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.

......

Quantum Probability and

Asymptotic Spectral Analysis of Growing Graphs

Nobuaki Obata

GSIS, Tohoku University

Roma, November 14, 2013

Nobuaki Obata (GSIS, Tohoku University) Quantum Probability and Asymptotic Spectral Analysis of Growing GraphsRoma, November 14, 2013 1 / 47

Plan

...1 Asymptotic Spectral Analysis of Graphs

...2 Quantum Probability

...3 Quantum Probabilistic Approach to Spectral Graph Theory

...4 Graph Products and Concepts of Independence

...5 Method of Quantum Decomposition

...6 Method of Quantum Decomposition for Growing Graphs

presentation: Roma, November 14, 2013


1. Asymptotic Spectral Analysis of Graphs


1.1. Graphs and Adjacency Matrices

.Definition (graph)..

......

A graph is a pair G = (V,E), where V is the set of vertices and E the set of edges.

We write x ∼ y (adjacent) if they are connected by an edge.

complete graph K5 2-dim lattice homogeneous treestar graph T4

.Definition (adjacency matrix)..

......

The adjacency matrix of a graph G = (V,E) is defined by

A = [Axy]x,y∈V Axy =

1, x ∼ y,

0, otherwise.

The adjacency matrix possesses all the information of a graph.


1.2. Spectra of Graphs

.Definition (spectrum and spectral distribution)..

......

Let G be a finite graph. The spectrum (eigenvalues) of G is the list of eigenvalues of the

adjacency matrix A.

Spec (G) =

(· · · λi · · ·· · · mi · · ·

)The spectral (eigenvalue) distribution of G is defined by

µG =1

|V |∑i

miδλi

...1 Spec (G) is a fundamental invariant of finite graphs.

...2 (isospectral problem) Non-isomorphic graphs may have the same spectra.

...3 Adjacency matrix, Laplacian matrix, distance matrix, Q-matrix, ... etc.

For algebraic graph theory or spectral graph theory see

N. Biggs: “Algebraic Graph Theory,” Cambridge UP, 1993.

D. M. Cvetkovic, M. Doob and H. Sachs: “Spectra of Graphs,” Academic Press, 1979.


1.3. Asymptotic Spectral Analysis of Large/Growing Graphs

.Our Aim is to extend the spectral graph theory to cover..

......

...1 infinite graphs — A is no longer a finite matrix and eigenvalues make no sense.

...2 growing graphs — interests in the asymptotic properties (asymptotic combinatorics)

G G G

Then the adjacency matrix A should be studied as

...1 an operator acting on a suitable Hilbert space:

Af(x) =∑y∈V

Axyf(y), f ∈ C0(V ) ⊂ ℓ2(V );

...2 a random variable in a suitable framework of probability theory


1.4. Illustration: Path Graphs Pn ad n → ∞

s1

s2

s3

p p p p p p p p p p sn− 1

sn

A =

0 1

1 0 1

1 0 1

. . .. . .

. . .

1 0 1

1 0

Spec (Pn) =

· · · 2 coskπ

n + 1· · ·

· · · 1 · · ·

1 ≤ k ≤ n

2

4

6

8

-2 -1 0 1 2

1

π√4 − x2


2. Quantum Probability

= Non-Commutative Probability

K. R. Parthasarathy:“An Introduction to Quantum Stochastic Calculus,”

Birkhauser, 1992.

P.-A. Meyer:“Quantum Probability for Probabilists,”

Lect. Notes in Math. Vol. 1538, Springer, 1993.

L. Accardi, Y. G. Lu and I. Volovich: “Quantum Theory and Its Stochastic Limit,”

Springer, 2002.


2.1. Quantum Probabilistic Model for Coin-toss

.Traditional Model for Coin-toss..

......

A random variable X on a probability space (Ω,F , P ) satisfying the property:

P (X = +1) = P (X = −1) =1

2

More essential is the probability distribution of X:

µX =1

2δ−1 +

1

2δ+1

.Moment sequence is one of the most fundamental characteristics..

......

Moment sequence Mm ⇐⇒ Probability distributions µ

Up to determinate moment problem

For a coin-toss we have

Mm(µX) =

∫ +∞

−∞xmµX(dx) =

1, if m is even,

0, otherwise.


2.1. Quantum Probabilistic Model for Coin-toss (cont)

.Let us observe:..

......

A =

[0 1

1 0

], e0 =

[0

1

], e1 =

[1

0

].

It is straightforward to see that

⟨e0, Ame0⟩ =

1, if m is even,

0, otherwise,

= Mm(µX) =

∫ +∞

−∞xmµX(dx).

...1 In other words, we have another model of coin-toss by means of the matrix A.

...2 Matrices form a non-commutative algebra so we say

.Quantum probabilistic (or non-commutative) model for cointoss..

......

A = ∗-algebra generated by A; φ(a) = ⟨e0, ae0⟩, a ∈ A,

We call A an algebraic realization of the random variable X.


2.2. Axioms: Quantum Probability

.Definition (Algebraic probability space)..

......

An algebraic probability space is a pair (A, φ), where A is a ∗-algebra over C with

multiplication unit 1A, and a state φ : A → C, i.e.,

(i) φ is linear; (ii) φ(a∗a) ≥ 0; (iii) φ(1A) = 1.

Each a ∈ A is called an (algebraic) random variable.

.Definition (Spectral distribution)..

......

For a real random variable a = a∗ ∈ A there exists a probability measure µ on

R = (−∞,+∞) such that

φ(am) =

∫ +∞

−∞xmµ(dx) ≡ Mm(µ), m = 1, 2, . . . .

This µ is called the spectral distribution of a (with respect to the state φ).

Existence of µ by Hamburger’s theorem using Hanckel determinants.

In general, µ is not uniquely determined (indeterminate moment problem).


2.3. Concept of Independence and Central Limit Theorem (classical)

.Classical independence..

......

If two random variables X,Y are independent, we have

E[XY Y XY XX] =

E[X4Y 3] = E[X4]E[Y 3]

.Classical central limit theorem (classical CLT)..

......

If X1, X2, . . . are independent, identically distributed, normalized (mean zero, variance

one) random variables, then1√N

N∑n=1

Xn obeys the standard normal law (Gaussian

distribution) in the limit:

limN→∞

P

(1√N

N∑n=1

Xn ≤ a

)=

1√2π

∫ a

−∞e−x2/2dx,

or equivalently, for all bounded continuous functions f ,

limN→∞

E

[f

(1√N

N∑n=1

Xn

)]=

1√2π

∫ +∞

−∞f(x)e−x2/2dx.


2.3. Concept of Independence and Central Limit Theorem (classical)

.Classical independence..

......

If two random variables X,Y are independent, we have

E[XY Y XY XX] = E[X4Y 3] = E[X4]E[Y 3]

.Classical central limit theorem (classical CLT)..

......


one) random variables, then1√N

N∑n=1

Xn obeys the standard normal law (Gaussian

distribution) in the limit:

limN→∞

P

(1√N

N∑n=1

Xn ≤ a

)=

1√2π

∫ a

−∞e−x2/2dx,

or equivalently, for all bounded continuous functions f ,

limN→∞

E

[f

(1√N

N∑n=1

Xn

)]=

1√2π

∫ +∞

−∞f(x)e−x2/2dx.


.Classical CLT in moment form..

......


one) random variables having moments of all orders, we have

limN→∞

E

[(1√N

N∑n=1

Xn

)m]=

1√2π

∫ +∞

−∞xme−x2/2dx, m = 1, 2, . . . .

...1 For the proof the factorization rule is essential

E[X1X3X1X2X1X3X1] = E[X41X2X

23 ] = E[X4

1 ]E[X2]E[X23 ]

...2 If X1, X2, X2, . . . are non-commutative, what can we do for

E[X1X3X1X2X1X3X1]?=E[X4

1X2X23 ]

?=E[X4

1 ]E[X2]E[X23 ]

...3 As a result, there are many

different factorization rules ⇐⇒ various concepts of independence


2.4. Four Concepts of Independence and Quantum CLTs

Consider an algebraic probability space and (A, φ) and a ∈ A1, b ∈ A2.

commutative free Boolean monotone

φ(aba) φ(a2)φ(b) φ(a2)φ(b) φ(a)2φ(b) φ(a2)φ(b)

φ(bab) φ(a)φ(b2) φ(a)φ(b2) φ(a)φ(b)2 φ(a)φ(b)2

φ(a)2φ(b2)

φ(abab) φ(a2)φ(b2) +φ(a2)φ(b)2 φ(a)2φ(b)2 φ(a2)φ(b)2

−φ(a)2φ(b)2

CLM Gaussian Wigner Bernoulli arcsine

CLM (Central Limit Measure) µ is obtained from a sequence of normalized random

variables an = a∗n which are commutative/free/Boolean/monotone independent:

limN→∞

φ

[(1√N

N∑n=1

an

)m]=

∫ +∞

−∞xmµ(dx), m = 1, 2, . . . .

Axiomatization: Ben Ghorbal–Schurmann (2002), Muraki (2002), Franz (2003)


2.5. Monotone Independence

.Definition (monotone independence)..

......

Let (A, φ) be an algebraic probability space and An a family of ∗-subalgebras. We

say that An is monotone independent if for a1 ∈ An1 , . . . , am ∈ Anm we have

φ(a1 · · · am) = φ(ai)φ(a1 · · · ai · · · am) (ai stands for omission)

holds when ni−1 < ni and ni > ni+1 happen for i ∈ 1, 2, . . . , n.

Computing φ(a1a2a3 · · · am) where a1 ∈ A2, a2 ∈ A1, a3 ∈ A4, . . .

φ(214343664435) = φ(4)φ(4)φ(66)φ(21334435)

= φ(4)φ(4)φ(66)φ(44)φ(213335)

= · · · = φ(4)φ(4)φ(66)φ(44)φ(2)φ(5)φ(333)φ(1)


.Monotone Central Limit Theorem (Muraki 2001)..

......

Let (A, φ) be an algebraic probability space and let an∞n=1 be a sequence of

algebraic random variables satisfying the following conditions:

(i) an are real, i.e., an = a∗n;

(ii) an are normalized, i.e., φ(an) = 0, φ(a2n) = 1;

(iii) an have uniformly bounded mixed moments, i.e., for each m ≥ 1 there exists

Cm ≥ 0 such that |φ(an1 . . . anm)| ≤ Cm for any choice of n1, . . . , nm.

(iv) an are monotone independent.

Then,

limN→∞

φ

(1√N

N∑i=1

ai

m)=

1

π

∫ +√

2

−√

2

xm√2 − x2

dx, m = 1, 2, . . . .

The probability measure in the right hand side is the normalized arcsine law.

Proof is to show

limN→∞

φ

1√N

N∑i=1

ai

2m =

(2m)!

2mm!m!=

1

π

∫ +√

2

−√

2

xm√2 − x2

dx


Arcsine Law as a Central Limit Measure for Monotone Independence

Arcsine law vs normal (Gaussian) law

1

π√

2 − x2

1√2π

e−x2/2

1.0

0.5

02 2 -4 -3 -2 -1 0 1 2 3 4

0.1

0.2

0.3

0.4

0.5


3. Quantum Probabilistic Approach to Spectral Graph Theory

A. Hora and N. Obata:

Quantum Probability and Spectral Analysis of Graphs,

Springer, 2007.


3.1. Quantum Probabilistic Approach

We deal with the adjacency algebra A(G) as an algebraic probability space equipped

with a state ⟨·⟩, and A as a real random variable.

.Main Problem (I)..

......

Given a graph G = (V,E) and a state ⟨·⟩ on A(G), find a probability measure µ on Rsatisfying

⟨Am⟩ =

∫ +∞

−∞xmµ(dx), m = 1, 2, . . . .

µ is called the spectral distribution of A in the state ⟨·⟩.

.Main Problem (II)..

......

Given a growing graph G(ν) = (V (ν), E(ν)) and a state ⟨·⟩ν on A(G(ν)), find a

probability measure µ on R satisfying

⟨(A(ν))m⟩ν ≈∫ +∞

−∞xmµ(dx), m = 1, 2, . . . .

µ is called the asymptotic spectral distribution of G(ν) in the state ⟨·⟩ν .


3.2. States on A(G)

.(I) Trace..

......

For a finite graph G we set

⟨a⟩tr =1

|V | Tr (a) =1

|V |∑x∈V

⟨δx , aδx⟩, a ∈ A(G)

Then (A(G), ⟨·⟩tr) becomes an algebraic probability space.

The spectral distribution µ of A, determined (uniquely) by

⟨Am⟩tr =

∫ +∞

−∞xmµ(dx), m = 1, 2, . . . ,

coincides with the eigenvalue distribution of A:

µ =1

|V |∑i

miδλi , Spec (G) = Spec (A) =

(· · · λi · · ·· · · mi · · ·

)


.(II) Vacuum State..

......

The vacuum state at o ∈ V is the vector state defined by

⟨a⟩o = ⟨δo , aδo⟩, a ∈ A(G).

For the adjacency matrix A we have

⟨Am⟩o = ⟨δo, Amδo⟩ = |m-step walks from o to o| =∫ +∞

−∞xmµ(dx)

.(III) Deformed Vacuum State..

......

The Q-matrix of G is defined by

Q = Qq = [q∂(x,y)]x,y∈V − 1 ≤ q ≤ 1.

Then the deformed vacuum state is defined by

⟨a⟩q = ⟨Qδo , aδo⟩, a ∈ A(G).

...1 ⟨·⟩q gives rise to a one-parameter deformation of vacuum states.

...2 In general, ⟨·⟩q is not necessarily positive (i.e., not a state), but one may hope it is

positive for q close to 0....3 (Open question) To detarmine the range of q for which Qq is positive definite.


4. Graph Products and Concepts of Independence


4.1. Independence and Graph Structures (I) Cartesian product

.Definition..

......

Let G1 = (V1, E1) and G2 = (V2, E2) be two graphs. For

(x, y), (x′, y′) ∈ V1 × V2 we write (x, y) ∼ (x′, y′) if one of the following conditions

is satisfied:

(i) x = x′ and y ∼ y′; (ii) x ∼ x′ and y = y′.

Then V1 × V2 becomes a graph in such a way that (x, y), (x′, y′) ∈ V1 × V2 are

adjacent if (x, y) ∼ (x′, y′). This graph is called the Cartesian product or direct

product of G1 and G2, and is denoted by G1 ×G2.

.Example (C4 × C3)..

......

(1,1’ (2,1’

1 2

34

1’ 2’

3’

(1,3’

(1,2’

(3,1’(4,1’

C C C C


.Theorem..

......

Let G = G1 ×G2 be the cartesian product of two graphs G1 and G2. Then the

adjacency matrix of G admits a decomposition:

A = A1 ⊗ I2 + I1 ⊗A2

as an operator acting on ℓ2(V ) = ℓ2(V1 × V2) ∼= ℓ2(V1) ⊗ ℓ2(V2). Moreover, the

right hand side is a sum of commutative independent random variables with respect to

the vacuum state.

The asymptotic spectral distribution is the Gaussian distribution by applying the

commutative central limit theorem..Example (Hamming graphs)..

......

The Hamming graph is the Cartesian product of complete graphs:

H(d,N) ∼= KN × · · · ×KN (d-fold cartesian power of complete graphs)

Ad,N =n∑

i=1

i−1︷︸︸︷I ⊗ · · · ⊗ I⊗B ⊗

n−i︷︸︸︷I ⊗ · · · ⊗ I

where B is the adjacency matrix of KN .

Hora (1998) obtained the limit distribution by a direct calculation of eigenvalues.Nobuaki Obata (GSIS, Tohoku University) Quantum Probability and Asymptotic Spectral Analysis of Growing GraphsRoma, November 14, 2013 24 / 47

4.2. Independence and Graph Structures (II) Comb product

.Definition..

......

Let G1 = (V1, E1) and G2 = (V2, E2) be two graphs. We fix a vertix o2 ∈ V2. For

(x, y), (x′, y′) ∈ V1 × V2 we write (x, y) ∼ (x′, y′) if one of the following conditions

is satisfied:

(i) x = x′ and y ∼ y′; (ii) x ∼ x′ and y = y′ = o2.

Then V1 × V2 becomes a graph, denoted by G1 ▷o2 G2, and is called the comb

product or the hierarchical product.

.Example (C4 ▷ C3)..

......

(1,1’ (2,1’

1 2

34

1’ 2’

3’

(1,3’

(1,2’

(3,1’(4,1’

C C C C


.Theorem..

......

As an operator on C0(V1) ⊗ C0(V2) the adjacency matrix of G1 ▷o2 G2 is given by

A = A1 ⊗ P2 + I1 ⊗A2

where P2 : C0(V2) → C0(V2) is the projection onto the space spanned by δo2 and I1

is the identity matrix acting on C0(V1).

.Theorem (Accardi–Ben Ghobal–O. IDAQP(2004))..

......

The adjacency matrix of the comb product G(1) ▷o2 G(2) ▷o3 · · · ▷on G(n) admits a

decomposition of the form:

A(1) ▷ A(2) ▷ · · · ▷ A(n)

=

n∑i=1

i−1︷︸︸︷I(1) ⊗ · · · ⊗ I(i−1) ⊗A(i) ⊗

n−i︷︸︸︷P (i+1) ⊗ · · · ⊗ P (n),

where P (i) the projection from ℓ2(V (i)) onto the one-dimensional subspace spanned by

δoi . Moreover, the right-hand side is a sum of monotone independent random variables

with respect to ψ ⊗ δo2 ⊗ · · · ⊗ δon , where ψ is an arbitrary state on B(ℓ2(V (1))).


4.3. Independence and Graph Structures (III) Star product

.Definition..

......

Let G1 = (V1, E1) and G2 = (V2, E2) be two graphs with distinguished vertices

o1 ∈ V1 and o2 ∈ V2. Define a subset of V1 × V2 by

V1 ⋆ V2 = (x, o2) ; x ∈ V1 ∪ (o1, y) ; y ∈ V2

The induced subgraph of G1 ×G2 spanned by V1 ⋆ V2 is called the star product of G1

and G2 (with contact vertices o1 and o2), and is denoted by G1 ⋆ G2 = G1 o1⋆o2 G2.

.Example (C4 ⋆ C3)..

......

(1,1’ (2,1’

1 2

34

1’ 2’

3’

(1,3’

(1,2’

(3,1’(4,1’

C C C C


.Theorem (O. IIS(2004))..

......

As an operator on C0(V1) ⊗ C0(V2) the adjacency matrix of G1 ⋆ G2 is given by

A = A(1) ⊗ P (2) + P (1) ⊗A(2), P (i) = |Φ(i)0 ⟩⟨Φ(i)

0 |.

where P (i) : C0(Vi) → C0(Vi) is the projection onto the space spanned by δoi .

.Theorem (O. IIS(2004))..

......

The adjacency matrix of the star product G(1) ⋆ G(2) ⋆ · · · ⋆ G(n) admits a

decomposition of the form:

A(1) ⋆ A(2) ⋆ · · · ⋆ A(n)

=

n∑i=1

i−1︷︸︸︷P (1) ⊗ · · · ⊗ P (i−1) ⊗A(i) ⊗

n−i︷︸︸︷P (i+1) ⊗ · · · ⊗ P (n),

where P (i) the projection from ℓ2(V (i)) onto the one-dimensional subspace spanned by

δoi . Moreover, the right-hand side is a sum of Boolean independent random variables

with respect to δo1 ⊗ δo2 ⊗ · · · ⊗ δon .


Asymptotic spectral distribution of graph products G(n) = G#G# . . .#G

independence monotone Boolean commutative free

CLM arcsine Bernoulli Gaussian Wigner

graph product comb star direct free

examples comb graph star graph integer lattice homogeneous tree


5. Method of Quantum Decomposition


5.1. A Simple Prototype: Coin-toss

Recall that the coin-toss is modelled by A =

[0 1

1 0

]and e0 =

[0

1

].A prototype of quantum decomposition..

......

A = A+ +A− =

[0 1

0 0

]+

[0 0

1 0

]

⟨e0, Ame0⟩ = ⟨e0, (A+ +A−)me0⟩ =∑

ϵ1,...,ϵm∈±

⟨e0, Aϵm · · ·Aϵ1e0⟩.

=⇒ Relations among non-commutative quantum components A± are useful.

se0

se1

G

@

@@R

@@R

@@R@

@@R

0

A+

1

A−

2 3 · · · m

⟨e0, Ame0⟩ =

1, if m is even,

0, otherwise.


5.2. Quantum Decomposition of the Adjacency Matrix

Fix an origin o ∈ V of G = (V,E).

Vn+1

Vn

Vn-1

V 1

V0

n+1

n

n-1

1

0

Φ

Φ

Φ

Φ

Φ

Γ (G)

Stratification (Distance Partition)

V =∞∪

n=0

Vn

Vn = x ∈ V ; ∂(o, x) = n

Associated Hilbert space Γ(G) ⊂ ℓ2(V )

Γ(G) =

∞∑n=0

⊕CΦn

Φn = |Vn|−1/2∑

x∈Vn

δx


5.2. Quantum Decomposition of the Adjacency Matrix (cont)

Vn+1

Vn

Vn−1

(A ) =1+yx

(A ) =1yx

(A ) =1yx−

y

x

y

y A = A+ +A− +A

(A+)∗ = A−, (A)∗ = A

In general, Γ(G) =

∞∑n=0

⊕CΦn is not invariant under the actions of Aϵ.

.Cases so far studied in detail..

......

...1 Γ(G) is invariant under Aϵ — distance-regular graphs

...2 Γ(G) is asymptotically invariant under Aϵ.


5.3. When Γ(G) is Invariant Under Aϵ

Vn+1

Vn−1

Vn

ω (x)

x

ω (x)

ω (x)

For x ∈ Vn we define

ωϵ(x) = |y ∈ Vn+ϵ ; y ∼ x|, ϵ = +,−,

Then, Γ(G) is invariant under Aϵ if and only if

(∗) ωϵ(x) is constant on each Vn .

Typical examples: distance-regular graphs

(in this case the constant in (∗) is independent

of the choice of o ∈ V ) e.g., homogeneous trees,

Hamming graphs, Johnson graphs, odd graphs, ...

.Theorem..

......

If Γ(G) is invariant under A+, A−, A, there exists a pair of sequences αn and

ωn such that

A+Φn =√ωn+1 Φn+1, A−Φn =

√ωn Φn−1, AΦn = αn+1Φn.

In other words, Γωn,αn = (Γ(G), Φn, A+, A−, A) is an interacting Fock

space associated with Jacobi parameters (ωn, αn).


5.4. Computing the Spectral Distribution

...1 By the interacting Fock space structure (Γ(G), A+, A−, A):

AΦn = (A+ +A +A−)Φn =√ωn+1 Φn+1 + αn+1Φn +

√ωn Φn−1.

...2 Associated with a probability distribution µ on R, the orthogonal polynomials

P0(x) = 1, . . . , Pn(x) = xn + · · · verify the three term recurrence relation:

xPn = Pn+1 + αn+1Pn + ωnPn−1 , P0 = 1, P1 = x− α1,

where (ωn, αn) is called the Jacobi parameters of µ.

...3 An isometry U : Γ(G) → L2(R, µ) is defined by Φn 7→ ∥Pn∥−1Pn .

...4 Then UAU∗ = x (multiplication operator) and

⟨Φ0, AmΦ0⟩ = ⟨UΦ0, UA

mU∗UΦ0⟩ = ⟨P0, xmP0⟩µ =

∫ +∞

−∞xmµ(dx).

.Theorem (Graph structure =⇒ (ωn, αn) =⇒ spectral distribution)..

......

If Γ(G) is invariant under A+, A−, A, the vacuum spectral distribution µ defined by

⟨Φ0, AmΦ0⟩ =

∫ +∞

−∞xmµ(dx), m = 1, 2, . . . ,

is a probability distribution on R that has the Jacobi parameters (ωn, αn).





√ωn Φn−1.




where (ωn, αn) is called the Jacobi parameters of µ....3 An isometry U : Γ(G) → L2(R, µ) is defined by Φn 7→ ∥Pn∥−1Pn ....4 Then UAU∗ = x (multiplication operator) and

⟨Φ0, AmΦ0⟩ = ⟨UΦ0, UA

mU∗UΦ0⟩ = ⟨P0, xmP0⟩µ =

∫ +∞

−∞xmµ(dx).


......


⟨Φ0, AmΦ0⟩ =

∫ +∞

−∞xmµ(dx), m = 1, 2, . . . ,

is a probability distribution on R that has the Jacobi parameters (ωn, αn).





√ωn Φn−1.




where (ωn, αn) is called the Jacobi parameters of µ....3 An isometry U : Γ(G) → L2(R, µ) is defined by Φn 7→ ∥Pn∥−1Pn ....4 Then UAU∗ = x (multiplication operator) and

⟨Φ0, AmΦ0⟩ = ⟨UΦ0, UA

mU∗UΦ0⟩ = ⟨P0, xmP0⟩µ =

∫ +∞

−∞xmµ(dx).


......


⟨Φ0, AmΦ0⟩ =

∫ +∞

−∞xmµ(dx), m = 1, 2, . . . ,

is a probability distribution on R that has the Jacobi parameters (ωn, αn).Nobuaki Obata (GSIS, Tohoku University) Quantum Probability and Asymptotic Spectral Analysis of Growing GraphsRoma, November 14, 2013 35 / 47

5.5. How to know µ from the Jacobi Parameters (ωn, αn)

We need to find a probability distribution µ for which the orthogonal polynomials satisfy


.Cauchy–Stieltjes transform..

......

Gµ(z) =

∫ +∞

−∞

µ(dx)

z − x=

1

z − α1 −ω1

z − α2 −ω2

z − α3 −ω3

z − α4 − · · ·

=1

z − α1 − ω1

z − α2 − ω2

z − α3 − ω3

z − α4 − · · ·

where the right-hand side is convergent in Im z = 0 if the moment problem is

determinate, e.g., if ωn = O((n logn)2) (Carleman’s test).


5.5. How to know µ from the Jacobi Parameters (cont)

Cauchy–Stieltjes transform

Gµ(z) =

∫ +∞

−∞

µ(dx)

z − x=

1

z − α1 −ω1

z − α2 −ω2

z − α3 −ω3

z − α4 − · · ·

.Stieltjes inversion formula..

......

The (right-continuous) distribution function F (x) = µ((−∞, x]) and the absolutely

continuous part of µ is given by

1

2F (t) + F (t− 0) − 1

2F (s) + F (s− 0)

= − 1

πlim

y→+0

∫ t

s

ImGµ(x+ iy)dx, s < t,

ρ(x) = − 1

πlim

y→+0ImGµ(x+ iy)


5.5. Illustration: Homogeneous tree Tκ (κ ≥ 2)

Stratification of T4

(1) Quantum decomposition: A = A+ +A−

A+Φ0 =√κΦ1, A+Φn =

√κ− 1Φn+1 (n ≥ 1)

A−Φ0 = 0, A−Φ1 =√κΦ0, A−Φn =

√κ− 1Φn−1 (n ≥ 2)

(2) Jacobi parameters: ω1 = κ, ω2 = ω3 = · · · = κ− 1, αn ≡ 0


(3) Cauchy–Stieltjes transform: (ω1 = κ, ω2 = ω3 = · · · = κ− 1)∫ +∞

−∞

µ(dx)

z − x= Gµ(z) =

1

z −ω1

z −ω2

z −ω3

z −ω4

z −ω5

z − · · ·

=(κ− 2)z − κ

√z2 − 4(κ− 1)

2(κ2 − z2)

(4) Spectral distribution: µ(dx) = ρκ(x)dx

ρκ(x) =κ

2π

√4(κ− 1) − x2

κ2 − x2

|x| ≤ 2√κ− 1

Kesten Measures (1959)

ρ4, ρ8, ρ12


6. Method of Quantum Decomposition for Growing Graphs

G G G


6.1. Growing Regular Graphs G(ν) = (V (ν), A(ν))

Here Γ(G(ν)) is not necessarily invariant but asymptotically invariant under Aϵν .

Vn+1

Vn−1

Vn

ω (x)

x

ω (x)

ω (x)

Statistics of ωϵ(x)

M(ωϵ|Vn) =1

|Vn|∑

x∈Vn

|ωϵ(x)|

Σ2(ωϵ|Vn) =1

|Vn|∑

x∈Vn

|ωϵ(x)| −M(ωϵ|Vn)

2L(ωϵ|Vn) = max|ωϵ(x)| ; x ∈ Vn.

Conditions for a growing regular graph G(ν) = (V (ν), E(ν))

(A1) limν deg(G(ν)) = ∞.

(A2) for each n = 1, 2, . . . ,

∃ limν

M(ω−|V (ν)n ) = ωn < ∞, lim

νΣ

2(ω−|V (ν)

n ) = 0, supν

L(ω−|V (ν)n ) < ∞.

(A3) for each n = 0, 1, 2, . . . ,

∃ limν

M(ω|V (ν)n )√

κ(ν)= αn+1 < ∞, lim

ν

Σ2(ω|V (ν)n )

κ(ν)= 0, sup

ν

L(ω|V (ν)n )√

κ(ν)< ∞.


.Theorem (QCLT in a general form [Hora-Obata: TAMS (2008)])..

......

Let G(ν) = (V (ν), E(ν)) be a growing regular graph satisfying

(A1) limν κ(ν) = ∞, where κ(ν) = deg(G(ν)).

(A2) for each n = 1, 2, . . . ,

∃ limν

M(ω−|V (ν)n ) = ωn < ∞, lim

νΣ

2(ω−|V (ν)

n ) = 0, supν

L(ω−|V (ν)n ) < ∞.

(A3) for each n = 0, 1, 2, . . . ,

∃ limν

M(ω|V (ν)n )√

κ(ν)= αn+1 < ∞, lim

ν

Σ2(ω|V (ν)n )

κ(ν)= 0, sup

ν

L(ω|V (ν)n )√

κ(ν)< ∞.

Then the asymptotic spectral distribution of the normalized Aν in the vacuum state is a

probability distribution µ determined by (ωn, αn):

limν

⟨( Aν√κ(ν)

)m⟩=

∫ +∞

−∞xmµ(dx), m = 1, 2, . . . .

Proof is to show that

limν

Aϵν√κ(ν)

= Bϵ (stochastically)

where (Γ, Ψn, B+, B−, B) is the interacting Fock space associated with the Jacobi

parameters (ωn, αn).Nobuaki Obata (GSIS, Tohoku University) Quantum Probability and Asymptotic Spectral Analysis of Growing GraphsRoma, November 14, 2013 42 / 47

Outline of Proof

(1)Aϵ√κ

Φn = γϵn+ϵΦn+ϵ + Sϵ

n+ϵ, ϵ ∈ +,−, , n = 0, 1, 2, . . . .

γ+n = M(ω−|Vn)

(|Vn|

κ|Vn−1|

)1/2

, γ−n = M(ω+|Vn)

(|Vn|

κ|Vn+1|

)1/2

, γn =

M(ω|Vn)√κ

.

(2) |Vn| =

n∏k=1

M(ω−|Vk)

−1

κn + O(κn−1).

(3) limν

γ+n =

√ωn, lim

νγ−n =

√ωn+1, lim

νγn = αn+1.

(4)

Aϵm

√κ

· · ·Aϵ1

√κ

Φn = γϵ1n+ϵ1

γϵ2n+ϵ1+ϵ2

· · · γϵmn+ϵ1+···+ϵm

Φn+ϵ1+···+ϵm

+m∑

k=1

γϵ1n+ϵ1

· · · γϵk−1

n+ϵ1+···+ϵk−1︸︷︷︸(k − 1) times

Aϵm

√κ

· · ·Aϵk+1

√κ︸︷︷︸

(m − k) times

Sϵkn+ϵ1+···+ϵk

.

(5) Estimate the error terms and show that

limν

⟨Φ

(ν)j ,

Aϵm√κ(ν)

· · ·Aϵk+1√κ(ν)

Sϵkn+ϵ1+···+ϵk

⟩= 0.


6.2. Illustration: ZN as N → ∞

...1 Asymptotic invariance of Γ(ZN) under Aϵ:

A+Φn =√

2N√n+ 1 Φn+1 + O(1),

A−Φn =√

2N√n Φn−1 + O(N−1/2).

...2 Normalized adjacency matrices:Aϵ√κ(N)

=Aϵ

√2N

...3 The interacting Fock space in the limit as N → ∞ is the Boson Fock space:

B+Ψn =√n+ 1 Ψn+1, B−Φn =

√n Ψn−1, B = 0.

...4 The asymptotic spectral distribution is the standard Gaussian distribution:

limN→∞

⟨Φ0,

(AN√2N

)m

Φ0

⟩= ⟨Ψ0, (B

+ +B−)mΨ0⟩

=1√2π

∫ +∞

−∞xme−x2/2dx.


6.3. Some Results: Asymptotic Spectral Distributions

graphs IFS vacuum state deformed vacuum state

Hamming graphs ωn = n Gaussian (N/d → 0) Gaussian

H(d,N) (Boson) Poisson (N/d → λ−1 > 0) or Poisson

Johnson graphs ωn = n2exponential (2d/v → 1) ‘Poissonization’ of

J(v, d) geometric (2d/v → p ∈ (0, 1)) exponential distribution

odd graphs ω2n−1 = n two-sided Rayleigh ?

Ok ω2n = n

homogeneous ωn = 1 Wigner semicircle free Poisson

trees Tκ (free)

integer lattices ωn = n Gaussian Gaussian

ZN (Boson)

symmetric groups ωn = n Gaussian Gaussian

Sn (Coxeter) (Boson)

Coxeter groups ωn = 1 Wigner semicircle free Poisson

(Fendler) (free)

Spidernets ω1 = 1 free Meixner law ?

S(a, b, c) ω2 = · · · = r


Quantum Probabilistic Approach to Spectral Analysis of Graphs

adjacency matrixcombinatorics

quantum decomposition

quantum CLT

interacting Fock space

Quantum

Classical

quantum

CLT

use of independence

orthogonal polynomials

spectral distribution

scaling limit

asymptotic

spectral

distribution µ

µνAν

Aν Aν Aν AνAν

Aν

Zνν

νX X

B

εε

Pn n n

B B

αω

α

Γ ( nα ) B B

linear algebra


Summary

...1 Quantum probability is an extension of classical probability based on non-comutative

algebras

1) algebraic probability space (A, φ) and random variables a ∈ A2) various concepts of independence

3) and associated quantum central limit theorems (QCLTs)

...2 Quantum probabilistic techniques are useful for spectral analysis of graphs

1) quantum decomposition and use of one-mode interacting Fock spaces

2) concepts of independence and product structures

3) asymptotic spectral distribution as a result of QCLT

...3 Some future directions

1) more on product structure of graphs and independence

2) extending the method of quantum decomposition beyond one-mode interacting

Fock spaces (orthogonal polynomials in one variable)

3) digraphs (directed graphs)

4) applications to dynamics on graphs, e.g., coupled oscillators, random walks (or

more generally Markov chains), quantum walks, etc.


quantum probability and asymptotic spectral …obata/research/file/presen...2013/11/14 · d. m....

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