11 A12 A23 A31 )
+ (n)3 E(2A12 A21 A13 A31 )
Michael La Croix
Combinatorial Models for Random Matrices
January 27, 2015
7 / 34
What’s being counted?
Example (for β ∈ {1, 2, 4})
i
tr(A4 ) =
X
Aij Ajk Akl Ali
A ij
A li
j
A jk
i,j,k,l
l
A kl
k
E tr(A4 ) = (n)1 E(A11 A11 A11 A11 ) + (n)2 E(2A11 A12 A22 A21 )
+ (n)2 E(4A11 A11 A12 A21 ) + (n)4 E(A12 A23 A34 A41 )
1
1
1
2
1
1
1
1
1
2
1
1
1
1
1
1
2
1
2
1
1
2
2
1
+ (n)2 E(A12 A21 A12 A21 ) + (n)3 E(4A11 A12 A23 A31 )
+ (n)3 E(2A12 A21 A13 A31 )
Michael La Croix
Combinatorial Models for Random Matrices
January 27, 2015
7 / 34
What’s being counted?
Example (for β ∈ {1, 2, 4})
i
tr(A4 ) =
X
Aij Ajk Akl Ali
A ij
A li
j
A jk
i,j,k,l
l
A kl
k
E tr(A4 ) = (n)1 E(A11 A11 A11 A11 ) + (n)2 E(2A11 A12 A22 A21 )
+ (n)2 E(4A11 A11 A12 A21 ) + (n)4 E(A12 A23 A34 A41 )
+ (n)2 E(A12 A21 A12 A21 ) + (n)3 E(4A11 A12 A23 A31 )
1
1
1
2
1
1
1
1
1
2
1
1
1
1
1
1
2
1
2
1
1
2
2
2
1
1
1
1
3
2
3
1
+ (n)3 E(2A12 A21 A13 A31 )
Michael La Croix
Combinatorial Models for Random Matrices
January 27, 2015
7 / 34
What’s being counted?
Example (for β ∈ {1, 2, 4})
i
tr(A4 ) =
X
Aij Ajk Akl Ali
A ij
A li
j
A jk
i,j,k,l
l
A kl
k
E tr(A4 ) = (n)1 E(A11 A11 A11 A11 )
+ (n)2 E(4A11 A11 A12 A21 )
+ (n)2 E(A12 A21 A12 A21 )
1
1
1
2
1
1
1
1
1
2
1
1
1
1
1
1
2
1
2
1
1
2
2
2
1
1
1
1
3
2
3
1
+ (n)3 E(2A12 A21 A13 A31 )
Michael La Croix
Combinatorial Models for Random Matrices
January 27, 2015
7 / 34
Expectations as Sums
Since the entries of A are centred Gaussians, Wick’s lemma applies
X Y
E(Ai1 j1 Ai2 j2 . . . Aik jk ) =
E(Au Av )
m (u,v)∈m
summed over perfect matchings of the multiset {i1 j1 , i2 j2 , . . . , ik jk }
Example
E(Au Av Aw Ax ) = E(Au Av )E(Aw Ax )
+ E(Au Aw )E(Av Ax )
+ E(Au Ax )E(Av Aw )
Michael La Croix
Combinatorial Models for Random Matrices
January 27, 2015
8 / 34
Expectations as Sums
Since the entries of A are centred Gaussians, Wick’s lemma applies
X Y
E(Ai1 j1 Ai2 j2 . . . Aik jk ) =
E(Au Av )
m (u,v)∈m
summed over perfect matchings of the multiset {i1 j1 , i2 j2 , . . . , ik jk }
E(A
A
A
A
E(A
A
) E(A
A
)
E(A
A
) E(A
A
)
E(A
A
) E(A
A
)
)
Example
E(Au Av Aw Ax ) = E(Au Av )E(Aw Ax )
+ E(Au Aw )E(Av Ax )
+ E(Au Ax )E(Av Aw )
Michael La Croix
Combinatorial Models for Random Matrices
January 27, 2015
8 / 34
Expectations as Sums
Since the entries of A are centred Gaussians, Wick’s lemma applies
X Y
E(Ai1 j1 Ai2 j2 . . . Aik jk ) =
E(Au Av )
m (u,v)∈m
summed over perfect matchings of the multiset {i1 j1 , i2 j2 , . . . , ik jk }
X
#{pairings consistent with p}
p a painting
=
X
#{paintings consistent with m}
m a matching
Michael La Croix
Combinatorial Models for Random Matrices
January 27, 2015
8 / 34
Count the polygon glueings in 2 different ways
(n)1
(n)1
(n)1
(n)1
(n)1
(n)1
(n)1
(n)1
(n)1
(n)1
(n)1
(n)1
0
0
(n)2
0
0
0
0
0
(n)2
0
0
0
2 n(n-1)
0
(n)2
0
0
0
0
0
(n)2
0
0
0
0
2 n(n-1)
0
0
(n)2
0
0
(n)2
0
0
0
0
0
0
2 n(n-1)
0
(n)2
0
0
(n)2
0
0
0
0
0
0
0
2 n(n-1)
0
(n)2
(n)2
0
0
0
0
0
0
(n)2
0
0
3 n(n-1)
0
0
(n)3
0
0
0
0
0
0
0
0
0
n(n-1)(n-2)
0
(n)3
0
0
0
0
0
0
0
0
0
0
n(n-1)(n-2)
n3
n3
n
n2
n2
n
n2
n2
n2
n
n
n
Michael La Croix
Combinatorial Models for Random Matrices
12 n
January 27, 2015
9 / 34
Polygon Glueings = Maps
Identifying the edges of a polygon creates a surface.
=
Its boundary is a graph embedded in the surface.
Michael La Croix
Combinatorial Models for Random Matrices
January 27, 2015
10 / 34
Polygon Glueings = Maps
=
=
Michael La Croix
Combinatorial Models for Random Matrices
January 27, 2015
10 / 34
Polygon Glueings = Maps
Extra polygons give extra faces (and possibles extra components)
This glueing contributes n11 to E p3,43 ,5,10 (A)
Michael La Croix
Combinatorial Models for Random Matrices
January 27, 2015
10 / 34
Equivalence of Maps
Two maps are equivalent if the embeddings are homeomorphic.
Michael La Croix
Combinatorial Models for Random Matrices
January 27, 2015
11 / 34
Vertices and Face
A map can be recovered from a neighbourhood of the
graph, or from its faces and surgery instructions.
Vertex and face degrees are interchanged by duality.
Michael La Croix
Combinatorial Models for Random Matrices
January 27, 2015
12 / 34
Duality
Michael La Croix
Combinatorial Models for Random Matrices
January 27, 2015
13 / 34
Outline
1
Random Matrices and Maps
What are Random Matrices?
Moments of random matrices count maps
What are Maps?
2
An Algebraic Perspective
Encoding Non-oriented Maps
Encoding Oriented Maps
Generalizing to Hypermaps
Generating Series
3
Combining Continuous and Discrete Perspectives
A Integration Formula for zonal polynomials
4
Generalizing β
Jack symmetric functions
A Recurrence
Michael La Croix
Combinatorial Models for Random Matrices
January 27, 2015
14 / 34
Three Involutions
Three natural involutions reroot a map.
Across Edge Around Vertex Along Edge
Michael La Croix
Combinatorial Models for Random Matrices
January 27, 2015
14 / 34
The Matchings Encode the Map
Each involution gives a perfect matching of flags.
Pairs of matchings recover vertices, edges, and faces.
Michael La Croix
Combinatorial Models for Random Matrices
January 27, 2015
15 / 34
Encodinge Non-oriented Maps
Mv
6’
5
5’
6
7’
2
7 8’
2’
3’
8
Me
1
3
1’
4’
4
Mf
Mv = {1, 3}, {1 , 3 }, {2, 5}, {2 , 5 }, {4, 8 }, {4 , 8}, {6, 7}, {60 , 70 }
Me = {1, 20 }, {10 , 4}, {2, 30 }, {3, 40 }, {5, 60 }, {50 , 8}, {6, 70 }, {7, 80 }
Mf = {1, 10 }, {2, 20 }, {3, 30 }, {4, 40 }, {5, 50 }, {6, 60 }, {7, 70 }, {8, 80 }
Michael La Croix
0
0
0
0
0
Combinatorial Models for Random Matrices
0
January 27, 2015
16 / 34
Encoding Oriented Maps (β = 2)
1
Orient and label the edges.
2
This induces labels on flags.
3
Clockwise circulations at each
vertex determine ν.
4
1
2
4
3
Face circulations are the cycles
of ν.
5
6
= (1 10 )(2 20 )(3 30 )(4 40 )(5 50 )(6 60 )
ν = (1 2 3)(10 4)(20 5)(30 50 6)(40 60 )
ν = φ = (1 4 60 30 )(10 2 5 6 40 )(20 3 50 )
Michael La Croix
Combinatorial Models for Random Matrices
January 27, 2015
17 / 34
Encoding Oriented Maps (β = 2)
1
Orient and label the edges.
2
This induces labels on flags.
3
Clockwise circulations at each
vertex determine ν.
4
Face circulations are the cycles
of ν.
1
4
2
3
5
6
= (1 10 )(2 20 )(3 30 )(4 40 )(5 50 )(6 60 )
ν = (1 2 3)(10 4)(20 5)(30 50 6)(40 60 )
ν = φ = (1 4 60 30 )(10 2 5 6 40 )(20 3 50 )
Michael La Croix
Combinatorial Models for Random Matrices
January 27, 2015
17 / 34
Encoding Oriented Maps (β = 2)
1’
1
Orient and label the edges.
2
This induces labels on flags.
3
Clockwise circulations at each
vertex determine ν.
4
Face circulations are the cycles
of ν.
4
2
1
2’
3
4’
6’
5
5’
3’
6
= (1 10 )(2 20 )(3 30 )(4 40 )(5 50 )(6 60 )
ν = (1 2 3)(10 4)(20 5)(30 50 6)(40 60 )
ν = φ = (1 4 60 30 )(10 2 5 6 40 )(20 3 50 )
Michael La Croix
Combinatorial Models for Random Matrices
January 27, 2015
17 / 34
Encoding Oriented Maps (β = 2)
1’
1
Orient and label the edges.
2
This induces labels on flags.
3
Clockwise circulations at each
vertex determine ν.
4
Face circulations are the cycles
of ν.
4
2
1
2’
3
4’
6’
5
5’
3’
6
= (1 10 )(2 20 )(3 30 )(4 40 )(5 50 )(6 60 )
ν = (1 2 3)(10 4)(20 5)(30 50 6)(40 60 )
ν = φ = (1 4 60 30 )(10 2 5 6 40 )(20 3 50 )
Michael La Croix
Combinatorial Models for Random Matrices
January 27, 2015
17 / 34
Hypermaps
An arbitrary triple of perfect matchings determines a hypermap with
cycles of Me ∪ Mf , Me ∪ Mv , and Mv ∪ Mf determining vertices, faces,
and hyperedges. Example
Hypermaps both specialize and generalize maps.
Example
,→
A hypermap can be represented as a bipartite map.
Michael La Croix
Combinatorial Models for Random Matrices
January 27, 2015
18 / 34
Hypermaps
An arbitrary triple of perfect matchings determines a hypermap with
cycles of Me ∪ Mf , Me ∪ Mv , and Mv ∪ Mf determining vertices, faces,
and hyperedges. Example
Hypermaps both specialize and generalize maps.
Example
,→
Subdivide edges to get a hypermap from a map.
Michael La Croix
Combinatorial Models for Random Matrices
January 27, 2015
18 / 34
The Hypermap Series
Definition
The hypermap series for a set H of hypermaps is the combinatorial sum
X
H(x, y, z) :=
xν(h) yφ(h) z(h)
h∈H
ν(h), φ(h), and (h) are vertex-, hyperface-, and hyperedge- degrees.
Example
ν = [23 , 32 ]
contributes 12 x32 x23 (y3 y4 y5 ) z26 .
Michael La Croix
φ = [3, 4, 5]
Combinatorial Models for Random Matrices
= [26 ]
January 27, 2015
19 / 34
Generating Series Algebraically
Instead of counting rooted maps, we can count labelled hypermaps.
This adds easily computable multiplicities.
These in turn are reduced to computing a multiplication table for
appropriate algebras.
C[S] for oriented hypermaps
A double-coset algebra for non-oriented hypermaps.
These can be evaluated via character theory.
Appropriate characters appear as coefficients of symmetric functions.
Standard enumerative techniques restrict the solution to connected
maps and remove factors introduced by the labelling.
Michael La Croix
Combinatorial Models for Random Matrices
January 27, 2015
20 / 34
Explicit Generating Series
Some hypermap series can be computed explicitly.
Theorem (Jackson and Visentin - 1990)
Matrix Derivation
When H is the set of connected oriented hypermaps,
!
X
∂
HO p(x), p(y), p(z); 0 = t ln
t|θ| Hθ sθ (x)sθ (y)sθ (z) ∂t
θ∈P
Theorem (Goulden and Jackson - 1996)
t=1.
Matrix Derivation
When H is the set of connected non-oriented hypermaps,
!
X
∂
|θ| 1
HA p(x), p(y), p(z); 1 = 2t ln
t
Zθ (x)Zθ (y)Zθ (z) ∂t
H2θ
θ∈P
Michael La Croix
Combinatorial Models for Random Matrices
t=1.
January 27, 2015
21 / 34
Outline
1
Random Matrices and Maps
What are Random Matrices?
Moments of random matrices count maps
What are Maps?
2
An Algebraic Perspective
Encoding Non-oriented Maps
Encoding Oriented Maps
Generalizing to Hypermaps
Generating Series
3
Combining Continuous and Discrete Perspectives
A Integration Formula for zonal polynomials
4
Generalizing β
Jack symmetric functions
A Recurrence
Michael La Croix
Combinatorial Models for Random Matrices
January 27, 2015
22 / 34
Why do Schur functions and zonal polynomials appear?
This depends on your perspective.
Continuous
Edge labelled maps can be encoded as permutations.
Schur functions come from characters of the symmetric group.
Discrete
maps can be counted by matrix integrals.
Schur functions are characters of Lie groups.
Face-Painted
Michael La Croix
Combinatorial Models for Random Matrices
January 27, 2015
22 / 34
Two Perspectives at Once
A Duck is a Duck is a Duck
Rabbit
Duck
Michael La Croix
Combinatorial Models for Random Matrices
January 27, 2015
23 / 34
An Integration Formula for the GOE
We can also build the generating series from matrix moments.
HA p(x), p(y), z|zi =z δi,2 ; 0
Z P
√ k
1
∂
pk (XM )pk (Y ) z tk − 41 tr(M 2 )
k
2k
= 2t ln e
e
dM ∂t
t=1 Z X
∂
Zθ (XM )Zθ (Y ) |θ|/2 |θ| − 1 tr(M 2 )
= 2t ln
z
t e 4
dM ∂t
hZθ , Zθ i2
θ∈P
t=1
Corollary (Goulden and Jackson - 1997)
When A is taken from GOEn ,
E(Zθ )(XA) = Zθ (X) [p2|θ|/2 ]Zθ
Michael La Croix
Combinatorial Models for Random Matrices
January 27, 2015
24 / 34
Example Using Zonal Polynomials
Z[14 ] =
1p[14 ]
−6p[2,12 ]
+3p[2,2]
+8p[3,1]
− 6p[4]
Z[2,12 ] =
1p[14 ]
−p[2,12 ]
− 2p[2,2]
−2p[3,1]
+ 4p[4]
Z[22 ] =
1p[14 ]
+2p[2,12 ]
+7p[2,2]
−8p[3,1]
− 2p[4]
Z[3,1] =
1p[14 ]
+ 5p[2,12 ]
− 2p[2,2]
+ 4p[3,1]
− 8p[4]
Z[4] =
1p[14 ]
+12p[2,12 ]
+12p[2,2]
+32p[3,1]
+48p[4]
[14 ]
θ
4
[2, 12 ]
3
[22 ]
2
2
[3, 1]
2
[4]
hpθ , pθ i2
4! · 2 = 384
2! · 2 · 2 = 32
2! · 2 · 2 = 32
3 · 2 = 12
4·2=8
hZθ , Zθ i2
2880
720
2880
2016
40320
Example
Z[4] , Z[4] = 12 · 384 + 122 · 32 + 122 · 32 + 322 · 12 + 482 · 8 = 40320
Michael La Croix
Combinatorial Models for Random Matrices
January 27, 2015
25 / 34
Example Using Zonal Polynomials
Z[14 ] =
1p[14 ]
−6p[2,12 ]
+3p[2,2]
+8p[3,1]
− 6p[4]
Z[2,12 ] =
1p[14 ]
−p[2,12 ]
− 2p[2,2]
−2p[3,1]
+ 4p[4]
Z[22 ] =
1p[14 ]
+2p[2,12 ]
+7p[2,2]
−8p[3,1]
− 2p[4]
Z[3,1] =
1p[14 ]
+ 5p[2,12 ]
− 2p[2,2]
+ 4p[3,1]
− 8p[4]
Z[4] =
1p[14 ]
+12p[2,12 ]
+12p[2,2]
+32p[3,1]
+48p[4]
[14 ]
θ
4
[2, 12 ]
3
[22 ]
2
2
[3, 1]
2
[4]
hpθ , pθ i2
4! · 2 = 384
2! · 2 · 2 = 32
2! · 2 · 2 = 32
3 · 2 = 12
4·2=8
hZθ , Zθ i2
2880
720
2880
2016
40320
Example
p[4] = − 6
8
8
8
8
8
Z 4 +4
Z 2 −2
Z 2 −8
Z[3,1] + 48
Z[4]
2880 [1 ]
720 [2,1 ]
2880 [2 ]
2016
40320
Michael La Croix
Combinatorial Models for Random Matrices
January 27, 2015
25 / 34
Example Using Zonal Polynomials
Z[14 ] =
1p[14 ]
−6p[2,12 ]
+3p[2,2]
+8p[3,1]
− 6p[4]
Z[2,12 ] =
1p[14 ]
−p[2,12 ]
− 2p[2,2]
−2p[3,1]
+ 4p[4]
Z[22 ] =
1p[14 ]
+2p[2,12 ]
+7p[2,2]
−8p[3,1]
− 2p[4]
Z[3,1] =
1p[14 ]
+ 5p[2,12 ]
− 2p[2,2]
+ 4p[3,1]
− 8p[4]
Z[4] =
1p[14 ]
+12p[2,12 ]
+12p[2,2]
+32p[3,1]
+48p[4]
Example
p[4] = − 6
8
8
8
8
8
Z 4 +4
Z 2 −2
Z 2 −8
Z[3,1] + 48
Z[4]
2880 [1 ]
720 [2,1 ]
2880 [2 ]
2016
40320
8
8
(3)(1y14 − 6y2 y12 + 3y22 + 8y3 y1 − 6y4 ) + 4 720
(−2)(1y14 − 1y2 y12 − 2y22 − 2y3 y1 + 4y4 )
E(p[4] (Y A)) = − 6 2880
8
8
(7)(1y14 + 2y2 y12 + 7y22 − 8y3 y1 − 2y4 ) − 8 2016
(−2)(1y14 + 5y2 y12 − 2y22 + 4y3 y1 − 8y4 )
− 2 2880
8
+ 48 40320
(12)(1y14 + 12y2 y12 + 12y22 + 32y3 y1 + 48y4 )
=2y2 y12 + y22 + 4y3 y1 + 5y4
Michael La Croix
Combinatorial Models for Random Matrices
January 27, 2015
25 / 34
3
n
3
n
n
[2,1,1] n2
[3,1] n2
[3,1] n
[ 4]
[1,2,1] n2
[1,3] n2
[3,1] n
[ 4]
[4] n
Michael La Croix
[4] n
Combinatorial Models for Random Matrices
[4] n2
January 27, 2015
[2,2]
26 / 34
Outline
1
Random Matrices and Maps
What are Random Matrices?
Moments of random matrices count maps
What are Maps?
2
An Algebraic Perspective
Encoding Non-oriented Maps
Encoding Oriented Maps
Generalizing to Hypermaps
Generating Series
3
Combining Continuous and Discrete Perspectives
A Integration Formula for zonal polynomials
4
Generalizing β
Jack symmetric functions
A Recurrence
Michael La Croix
Combinatorial Models for Random Matrices
January 27, 2015
27 / 34
A Generalized Hypermap Series
A common generalization involves Jack symmetric functions,
Definition
.
b -Conjecture (Goulden and Jackson - 1996)
H p(x), p(y), p(z); b
(1+b)
(1+b)
(1+b)
X
J
(x)Jθ
(y)Jθ
(z)
∂
:= (1 + b)t ln
t|θ| θ
∂t
hJθ , Jθ i1+b [p1|θ| ]Jθ
θ∈P
X X
=
cν,φ, (b)pν (x)pφ (y)p (z),
!
t=1
n≥0 ν,φ,`n
enumerates rooted hypermaps with cν,φ, (b) =
X
b β(h) for some β.
h∈Hν,φ,
Michael La Croix
Combinatorial Models for Random Matrices
January 27, 2015
27 / 34
General Gaussian Ensembles
In terms of Jack functions, we get:
Corollary (Goulden and Jackson - 1997)
When A is taken from GUEn ,
(1)
(1)
(1)
E(Jθ (XA)) = Jθ (X) [p2|θ|/2 ]Jθ
Corollary (Implicit in Jackson - 1995)
When A is taken from GOEn ,
(2)
(2)
(2)
E(Jθ (XA)) = Jθ (X) [p2|θ|/2 ]Jθ
X = diag(x1 , x2 , . . . , xn )
Michael La Croix
Combinatorial Models for Random Matrices
January 27, 2015
28 / 34
For general β, integrate over eigenvalues
We can generalize the eigenvalue density.
Definition
For a function f : Rn → R, define an expectation operator h·i by
Z
2
− 1 p (λ)
|V (λ)| 1+b f (λ)e 2(1+b) 2 dλ,
hf i1+b := c1+b
Rn
with c1+b chosen such that h1i1+b = 1.
Theorem (Okounkov - 1997)
If n is a positive integer, 1 + b > 0, and θ ` 2n, then
D
E
(1+b)
(1+b)
(1+b)
(In )[p[2n ] ]Jθ
.
(λ)
= Jθ
Jθ
1+b
Michael La Croix
Combinatorial Models for Random Matrices
January 27, 2015
29 / 34
Expectations of power-sums are polynomials
The eigenvalues of A are all real with joint density proportional to
!
n
2
Y
X
β
λ
i
|λi − λj |β exp −
2
2
1≤i<j≤n
i=1
Theorem
For every θ, E(pθ (λ))β is a polynomial in the variables n and b =
2
β
− 1.
Example
E(p4 (λ))β = (1 + b + 3b2 )n + 5bn2 + 2n3
Michael La Croix
Combinatorial Models for Random Matrices
January 27, 2015
30 / 34
Expectations of power-sums are polynomials
The eigenvalues of A are all real with joint density proportional to
!
n
Y
β X λ2i
β
|λi − λj | exp −
2
2
1≤i<j≤n
i=1
Theorem
For every θ, E(pθ (λ))β is a polynomial in the variables n and b =
2
β
− 1.
Example
E(p5,3 (λ))β = (75b + 150b2 + 180b3 + 105b4 )n
+ (60 + 120b + 300b2 + 240b3 )n2
+ (180b + 180b2 )n3 + (45 + 45b)n4
Michael La Croix
Combinatorial Models for Random Matrices
January 27, 2015
30 / 34
Algebraic and Combinatorial Recurrences agree
An Algebraic Recurrence
Derivation
hpj+2 pθ i = b(j + 1) hpj pθ i + α
X
imi (θ) pi+j pθ\i +
i∈θ
j
X
Example
hpl pj−l pθ i.
l=0
A Combinatorial Recurrence
It corresponds to a combinatorial recurrence for counting polygon glueings.
i
j-l
i+j
j
j+2
Michael La Croix
j+2
Combinatorial Models for Random Matrices
j+2
January 27, 2015
l
31 / 34
Root Edge Deletion
A rooted map with k edges can be thought of as a sequence of k maps.
ab n
2
bn
2 2
bn
n
2
abn
an
n
abn
2
2
2
Consecutive submaps differ in genus by 0, 1, or 2, and these steps are
marked by 1, b, and a to assign a weight to a rooted map.
Michael La Croix
Combinatorial Models for Random Matrices
January 27, 2015
32 / 34
3
[2,1,1] bn2
[3,1] bn2
[3,1] b2n
[ 4]
n
3
[1,2,1] bn2
[1,3] bn2
[3,1] b2n
[ 4]
n
[4] b2n
[4] bn2
[2,2]
n
Michael La Croix
[4] bn
Combinatorial Models for Random Matrices
January 27, 2015
33 / 34
Other Ensembles
We also get combinatorial interpretations for the moments of:
The Laguerre Unitary Ensemble (LUE)
The Laguerre Orthogonal Ensemble (LOE)
The Complex Ginibre Ensemble.
Complex Symmetric Matrices.
Michael La Croix
Combinatorial Models for Random Matrices
January 27, 2015
34 / 34
The End
Thank You
Michael La Croix
Combinatorial Models for Random Matrices
January 27, 2015
34 / 34
Outline
5
Appendix
6
Need Placement
Michael La Croix
Combinatorial Models for Random Matrices
January 27, 2015
35 / 34
Finding a partial differential equation
Root-edge type
Schematic
Contribution to M
Cross-border
z
X
(i + 1)bri+2
i≥0
Border
z
i+1
XX
rj yi−j+2
i≥0 j=1
Handle
z
X
(1 + b)jri+j+2
i,j≥0
Bridge
z
X
ri+j+2
i,j≥0
Michael La Croix
Combinatorial Models for Random Matrices
∂
M
∂ri
∂
M
∂ri
∂
M
∂ri
∂2
M
∂ri ∂yj
∂
M
∂rj
January 27, 2015
35 / 34
Finding a partial differential equation
Root-edge type
Schematic
Contribution to M
Cross-border
z
X
(i + 1)bri+2
i≥0
Border
z
i+1
XX
rj yi−j+2
i≥0 j=1
Handle
z
X
(1 + b)jri+j+2
i,j≥0
Bridge
z
X
ri+j+2
i,j≥0
Michael La Croix
Combinatorial Models for Random Matrices
∂
M
∂ri
∂
M
∂ri
∂
M
∂ri
∂2
M
∂ri ∂yj
∂
M
∂rj
January 27, 2015
35 / 34
Finding a partial differential equation
Root-edge type
Schematic
Contribution to M
Cross-border
z
X
(i + 1)bri+2
i≥0
Border
z
i+1
XX
rj yi−j+2
i≥0 j=1
Handle
z
X
(1 + b)jri+j+2
i,j≥0
Bridge
z
X
ri+j+2
i,j≥0
Michael La Croix
Combinatorial Models for Random Matrices
∂
M
∂ri
∂
M
∂ri
∂
M
∂ri
∂2
M
∂ri ∂yj
∂
M
∂rj
January 27, 2015
35 / 34
Finding a partial differential equation
Root-edge type
Schematic
Contribution to M
Cross-border
z
X
(i + 1)bri+2
i≥0
Border
z
i+1
XX
rj yi−j+2
i≥0 j=1
Handle
z
X
(1 + b)jri+j+2
i,j≥0
Bridge
z
X
ri+j+2
i,j≥0
Michael La Croix
Combinatorial Models for Random Matrices
∂
M
∂ri
∂
M
∂ri
∂
M
∂ri
∂2
M
∂ri ∂yj
∂
M
∂rj
January 27, 2015
35 / 34
Finding a partial differential equation
Root-edge type
Schematic
Contribution to M
Cross-border
z
X
(i + 1)bri+2
i≥0
Border
z
i+1
XX
rj yi−j+2
i≥0 j=1
Handle
z
X
(1 + b)jri+j+2
i,j≥0
Bridge
z
X
ri+j+2
i,j≥0
Michael La Croix
Combinatorial Models for Random Matrices
∂
M
∂ri
∂
M
∂ri
∂
M
∂ri
∂2
M
∂ri ∂yj
∂
M
∂rj
January 27, 2015
35 / 34
Painted Faces
Return
There are N #f ways to
paint an f faced map.
Repaint N = 1
Repaint N = 2
Repaint N = 3
Repaint N = 4
Show Neighbourhoods
Show Map
Michael La Croix
Combinatorial Models for Random Matrices
January 27, 2015
36 / 34
A Recurrence behind the theorem
1
− 2(1+b)
p2 (x)
Set Ω := e
Integrate
2
|V (x)| 1+b , so that hf i = E(f (x)) = cb,n
R
Rn
f Ω dx.
p (x)
2
∂ j+1
∂ j+1
− 2
x1 pθ (x)Ω =
x1 pθ (x)|V (x)| 1+b e 2(1+b)
∂x1
∂x1
= (j + 1)xj1 pθ (x)Ω +
X
imi (θ)xi+j
1 pθ\i (x)Ω +
2
1+b
N j+1
X
x
p
θ (x)
x1 −xi Ω
1
−
j+2
1
1+b x1 pθ (x)Ω
i=2
i∈θ
to get
An Algebraic recurrence
hpj+2 pθ i = b(j + 1) hpj pθ i + α
Return
X
j
X
imi (θ) pi+j pθ\i +
hpl pj−l pθ i.
i∈θ
Michael La Croix
Combinatorial Models for Random Matrices
l=0
January 27, 2015
37 / 34
Recurrence Example
hpj+2 pθ i = b(j + 1) hpj pθ i + (1 + b)
X
j
X
imi (θ) pi+j pθ\i +
hpl pj−l pθ i
i∈θ
l=0
Example
h1i = 1
hp0 i = n
hp2 i = b hp0 i + hp0 p0 i = bn + n2
hp1 p1 i = (1 + b) hp0 i = (1 + b)n
hp4 i = 3b hp2 i + hp0 p2 i + hp1 p1 i + hp2 p0 i = (1 + b + 3b2 )n + 5bn2 + 2n3
hp3 p1 i = 2b hp1 p1 i + (1 + b) hp2 i + hp0 p1 p1 i + hp1 p0 p1 i = (3b + 3b2 )n + (3 + 3b)n2
hp2 p2 i = b hp0 p2 i + 2(1 + b) hp2 i + hp0 p0 p2 i = 2b(1 + b)n + (2 + 2b + b2 )n2 + 2bn3 + n4
hp2 p1,1 i = b hp0 p1,1 i + 2(1 + b) hp1,1 i + hp0 p0 p1,1 i = 2(1 + b)2 n + (b + b2 )n2 + (1 + b)n3
hp1 p3 i = 3(1 + b) hp2 i = (3b + 3b2 )n + (3 + 3b)n2
hp1 p2,1 i = 2(1 + b) hp1,1 i + (1 + b) hp0 p2 i = (2 + 4b + 2b2 )n + (b + b2 )n2 + (1 + b)n3
hp1,1,1,1 i = 3(1 + b) hp0 p1,1 i = (1 + 2b + b2 )n2
Return
Michael La Croix
Combinatorial Models for Random Matrices
January 27, 2015
38 / 34
Jack Symmetric Functions
With respect to the inner product defined by
hpλ (x), pµ (x)iα = δλ,µ
|λ|! `(λ)
α ,
|Cλ |
Jack symmetric functions are the unique family satisfying:
D
E
(α)
(α)
(P1) (Orthogonality) If λ 6= µ, then Jλ , Jµ
= 0.
α
(α)
(P2) (Triangularity) Jλ
=
X
vλµ (α)mµ , where vλµ (α) is a rational
µ4λ
function in α, and ‘4’ denotes the natural order on partitions.
(P3) (Normalization) If |λ| = n, then vλ,[1n ] (α) = n!.
Return
Michael La Croix
Combinatorial Models for Random Matrices
January 27, 2015
39 / 34
Jack Symmetric Functions
(α)
Jack symmetric functions, are a one-parameter family, denoted by {Jθ }θ ,
that generalizes both Schur functions and zonal polynomials.
Proposition (Stanley - 1989)
Jack symmetric functions are related to Schur functions and zonal
polynomials by:
D
E
(1)
(1)
(1)
Jλ = Hλ sλ ,
Jλ , Jλ
= Hλ2 ,
1
E
D
(2)
(2)
(2)
and
= H2λ ,
Jλ = Zλ ,
Jλ , Jλ
2
where 2λ is the partition obtained from λ by multiplying each part by two.
Return
Michael La Croix
Combinatorial Models for Random Matrices
January 27, 2015
40 / 34
Example
Mf
ν = [23 ]
= [32 ]
Mv
φ = [6]
Me
Return
Michael La Croix
Combinatorial Models for Random Matrices
January 27, 2015
41 / 34
Oriented Derivation (Matrices over C)
With X = diag (x1 , x2 , . . . , xM ), Y = diag (y1 , y2 , . . . , yN ), Z = diag (z1 , z2 , . . . , zO )
aij
yj
HA p(x), p(y), p(z), t ; 0
Z
xi
∂
∗ ∗ ∗
∗
∗
∗
= t ln e[tr(XAY BZC)+tr(C B A )]t e− tr(AA +BB +CC ) dA dB dC
zk
∂t
Z X
c
ki
∂
sθ (XAY BZC)sφ (C ∗ B ∗ A∗ ) |θ|+|φ| − tr(AA∗ +BB ∗ +CC ∗ )
= t ln
t
e
dA dB dC
([p1|θ| ]sθ )−1 ([p1|φ| ]sφ )−1
∂t
θ,φ∈P
Z X
∂
sθ (X)sθ (Y )sθ (Z)sθ (ABC)sθ (C ∗ B ∗ A∗ ) 2|θ| − tr(AA∗ +BB ∗ +CC ∗ )
= t ln
t e
dA dB dC
sθ (IM )sθ (IN )sθ (IO )([p1|θ| ]sθ )−2
∂t
2
bjk
a*ij
θ∈P
=t
b*jk
X sθ (X)sθ (Y )sθ (Z)
∂
ln
t2|θ|
∂t
[p1|θ| ]sθ
θ∈P
Z
Since
CM ×N
cki*
sθ (XAY A∗ )e− tr(AA
Michael La Croix
∗)
dA =
sθ (X)sθ (Y )
[p1|θ| ]sθ
Combinatorial Models for Random Matrices
Return
January 27, 2015
42 / 34
Non-Oriented Derivation (Matrices over R)
With
√
√ √
√
X = diag ( x1 , x2 , . . . , xM ), Y = diag (y1 , y2 , . . . , yN ), Z = diag (z1 , z2 , . . . , zO )
aij
HA p(x), p(y), p(z), t; 1
 xi
√
Z
√
√
t
∂
T XBZB T ) − 1 tr(AAT +BB T )
tr(
XAY
A
= 2t ln e 2
e 2
dA dB
bil
∂t
√
√
Z X
T
T
∂
Zθ ( XAY A XBZB ) |θ| − 1 tr(AAT +BB T )
= 2t ln
t e 2
dA dB
∂t
hZθ , Zθ i2
θ∈P
Z X
∂
Zθ (Y )Zθ (XBZB T ) |θ| − 1 tr(BB T )
= 2t ln
dB
t e 2
∂t
hZθ , Zθ i2
yj
zl
akj
 xk
√
bkl
θ∈P
X Zθ (X)Zθ (Y )Zθ (Z)
∂
= 2t ln
t|θ|
∂t
hZθ , Zθ i2
θ∈P
Z
Since
RM ×N
1
Zθ (XAY AT )e− 2 tr(AA
Michael La Croix
T)
dA = Zθ (X)Zθ (Y )
Combinatorial Models for Random Matrices
January 27, 2015
Return
43 / 34
Outline
5
Appendix
6
Need Placement
Michael La Croix
Combinatorial Models for Random Matrices
January 27, 2015
44 / 34
Polynomiality and half-integers - combinatorial vs bidiagonal vs anti-GUE
Michael La Croix
Combinatorial Models for Random Matrices
January 27, 2015
44 / 34
How do we deal with additional β
Both our combinatorial and analytic descriptions, when specialized, satisfy
the same partial differential equation
Michael La Croix
Combinatorial Models for Random Matrices
January 27, 2015
45 / 34
An S3 Action on Hypermaps
Every permutation of the matchings gives a hypermap.
Michael La Croix
Combinatorial Models for Random Matrices
January 27, 2015
46 / 34