[PPT] - Asymptotic Spectral Analysis of Growing Graphs Nobuaki Obata PowerPoint Presentation

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Asymptotic Spectral Analysis of Growing Graphs

Nobuaki Obata

Graduate School of Information Sciences Tohoku University www.math.is.tohoku.ac.jp/˜obata

SJTU, Shanghai, China, 2018.11.15–18

Nobuaki Obata (Tohoku University) Asymptotic Spectral Analysis SJTU, Shanghai, China, 2018.11.15–18 1 / 80

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Introducing myself...

Sendai Tianjin

1

Tohoku University — The 3rd oldest national University of Japan, founded in 1907.

2

Graduate School of Information Sciences (GSIS) — One of the 17 Graduate Schools, founded in 1993.

3

Nobuaki Obata — Serving as Professor since 2001. Before then I was a member of Department of Mathematics in Nagoya University.

4

Major research interests — Spectral analysis of graphs, Random graphs, Quantum probability, Quantum white noise analysis, and any topics related to network science.

Nobuaki Obata (Tohoku University) Asymptotic Spectral Analysis SJTU, Shanghai, China, 2018.11.15–18 2 / 80

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Main Theme: Asymptotic Spectral Analysis of Growing Graphs

▶ Spectral analysis of graphs A Axy [ ] G = V, E ( ) dx = f x dx ( ) ( ) ▶ Growing graphs

Nobuaki Obata (Tohoku University) Asymptotic Spectral Analysis SJTU, Shanghai, China, 2018.11.15–18 3 / 80

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My Motivations and Backgrounds (1) Statistics in large scale discrete systems

1

A. M. Vershik’s asymptotic combinatorics (1970s–)

[1] Asymptotic combinatorics and algebraic analysis (ICM 1994) ... the study of asymptotic problems in combinatorics is stimulated enormously by taking into account the various approaches from different branches of mathematics. ... The main question in this context is: What kind of limit behavior can have a combinatorial object when it “grows” ? [2] Between “very large” and “infinite” (Bedlewo 2012) [3] Takagi lecture of Mathematical Society of Japan (Tohoku University, 2015) [4] see also A. Hora: The limit shape problem for emsembles of Young diagrams, Springer 2017.

2

Complex networks — modelling real world large networks [1] A.-L. Barab´ asi and R. Albert (1999) — scale free networks [2] D. J. Watts and S. H. Strogatz (1998) — small world networks [3] F. Chung and L. Lu (2006), R. Durrett (2007), L. Lovasz (2012).

Nobuaki Obata (Tohoku University) Asymptotic Spectral Analysis SJTU, Shanghai, China, 2018.11.15–18 4 / 80

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My Motivations and Backgrounds (2) Quantum probability = Noncommutative Probability = Algebraic Probability

1

J. von Neumann: Mathematische Grundlagen der Quantenmechanik (1932)

Mathematical theory for the probabilistic interpretation in quantum mechanics in terms of operators on Hilbert spaces.

2

The term quantum probability was introduced actively by L. Accardi (Roma) around 1978.

3

R. Hudson and K. R. Parthasarathy (1984) initiated quantum Ito calculus.

4

P.-A. Meyer: Quantum Probability for Probabilists, LNM 1538 (1993).

5

N. Obata: Quantum probability + graph theory and network science since 1998.

[1] A. Hora and N. Obata: Quantum Probability and Spectral Analysis of Graphs, Springer, 2007. [2] N. Obata: Spectral Analysis of Growing Graphs. A Quantum Probability Point

f View, Springer, 2017.

Nobuaki Obata (Tohoku University) Asymptotic Spectral Analysis SJTU, Shanghai, China, 2018.11.15–18 5 / 80

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Plan

1

Spectral Distributions of Graphs

2

Method of Quantum Decomposition

3

Asymptotic Spectral Analysis of Growing Regular Graphs

4

Graph Products and Concepts of Independence

5

Summary and Perspectives ▶ Main Reference

N. Obata: Spectral Analysis of Growing Graphs

— A Quantum Probability Point of View, Springer, 2017.

Nobuaki Obata (Tohoku University) Asymptotic Spectral Analysis SJTU, Shanghai, China, 2018.11.15–18 6 / 80

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Spectral Distributions of Graphs

1. Spectral Distributions of Graphs

[Chapters 1–3]

Nobuaki Obata (Tohoku University) Asymptotic Spectral Analysis SJTU, Shanghai, China, 2018.11.15–18 7 / 80

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Spectral Distributions of Graphs

1.1. Quantum Probability — Algebraic Probability Spaces Definition

A pair (A, φ) is called an algebraic probability space if A is a unital ∗-algebra over C and φ a state on it, i.e., (i) φ : A → C is a linear function; (ii) positive, i.e., φ(a∗a) ≥ 0; (iii) normalized, i.e., φ(1A) = 1.

Definition

Each a ∈ A is called an (algebraic) random variable. It is called real if a = a∗. ▶ (Ω, F, P ): classical (Kolmogorovian) probability space A = L∞−(Ω, F, P ) = ∩

1≤p<∞

Lp(Ω, F, P ) = {X : Ω → C ; E[|X|m] < ∞ for all m ≥ 1} φ(X) = E[X], X ∈ A.

Nobuaki Obata (Tohoku University) Asymptotic Spectral Analysis SJTU, Shanghai, China, 2018.11.15–18 8 / 80

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Spectral Distributions of Graphs

1.1. Quantum Probability — Statistics Definition

(1) For a random variable a ∈ A, its mixed moments are defined by φ(aϵm · · · aϵ2aϵ1), ϵ1, ϵ2 . . . , ϵm ∈ {1, ∗}. (2) For a real random variable a = a∗ ∈ A the mixed moments are reduced to the moment sequence: φ(am), m = 1, 2, . . . .

Definition

(1) Two algebraic random variables a in (A, φ) and b in (B, ψ) are called stochastically equivalent if their all mixed moments coincide: φ(aϵm · · · aϵ2aϵ1) = ψ(bϵm · · · bϵ2bϵ1). (2) Two real random variables a = a∗ in (A, φ) and b = b∗ in (B, ψ) are stochastically equivalent if φ(am) = ψ(bm), m = 1, 2, . . . .

Nobuaki Obata (Tohoku University) Asymptotic Spectral Analysis SJTU, Shanghai, China, 2018.11.15–18 9 / 80

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Spectral Distributions of Graphs

1.1. Quantum Probability — Spectral Distributions Theorem (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 in the state φ.

1

Existence proof is by Hamburger’s theorem using Hanckel determinants.

2

µ is not uniquely determined in general (determinate moment problem).

3

µ is unique, for example, if

∞

∑

m=1

M

−

1 2m

2m

= +∞ (Carleman’s moment test) = ⇒ Details omitted (see Chapters 4-5)

Nobuaki Obata (Tohoku University) Asymptotic Spectral Analysis SJTU, Shanghai, China, 2018.11.15–18 10 / 80

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Spectral Distributions of Graphs

1.1. Quantum Probability vs Classical Probability

Classical Probability Quantum Probability probability space (Ω, F, P ) (A, φ) random variable X : Ω → R a = a∗ ∈ A expectation E[X] = ∫

Ω

X(ω) P (dω) φ(a) distribution µX((−∞, x]) = P (X ≤ x) NA moments E[Xm] φ(am) E[Xm] = ∫ +∞

−∞

xmµX(dx) φ(am) = ∫ +∞

−∞

xmµa(dx)

Definition (algebraic realization)

An algebraic random variable a = a∗ in an algebraic probability space (A, φ) is called an algebraic realization of a classical random variable X if their moments coincide: φ(am) = E[Xm], m = 1, 2, . . . .

Nobuaki Obata (Tohoku University) Asymptotic Spectral Analysis SJTU, Shanghai, China, 2018.11.15–18 11 / 80

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Spectral Distributions of Graphs

1.2. Matrix Algebras — States on M(n, C)

Equipped with the usual matrix operations, A = M(n, C) = {a = [aij] ; aij ∈ C} becomes a unital ∗-algebra over C. (i) the normalized trace: φ(a) = 1 nTr (a) = 1 n

n

∑

i=1

aii , a = [aij]. (ii) a vector state: φ(a) = ⟨ξ, aξ⟩, ξ ∈ Cn, ∥ξ∥ = 1.

Lemma (exercise)

A general form of a state on M(n, C) is given by φ(a) = Tr (ρa), where ρ is a density matrix, i.e., ρ = ρ∗ ≥ 0 and Tr (ρ) = 1.

Nobuaki Obata (Tohoku University) Asymptotic Spectral Analysis SJTU, Shanghai, China, 2018.11.15–18 12 / 80

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Spectral Distributions of Graphs

1.2. Matrix Algebras — A Model of 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 2 δ−1 + 1 2 δ+1

Lemma (Moment sequence)

Let X be the coin toss defined as above. Then we have E[Xm] = Mm(µX) = ∫ +∞

−∞

xmµX(dx) =    1, if m is even, 0,

therwise.

Nobuaki Obata (Tohoku University) Asymptotic Spectral Analysis SJTU, Shanghai, China, 2018.11.15–18 13 / 80

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Spectral Distributions of Graphs

1.2. Matrix Algebras — A Model of Coin Toss (cont)

1

Set A = [ 1 1 ] , e0 = [ 1 ] , e1 = [ 1 ] .

2

Define an algebraic probability space (A, φ) by A = ∗-algebra generated by A; φ(a) = ⟨e0, ae0⟩, a ∈ A.

3

It is straightforward to see that φ(Am) = ⟨e0, Ame0⟩ =    1, if m is even, 0,

therwise,

Thus, φ(Am) = E[Xm], m = 1, 2, . . . .

4

Namely, A is an algebraic realization of the coin toss X.

Nobuaki Obata (Tohoku University) Asymptotic Spectral Analysis SJTU, Shanghai, China, 2018.11.15–18 14 / 80

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Spectral Distributions of Graphs

1.2. Matrix Algebras — A Model of Coin Toss (cont)

▶ Why quantum probabilistic model? A = [ 1 1 ] , φ(Am) = ⟨e0, Ame0⟩ =    1, if m is even, 0,

therwise,

1

We have a natural decomposition: A = [ 1 1 ] = [ 1 ] + [ 1 ] = A+ + A−

2

Suggests a method of computing the moment: φ(Am) = ⟨e0, (A+ + A−)me0⟩ = ∑

ϵ1,...,ϵm

⟨e0, Aϵm · · · Aϵ2Aϵ1e0⟩

3

This becomes a problem of combinatorics (counting paths). = ⇒ We are led to the concept of quantum decomposition.

Nobuaki Obata (Tohoku University) Asymptotic Spectral Analysis SJTU, Shanghai, China, 2018.11.15–18 15 / 80

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Spectral Distributions of Graphs

Classical Probability vs Quantum Probability

Classical Probability Quantum Probability probability space (Ω, F, P ) (A, φ) random variable X : Ω → R a = a∗ ∈ A expectation E[X] = ∫

Ω

X(ω) P (dω) φ(a) moments E[Xm] φ(am) distribution µX((−∞, x]) = P (X ≤ x) NA E[Xm] = ∫ +∞

−∞

xmµX(dx) φ(am) = ∫ +∞

−∞

xmµa(dx) independence E[XmY n] = E[Xm]E[Y n] many variants LLN lim

n→∞

1 n

n

∑

k=1

Xk many variants CLT lim

n→∞

1 √n

n

∑

k=1

Xk many variants

Nobuaki Obata (Tohoku University) Asymptotic Spectral Analysis SJTU, Shanghai, China, 2018.11.15–18 16 / 80

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Spectral Distributions of Graphs

1.3. Graphs and Matrices Definition (graph)

A (finite or infinite) 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.

Definition (adjacency matrix)

The adjacency matrix A = [Axy] is defined by Axy = { 1, x ∼ y, 0,

therwise.

Assumption 1 [connected] Any pair of distinct vertices are connected by a walk. Assumption 2 [locally finite] degG(x) = (degree of x) < ∞ for all x ∈ V .

Definition (adjacency algebra)

Let G = (V, E) be a graph. The ∗-algebra generated by the adjacency matrix A is called the adjacency algebra of G and is denoted by A(G). In fact, A(G) is the set of polynomials in A.

Nobuaki Obata (Tohoku University) Asymptotic Spectral Analysis SJTU, Shanghai, China, 2018.11.15–18 17 / 80

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Spectral Distributions of Graphs

1.4. Spectra of Finite Graphs Definition (Spectrum)

Let G be a finite graph and A the adjacency matrix. Let ev(A) = {λ1, . . . , λs} be the set of eigenvalues, where λi ̸= λj for i ̸= j. Letting mi be the multiplicity of λi. The multi-set Spec (G) = {λ1(m1), . . . , λs(ms)}, is called the spectrum (or eigenvalues) of G. Example: path Pn

s

1 s

2 s

3 ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ s

n − 1

s

n

A =           1 1 1 1 1 ... ... ... 1 1 1           Spec (Pn) = { 2 cos kπ n + 1 ; 1 ≤ k ≤ n }

Nobuaki Obata (Tohoku University) Asymptotic Spectral Analysis SJTU, Shanghai, China, 2018.11.15–18 18 / 80

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Spectral Distributions of Graphs

1.5. Spectral Distributions of Graphs Definition (Spectral (or Eigenvalue) distribution)

With Spec (G) = {λ1(m1), . . . , λs(ms)} we associate a probability distribution defined by µ = 1 |V |

s

∑

i=1

miδλi . Or if the eigenvalues of A are listed as λ1 ≤ λ2 ≤ · · · ≤ λ|V |, we have µ = 1 |V |

|V |

∑

i=1

δλi . ▶ δλ is the delta function (delta measure/point mass): ∫ +∞

−∞

f(x)δλ(dx) = f(λ), f ∈ Cb(R). Hence ∫ +∞

−∞

f(x)µ(dx) = 1 |V |

|V |

∑

i=1

f(λi), f ∈ Cb(R).

Nobuaki Obata (Tohoku University) Asymptotic Spectral Analysis SJTU, Shanghai, China, 2018.11.15–18 19 / 80

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Spectral Distributions of Graphs

1.5. Spectral Distributions of Graphs — Examples Pn and Kn

Spec (Pn) = { 2 cos kπ n + 1 ; 1 ≤ k ≤ n } µ = 1 n

n

∑

k=1

δ2 cos

kπ n+1

2

2

Spec (Kn) = {−1(n − 1), n − 1(1)} µ = 1 n δn−1 + n − 1 n δ−1

n

Nobuaki Obata (Tohoku University) Asymptotic Spectral Analysis SJTU, Shanghai, China, 2018.11.15–18 20 / 80

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Spectral Distributions of Graphs

1.6. Asymptotic Spectral Distributions — Motivating Example Pn

Spec (Pn) = { 2 cos kπ n + 1 ; 1 ≤ k ≤ n } µn = 1 n

n

∑

k=1

δ2 cos

kπ n+1

2

2

We are interested in n → ∞. Let f ∈ Cb(R). ∫ +∞

−∞

f(x)µn(dx) = 1 n

n

∑

k=1

f ( 2 cos kπ n + 1 ) → ∫ 1 f(2 cos πt)dt = ∫ +2

−2

f(x) dx π√4 − x2 . Thus, the limit distribution is obtained.

Nobuaki Obata (Tohoku University) Asymptotic Spectral Analysis SJTU, Shanghai, China, 2018.11.15–18 21 / 80

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Spectral Distributions of Graphs

1.6. Asymptotic Spectral Distributions — Motivating Example Pn (cont)

2

2 2 4 6

2
1

1 2

Arcsine law dx π √ 4 − x2

Nobuaki Obata (Tohoku University) Asymptotic Spectral Analysis SJTU, Shanghai, China, 2018.11.15–18 22 / 80

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Spectral Distributions of Graphs

1.6. Asymptotic Spectral Distributions — Motivating Example Kn

Spec (Kn) = {−1(n − 1), n − 1(1)} and µn = 1 n δn−1 + n − 1 n δ−1 ▶ For f ∈ Cb(R) we have ∫ +∞

−∞

f(x)µn(dx) = 1 nf(n − 1) + n − 1 n f(−1) → f(−1). Thus, µn → δ−1 but ...?? ▶ Need normalization: µ → ˜ µ with mean(˜ µ) = 0 and var(˜ µ) = 1. ∫ f(x)˜ µ(dx) = ∫ f (x − m σ ) µ(dx), m = mean(µ), σ2 = var(µ). ▶ After normalization we have ∫ +∞

−∞

f(x)˜ µn(dx) = 1 nf ( n − 1 √n − 1 ) + n − 1 n f ( −1 √n − 1 ) → f(0). Thus, we have ˜ µn → δ0. This is reasonable. Note: For Kn we have var(µn) = n − 1 → ∞ while for Pn we have var(µn) = 2(n − 1) n → 2.

Nobuaki Obata (Tohoku University) Asymptotic Spectral Analysis SJTU, Shanghai, China, 2018.11.15–18 23 / 80

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Spectral Distributions of Graphs

1.7. Quantum Probabilistic Approach — States on A(G)

Now, equipped with a state φ consider (A(G), φ) as an algebraic probability space.

(1) Trace

φ(a) = ⟨a⟩tr = 1 |V | Tr (a) = 1 |V | ∑

x∈V

⟨δx , aδx⟩, a ∈ A. The spectral distribution of A coincides with the eigenvalue distribution of G, namely, letting µ be the eigenvalue distribution of G, we have ⟨Am⟩tr = ∫ +∞

−∞

xmµ(dx), m = 1, 2, . . . .

(2) Vacuum state (at a fixed origin o ∈ V )

φ(a) = ⟨a⟩o = ⟨δo , aδo⟩, a ∈ A(G). Let µ be the spectral distribution of A. Then we have ⟨Am⟩o = ⟨δo, Amδo⟩ = ∫ +∞

−∞

xmµ(dx) = |{m-step walks from o to o}|.

Nobuaki Obata (Tohoku University) Asymptotic Spectral Analysis SJTU, Shanghai, China, 2018.11.15–18 24 / 80

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Spectral Distributions of Graphs

1.7. Quantum Probabilistic Approach — Main Problem Main Problem

Given a graph G = (V, E) (resp. a growing graph) and a state ⟨·⟩ on A(G), find a probability measure µ on R satisfying ⟨Am⟩ = ∫ +∞

−∞

xmµ(dx)

r

⟨(A − ⟨A⟩ Σ(A) )m⟩ → ∫ +∞

−∞

xmµ(dx). The above µ is called the (asymptotic) spectral distribution of A in the state ⟨·⟩.

Quantum Probabilistic Approaches — Use of Non-Commutativity

1

Method of quantum decomposition: A = A+ + A− + A◦

2

Use of various concepts of independence: A = B1 + B2 + · · · + Bn where Bk are “independent” identically distributed random variables.

Nobuaki Obata (Tohoku University) Asymptotic Spectral Analysis SJTU, Shanghai, China, 2018.11.15–18 25 / 80

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Spectral Distributions of Graphs

Of course, we may focus on generalizations of graphs Lemma

A matrix A with index set V × V is the adjacency matrix of a graph on V if and only if (i) (A)xy ∈ {0, 1}; (ii) (A)xy = (A)yx; (iii) (A)xx = 0.

1

Graph with loops. Dropping (iii) allows a loop connecting a vertex with itself.

2

Multigraph. Relaxing (i) as (A)xy ∈ {0, 1, 2, . . . } allows a multi-edge.

3

Digraph (directed graph). Dropping (ii) gives rise to orientation of edges, namely, (A)xy = 1 ⇔ x → y.

4

Network. In a broad sense, an arbitrary matrix A with index set V × V gives rise

to a network, where each directed edge x → y is associated with the value (A)xy whenever (A)xy ̸= 0. A transition diagram of a Markov chain is an example.

Nobuaki Obata (Tohoku University) Asymptotic Spectral Analysis SJTU, Shanghai, China, 2018.11.15–18 26 / 80

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Spectral Distributions of Graphs

and more matrices associated to graphs ...

▶ Matrices with index set V × V :

1

Adjacency matrix: A = [Axy]

2

Combinatorial Laplacian: L = D − A, where D = [δxy deg x] (degree matrix).

3

Signless Laplacian: D + A

4

Transition matrix: T = [Txy], where Txy = deg(x)−1Axy.

5

Normalized transition matrix: ˆ T = D1/2T D−1/2.

6

Random walk Laplacian: I − T = D−1L

7

Normalized Laplacian: ˆ L = D−1/2LD−1/2 = I − ˆ T

8

Distance matrix: D = [dG(x, y)]

9

Q-matrix: Q = [qd(x,y)] ▶ Other matrices with index set V × E: incidence matrix, oriented incidence matrix (coboundary matrix), ...

Nobuaki Obata (Tohoku University) Asymptotic Spectral Analysis SJTU, Shanghai, China, 2018.11.15–18 27 / 80

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Method of Quantum Decomposition

2. Method of Quantum Decomposition

[Chapters 4–6]

Nobuaki Obata (Tohoku University) Asymptotic Spectral Analysis SJTU, Shanghai, China, 2018.11.15–18 28 / 80

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Method of Quantum Decomposition

2.1. Fock Spaces Associated to Graphs — Stratification

1

Fix an origin o ∈ V of G = (V, E).

2

Stratification (Distance Partition) V =

∞

∪

n=0

Vn , Vn = {x ∈ V ; d(o, x) = n}

V

n+1

Vn Vn-1 V1 V0

n+1 n n-1 1

(G):

V :

3

Associated Hilbert space Γ(G) ⊂ ℓ2(V ) Φn = |Vn|−1/2 ∑

x∈Vn

δx , Γ(G) =

∞

∑

n=0

⊕CΦn .

Nobuaki Obata (Tohoku University) Asymptotic Spectral Analysis SJTU, Shanghai, China, 2018.11.15–18 29 / 80

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Method of Quantum Decomposition

2.1. Fock Spaces Associated to Graphs — Quantum Decomposition

( ) ( ) Vn+1 Vn Vn1

+

y

x y y

A A A

yx

( )

yx yx

4 Quantum decomposition A = A+ + A− + A◦, (A+)∗ = A−, (A◦)∗ = A◦. 5 Γ(G) is not necessarily invariant under the actions of Aϵ. ▶ Cases we have studied: (Case 1) Γ(G) is invariant (typically, distance-regular graphs). (Case 2) Γ(G) is asymptotically invariant.

Nobuaki Obata (Tohoku University) Asymptotic Spectral Analysis SJTU, Shanghai, China, 2018.11.15–18 30 / 80

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Method of Quantum Decomposition

2.1. Fock Spaces Associated to Graphs — Invariance 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

Theorem

If Γ(G) is invariant under A+, A−, A◦, there exist a pair of sequences {αn} and {ωn} such that A+Φn = √ ωn+1 Φn+1, A−Φn = √ ωn Φn−1, A◦Φn = αn+1Φn. In other words, (Γ(G), {Φn}, A+, A−, A◦) is an interacting Fock space (IFS).

Nobuaki Obata (Tohoku University) Asymptotic Spectral Analysis SJTU, Shanghai, China, 2018.11.15–18 31 / 80

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Method of Quantum Decomposition

2.2. Interacting Fock Spaces — Vacuum Distributions Definition (Motivated by QED models in physics (Accardi))

(Γ, {Φn}, A+, A−, A◦) is called an interacting Fock space (IFS) associated with Jacobi coefficients ({ωn}, {αn}) if {Φn} is a CONS of a Hilbert space Γ and A+Φn = √ ωn+1 Φn+1, A−Φn = √ ωn Φn−1, A◦Φn = αn+1Φn. We call A+, A− and A◦ the creation, annihilation and conservation operators, respectively.

n+1

n

n

n+1

n

n+1 n

like a birth-and-death process....

Nobuaki Obata (Tohoku University) Asymptotic Spectral Analysis SJTU, Shanghai, China, 2018.11.15–18 32 / 80

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Method of Quantum Decomposition

2.2. Interacting Fock Spaces — Vacuum Distributions Definition (Motivated by QED models in physics (Accardi))

(Γ, {Φn}, A+, A−, A◦) is called an interacting Fock space (IFS) associated with Jacobi coefficients ({ωn}, {αn}) if if {Φn} is a CONS of a Hilbert space Γ and A+Φn = √ ωn+1 Φn+1, A−Φn = √ ωn Φn−1, A◦Φn = αn+1Φn. We call A+, A− and A◦ the creation, annihilation and conservation operators, respectively. ▶ Vacuum spectral distribution is the probability distribution µ characterized by ⟨Φ0, (A+ + A− + A◦)mΦ0⟩ = ∫ +∞

−∞

xmµ(dx), m = 1, 2, . . . .

1

(Boson Fock space) A−A+ − A+A− = I ⇒ µ ∼ N(0, 1)

2

(Free Fock space) A−A+ = I ⇒ µ(dx) = 1 2π √ 4 − x2 dx (semi-circle law)

3

(Fermion Fock space) A−A+ + A+A− = I ⇒ µ = 1 2δ−1 + 1 2δ+1

Nobuaki Obata (Tohoku University) Asymptotic Spectral Analysis SJTU, Shanghai, China, 2018.11.15–18 33 / 80

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Method of Quantum Decomposition

2.2. Interacting Fock Spaces — Orthogonal Polynomials

µ(dx): a probability distribution with finite moments of all orders Define an inner product by ⟨f, g⟩ = ∫ +∞

−∞

f(x)g(x)µ(dx), f, g ∈ L2(R, µ; R).

Definition (Orthogonal polynomials)

Applying the Gram-Schmidt orthogonalization to 1, x, x2, . . . , xn, . . . we obtain a sequence of polynomials: P0(x) = 1, P1(x) = x− ⟨x, P0⟩ ⟨P0, P0⟩ P0(x), Pn(x) = xn −

n−1

∑

k=0

⟨xn, Pk⟩ ⟨Pk, Pk⟩ Pk(x). We call {Pn(x)} the orthogonal polynomials associated to µ. Note: The orthogonalization process stops at n = d if ⟨Pd, Pd⟩ = 0 happens. In that case we consider {P0(x), P1(x), . . . , Pd−1(x)} as the orthogonal polynomials. That happens if and only if |supp µ| = d.

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Method of Quantum Decomposition

2.2. Interacting Fock Spaces — Jacobi Parameters Theorem (Three-term recurrence relation — exercise)

There exist two sequences ω1, ω2, · · · > 0 and α1, α2, · · · ∈ R, called Jacobi parameters, such that P0 = 1, P1 = x − α1, xPn = Pn+1 + αn+1Pn + ωnPn−1 .

Theorem (Cauchy–Stieltjes transform)

If µ is a unique solution to the determinate moment problem, e.g., if ∑∞

n=1 ω−1/2 n

= ∞ (Carleman’s test), we have 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}.

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Method of Quantum Decomposition

2.2. Interacting Fock Spaces — Inversion Formula

({ωn}, {αn}): Jacobi parameters ▶ 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 2{F (t) + F (t − 0)} − 1 2{F (s) + F (s − 0)} = − 1 π lim

y→+0

∫ t

s

Im Gµ(x + iy)dx, s < t, ρ(x) = − 1 π lim

y→+0 Im Gµ(x + iy)

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Method of Quantum Decomposition

2.2. Interacting Fock Spaces — Calculating the Spectral Distribution Theorem

Let (Γ(G), A+, A−, A◦) be an interacting Fock space given by A+Φn = √ ωn+1 Φn+1, A−Φn = √ ωn Φn−1, A◦Φn = αn+1Φn. Then the vacuum spectral distribution of A = A+ + A◦ + A− is a probability distribution µ of which the orthogonal polynomials are given by P0 = 1, P1 = x − α1, xPn = Pn+1 + αn+1Pn + ωnPn−1 . Namely, we have ⟨Φ0, AmΦ0⟩ = ⟨Φ0, (A+ + A◦ + A−)mΦ0⟩ = ∫ +∞

Method of Quantum Decomposition

(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) (5) Wigner’s semicircle law (free CLT) lim

κ→∞

√κ ρκ(√κ x) = 1 2π √ 4 − x2

κ = 4 κ = 8 κ = 12

0.1 2 4 6

2
4
6

ρ4, ρ8, ρ12

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Method of Quantum Decomposition

2.3. Calculating the Vacuum Spectral Distribution — Spidernets

S(4, 6, 3) S(a, b, c) deg(x) =    a x = o (origin) b x ̸= o        ω−(x) = 1 ω+(x) = c ω◦(x) = b − 1 − c for x ̸= o. ▶ The spectral distribution is given by the free Meixner law.

Definition (Free Meixner law with parameters p > 0, q ≥ 0, a ∈ R)

∫ +∞

−∞

µ(dx) z − x = 1 z − p z − a − q z − a − q z − a − q z − a − · · · We know that µ = (absolutely continuous part) + (at most 2 point masses).

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Method of Quantum Decomposition

2.3. Calculating the Vacuum Spectral Distribution — DRGs Definition

A graph G = (V, E) is called distance regular if the intersection numbers: pk

i,j = |{z ∈ V ; d(x, z) = i, d(y, z) = j}|,

is constant for all pairs x, y such that d(x, y) = k. Examples: Hamming graphs, Johnson graphs, odd graphs, homogeneous trees, ...

Theorem

Let G be a distance-regular graph with a fixed origin o ∈ V , and A = A+ + A− + A◦ the quantum decomposition. Then A+Φn = √ ωn+1 Φn+1, A−Φn = √ ωn Φn−1, A◦Φn = αn+1Φn, where ωn = pn

1,n−1pn−1 1,n

and αn = pn−1

1,n−1.

Remark: For a finite distance-regular graph we have ⟨Am⟩tr = ⟨Am⟩o = ⟨δo, Amδo⟩, m = 1, 2, . . . .

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Asymptotic Spectral Analysis of Growing Regular Graphs

3. Asymptotic Spectral Analysis of Growing Regular Graphs

[Chapters 4–6]

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Asymptotic Spectral Analysis of Growing Regular Graphs

Main Theme: Asymptotic Spectral Analysis of Growing Graphs

▶ Spectral analysis of graphs A Axy [ ] G = V, E ( ) dx = f x dx ( ) ( ) ▶ Growing graphs ▶ Asymptotics of the spectral distributions µn as n → ∞

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Asymptotic Spectral Analysis of Growing Regular Graphs

3.1. Growing Family of DRGs — An Example: H(d, N)

H(d, N) = KN × · · · × KN (d times): Hamming graph p0

1,1 = deg H(d, N) = d(N − 1),

pn

1,n−1 = n,

pn−1

1,n

= (d − n)(N − 1), pn−1

1,n−1 = (n − 1)(N − 2).

Theorem

Let µd,N be the vacuum spectral distribution of H(d, N) (in coincidence with the eigenvalue distribution). Then the Jacobi parameters of µd,N are given by ωn = pn

1,n−1pn−1 1,n

= n(d − n + 1)(N − 1), 1 ≤ n ≤ d, αn = pn−1

1,n−1 = (n − 1)(N − 2),

1 ≤ n ≤ d + 1. The IFS structure: A+Φn = √ ωn+1 Φn+1 = √ (n + 1)(d − n)(N − 1) Φn+1, A−Φn = √ ωn Φn−1 = √ n(d − n + 1)(N − 1) Φn−1, A◦Φn = αn+1 Φn = n(N − 2)Φn, ▶ In fact, the vacuum distribution of A is the binomial distribution.

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Asymptotic Spectral Analysis of Growing Regular Graphs

3.1. Growing Family of DRGs — An Example: H(d, N) (cont)

A+Φn = √ ωn+1 Φn+1 = √ (n + 1)(d − n)(N − 1) Φn+1, A−Φn = √ ωn Φn−1 = √ n(d − n + 1)(N − 1) Φn−1, A◦Φn = αn+1 Φn = n(N − 2)Φn, ▶ What happens when N → ∞ and d → ∞? ▶ Normalization by mean(A) = ⟨A⟩ = 0 and Σ2(A) = ⟨A2⟩ = d(N − 1). A+ √ d(N − 1) Φn = √ (n + 1) ( 1 − n d ) Φn+1, A− √ d(N − 1) Φn = √ n ( 1 − n − 1 d ) Φn−1, A◦ √ d(N − 1) Φn = n √ N − 2 d √ N − 2 N − 1 Φn, ▶ Proper scaling limit: N → ∞, d → ∞, N d → τ ≥ 0.

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Asymptotic Spectral Analysis of Growing Regular Graphs

3.1. Growing Family of DRGs — An Example: H(d, N) (cont)

▶ Taking the limit as N → ∞, d → ∞ and N d → τ ≥ 0, we have A+ √ d(N − 1) Φn = √ (n + 1) ( 1 − n d ) Φn+1 → √ n + 1 “Φn+1” , A− √ d(N − 1) Φn = √ n ( 1 − n − 1 d ) Φn−1 → √ n “Φn−1” , A◦ √ d(N − 1) Φn = n √ N − 2 d √ N − 2 N − 1 Φn → n √ τ “Φn”. ▶ Recall the Boson Fock space (Γ, {Ψn}, B+, B−) is defined by B+Ψn = √ n + 1 Ψn+1, B−Ψn = √ n Ψn−1. ▶ Note also that B+B−Ψn = nΨn .

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Asymptotic Spectral Analysis of Growing Regular Graphs

3.1. Growing Family of DRGs — QCLT for H(d, N) Theorem (Quantum central limit theorem for H(d, N))

( A+ √ d(N − 1) , A− √ d(N − 1) , A◦ √ d(N − 1) )

m

− → (B+, B−, √ τ B+B−), as N → ∞, d → ∞ and N d → τ ≥ 0. In particular, ⟨ δo, ( A √ d(N − 1) )m δo ⟩ → ⟨ Ψ0, (B+ + B− + √ τ B+B−)mΨ0 ⟩ . ▶ By observing moments or generating functions, we know ⟨ Ψ0, (B+ + B− + √ τ B+B−)mΨm ⟩ = ∫ +∞

−∞

xmµ(dx), where µ =    N(0, 1), τ = 0, affine transformed Po(τ −1), τ > 0. ▶ This is the asymptotic spectral (= eigenvalue) distribution of H(d, N).

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Asymptotic Spectral Analysis of Growing Regular Graphs

3.1. Growing Family of DRGs — General Results

{Gν}: a growing family of drg’s with adjacency matrices Aν Normalization: ⟨A⟩ = ⟨δo, Aδo⟩ = 0 and Σ2(A) = ⟨A2⟩ = deg(o) = p0

11.

Aν − ⟨Aν⟩ Σν = A+

ν

Σν + A◦

ν

Σν + A−

ν

Σν .

Theorem (Quantum CLT for growing family of DRGs)

Assume that for all n = 1, 2, . . . the limits ωn = lim

ν

pn

1,n−1(ν)pn−1 1,n (ν)

p0

1,1(ν)

, αn = lim

ν

pn−1

1,n−1(ν)

√ p0

1,1(ν)

,

exist. Let (Γ, {Φn}, B+, B−, B◦) be the interacting Fock space associated with

({ωn}, {αn}). Then we have (A+

ν

Zν , A−

ν

Zν , A◦

ν

Zν )

m

− → (B+, B−, B◦).

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Asymptotic Spectral Analysis of Growing Regular Graphs

3.2. Case of Asymptotic Invariance

V

n+1

Vn Vn-1 V1 V0 ▶ Fock space associated to G = (V, E): Φn = |Vn|−1/2 ∑

x∈Vn

δx , Γ(G) =

∞

∑

n=0

⊕CΦn . ▶ Quantum decomposition: A = A+ + A− + A◦ (Case 1) Γ(G) is invariant under the actions of A+, A−, A◦. — typically, distance-regular graphs (Case 2) Γ(G) is asymptotically invariant.

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Asymptotic Spectral Analysis of Growing Regular Graphs

3.2. Case of Asymptotic Invariance — An Example ZN as N → ∞

1

Γ(ZN) is asymptotically invariant 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

N

Σ(AN) = Aϵ

N

√ 2N → Bϵ

3

The interacting Fock space in the limit: B+Ψn = √ n + 1 Ψn+1, B−Φn = √ n Ψn−1, B◦ = 0. This is Boson Fock space!

4

The asymptotic spectral distribution is the standard Gaussian distribution: lim

N→∞

⟨ Φ0, ( AN √ 2N )m Φ0 ⟩ = ⟨Ψ0, (B+ + B−)mΨ0⟩ = 1 √ 2π ∫ +∞

−∞

xme−x2/2dx.

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Asymptotic Spectral Analysis of Growing Regular Graphs

3.2. Case of Asymptotic Invariance — Growing Regular Graphs

▶ Γ(G) is not necessarily invariant but asymptotically invariant under Aϵ. 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) }2 L(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, . . . ,

√ κ(ν) , A−

ν

√ κ(ν) , A◦

ν

√ κ(ν) )

m

− → (B+, B−, B◦) In particular, the asymptotic spectral distribution of the normalized Aν in the vacuum state is a probability distribution determined by ({ωn}, {αn}).

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Asymptotic Spectral Analysis of Growing Regular Graphs

Some concrete examples: 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)

r Poisson

Johnson graphs ωn = n2

exponential (2d/v → 1)

‘Poissonization’ of J(v, d)

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

dd 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 (free Meixner law) S(a, b, c)

ω2 = · · · = q

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Graph Products and Concepts of Independence

4. Graph Products and Concepts of Independence

[Chapter 7]

▶ A binary operation or a product of graphs:

#

G G

1 2

G1 G2 #

(G1, G2) → Φ(G1, G2) = G1#G2 (A1, A2) → Φ(A1, A2) = A[G1#G2] (µ1, µ2) → Φ(µ1, µ2) = µ1#µ2 (convolution)

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Graph Products and Concepts of Independence

Summary: Graph Products and Convolutions

products G1#G2 A[G1#G2] spectral distribution Cartesian G1 ×C G2 A1 ⊗ I2 + I1 ⊗ A2 µ1 ∗ µ2 monotone G1 ▷ G2 A1 ⊗ P2 + I2 ⊗ A2 µ1 ▷ µ2 star G1 ⋆ G2 A1 ⊗ P2 + P1 ⊗ A2 µ1 ⊎ µ2 lexicographic G1 ▷L G2 A1 ⊗ J2 + P1 ⊗ A2 D(µ1) ▷ µ2 Kronecker G1 ×K G2 A1 ⊗ A2 µ1 ∗M µ2 strong G1 ×S G2 A1 ⊗ I2 + I1 ⊗ A2 +A1 ⊗ A2 S−1(Sµ1 ∗M Sµ2) free G1 ∗ G2 A1 ∗ A2 µ1 ⊞ µ2

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Graph Products and Concepts of Independence

4.1. Graph Products — Cartesian Product Definition

Let G1 = (V1, E1) and G2 = (V2, E2) be two graphs. The Cartesian product or direct product of G1 and G2, denoted by G1 × G2, is a graph on V = V1 × V2 with adjacency relation: (x, y) ∼ (x′, y′) ⇐ ⇒    x = x′ y ∼ y′

r

   x ∼ x′ y = y′.

Example (C4 × C3)

(2,1’ 1 2 3 4 1’ 2’ 3’ (1,3’ (1,2’ (3,1’ (4,1’

C C C × C

3 4 3 4

(1,1’

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Graph Products and Concepts of Independence

4.1. Graph Products — Comb Product Definition

Let G1 = (V1, E1) and G2 = (V2, E2) be two graphs. We fix a vertex 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 ▷o2 C3 with o2 = 1′)

(1,1’ (2,1’ 1 2 3 4 1’ 2’ 3’ (1,3’ (1,2’ (3,1’ (4,1’

C C C C

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Graph Products and Concepts of Independence

4.1. Graph Products — Star Product Definition

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

1 ∈ 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 3 4 1’ 2’ 3’ (1,3’ (1,2’ (3,1’ (4,1’

C C C C

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Graph Products and Concepts of Independence

4.1. Graph Products — Adjacency Matrices

G1 = (V1, E1), G2 = (V2, E2): two graphs G = G1#G2: a graph product and assume that V [G] = V1 × V2 Ai: adjacency matrices of Gi acting on ℓ2(Vi), (i = 1, 2) = ⇒ A = A[G1#G2] acts on ℓ2(V ) = ℓ2(V1 × V2) ∼ = ℓ2(V1) ⊗ ℓ2(V2).

Theorem

1

[Cartesian product] A = A1 ⊗ I2 + I1 ⊗ A2 .

2

[comb product] A = A1 ⊗ P2 + I1 ⊗ A2 .

3

[star product] A = A1 ⊗ P2 + P1 ⊗ A2 . Here, Pi is the rank one projection corresponding to oi.

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Graph Products and Concepts of Independence

4.2. Independence — Factorization Rule for Mixed Moments

▶ If two random variables X, Y are independent (in the classical sense), then E[XY ] = E[X]E[Y ]. Hence, E[

X appears m times Y appears n times

X · · · X · · · Y · · · Y · · · X] = E[XmY n].

▶ Let a = a∗, b = b∗ be two algebraic random variables in (A, φ). Consider φ(

a appears m times b appears n times

a · · · b · · · b · · · a · · · a) =???

▶ There are many variants of factorization rules. = ⇒ Many concepts of independence in quantum probability.

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Graph Products and Concepts of Independence

4.2. Independence — Adjacency Matrices of Graph Products

▶ Let φi be the vacuum state at oi and consider the product state φ = φ1 ⊗ φ2. = ⇒ A = A[G1#G2] is a random variable in (A(G1#G2), φ).

Theorem

1

[Cartesian product] A = A1 ⊗ I2 + I1 ⊗ A2 is a sum of commutative independent random variables.

2

[comb product] A = A1 ⊗ P2 + I1 ⊗ A2 is a sum of monotone independent random variables.

3

[star product] A = A1 ⊗ P2 + P1 ⊗ A2 is a sum of Boolean independent random variables.

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Graph Products and Concepts of Independence

4.2. Independence — Factorization Rule for Mixed Moments (cont)

▶ Illustration by φ(a2a1a4a3a4a3a6a6a4a4a3a5) = φ(214343664435)

1

[commutative independence] φ(214343664435) = φ(1)φ(2)φ(33)φ(44)φ(5)φ(62)

2

[monotone independence]

2 1 3 1 3 3 4 4 6 6 4 4 3 3 3 4 4 3 3 3 1 5

φ(214343664435) = φ(2)φ(4)φ(4)φ(66)φ(133443)φ(5) = φ(2)φ(4)φ(4)φ(66)φ(44)φ(1333)φ(5) = φ(2)φ(4)φ(4)φ(66)φ(44)φ(333)φ(1)

3

[Boolean independence] φ(214343664435) = φ(2)φ(1)φ(4)φ(3)φ(4)φ(3)φ(66)φ(44)φ(3)φ(5)

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Graph Products and Concepts of Independence

4.3. Quantum Central Limit Theorems

Let {an} be real random variables in (A, φ) such that φ(an) = 0, φ(a2

n) = 1.

▶ The CLT gives the limit distribution µ appearing in the following φ [( 1 √n

n

∑

k=1

ak )m] → ∫ +∞

−∞

xmµ(dx).

Theorem (QCLT)

1

[commutative CLT] If a1, a2, . . . are commutative independent, we have µ ∼ N(0, 1).

2

[monotone CLT] If a1, a2, . . . are monotone independent, we have µ(dx) = dx π √ 2 − x2 (normalized arcsine law)

3

[Boolean CLT] If a1, a2, . . . are Boolean independent, we have µ = 1 2 δ+1 + 1 2 δ−1 (normalized Bernoulli distribution)

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Graph Products and Concepts of Independence

4.3. QCLTs — Applications to Growing Graphs G#n as n → ∞ Theorem (CLT for Cartesian product graphs)

For the n-fold Cartesian product G(n) = G × · · · × G (n-times), lim

n→∞

⟨( A(n) √n √ deg(o) )m⟩ = ∫ +∞

−∞

xm 1 √ 2π e−x2/2dx.

Theorem (CLT for comb product graphs)

For the n-fold monotone product graph G(n) = G ▷o G ▷o · · · ▷o G (n-times), lim

n→∞

⟨( A(n) √n √ deg(o) )m⟩ = ∫ +

√ 2 − √ 2

xm dx π √ 2 − x2 , m = 1, 2, . . . .

Theorem (CLT for star product graphs)

For the n-fold star product graph G(n) = G ⋆ G ⋆ · · · ⋆ G (n-times) we have lim

n→∞

⟨( A(n) √n √ deg(o) )m⟩ = ∫ +∞

−∞

xm 1 2(δ−1 + δ+1)(dx), m = 1, 2, . . . .

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4.4. Kronecker Product — Application to Counting Walks

products G1#G2 A[G1#G2] spectral distribution Cartesian G1 ×C G2 A1 ⊗ I2 + I1 ⊗ A2 µ1 ∗ µ2 monotone G1 ▷ G2 A1 ⊗ P2 + I2 ⊗ A2 µ1 ▷ µ2 star G1 ⋆ G2 A1 ⊗ P2 + P1 ⊗ A2 µ1 ⊎ µ2 lexicographic G1 ▷L G2 A1 ⊗ J2 + P1 ⊗ A2 D(µ1) ▷ µ2 Kronecker G1 ×K G2 A1 ⊗ A2 µ1 ∗M µ2 strong G1 ×S G2 A1 ⊗ I2 + I1 ⊗ A2 +A1 ⊗ A2 S−1(Sµ1 ∗M Sµ2) free G1 ∗ G2 A1 ∗ A2 µ1 ⊞ µ2

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Graph Products and Concepts of Independence

4.4. Kronecker Product — Definition Definition (Kronecker product)

Let G1 = (V1, E1) and G2 = (V2, E2) be graphs. The Kronecker product G1 ×K G2 is a graph on V = V1 × V2 with the adjacency relation: (x, y) ∼K (x′, y′) ⇐ ⇒ x ∼ x′, y ∼ y′. In other words, the adjacency matrix A = A[G1 ×K G2] is given by A = A1 ⊗ A2 .

1

If |V1| ≥ 2 and |V2| ≥ 2, then G1 ×K G2 has at most two connected components.

2

Let P1 = K1 (single-vertex graph). Then for any graph G = (V, E) the Kronecker product P1 ×K G is a graph on V with no edges, i.e., an empty graph on V .

3

G1 ×K G2 is a subgraph (not necessarily induced subgraph) of the distance-2 graph of G1 ×C G2.

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4.4. Kronecker Product — Examples

P3 ×K P3 P4 ×K P5 C4 ×K P5 C5 ×K P5

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4.4. Kronecker Product — Spectral Distribution by Mellin Convolution Theorem

For i = 1, 2 let Gi = (Vi, Ei) be a graph with a distinguished vertex oi. Let µi be the spectral distribution of the adjacency matrix Ai of Gi in the vector state at oi. Assume that µi is symmetric. Then, for their spectral distributions we have µ(G1 ×K G2) = µ1 ∗M µ2. Mellin convolution

1

For symmetric probability distributions µ, ν on R we define ∫

R

h(x)µ ∗M ν(dx) = ∫

R

∫

R

h(xy)µ(dx)ν(dy), h ∈ Cb(R).

2

If µ(dx) = f(x)dx and ν(dx) = g(x)dx with symmetric density functions, then µ ∗M ν admits a symmetric density function 2f ⋆ g(x), where f ⋆ g(x) = ∫ ∞ f(y)g (x y )dy y = ∫ ∞ f (x y ) g(y)dy y , x > 0.

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4.5. Restricted Lattices

▶ Z ×K Z: a graph on Z2 = {(u, v) ; u, v ∈ Z} with adjacency relation: (u, v) ∼K (u′, v′) ⇐ ⇒ u′ = u ± 1 and v′ = v ± 1. ▶ Z ×C Z (2d interger lattice): a graph on Z2 with adjacency relation: (x, y) ∼ (x′, y′) ⇐ ⇒    x′ = x ± 1, y′ = y,

r

   x′ = x, y′ = y ± 1.

u v x y

1

Z ×K Z has two connected components, each of which is isomorphic to Z ×C Z.

2

Let (Z ×K Z)O denote the connected component of Z ×K Z containing O = (0, 0). Then (Z ×K Z)O ∼ = Z ×C Z.

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4.5. Restricted Lattices — As Kronecker Products

For a subset D ⊂ Z2 let L[D] denote the lattice restricted to D, i.e., the induced subgraph of Z ×C Z spanned by the vertices in D.

1

L{(x, y) ∈ Z2 ; x ≥ y ≥ x − (n − 1)} ∼ = (Pn ×K Z)O for n ≥ 2.

2

L{(x, y) ∈ Z2 ; x ≥ y} ∼ = (Z+ ×K Z)O. (P5 ×K Z)O (Z+ ×K Z)O

x y u v x y u v

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4.5. Restricted Lattices — As Kronecker Products (cont)

1

L { (x, y) ∈ Z2 ; 0 ≤ x + y ≤ m − 1, 0 ≤ x − y ≤ n − 1 } ∼ = (Pm ×K Pn)O for m ≥ 2 and n ≥ 2.

2

L{(x, y) ∈ Z2 ; x ≥ y ≥ −x} ∼ = (Z+ ×K Z+)O,

u v x y

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4.5. Restricted Lattices — Counting Walks

Wk(o; G) = |{o → o : k-step walk}|

1

Z.

O

W2m(0; Z) = ( 2m m ) , W2m+1(0; Z) = 0.

2

Z+ = {0, 1, 2, . . . }.

O

W2m(0; Z+) = Cm = 1 m + 1 ( 2m m ) , W2m+1(0; Z+) = 0, where Cm is the renown Catalan number.

Nobuaki Obata (Tohoku University) Asymptotic Spectral Analysis SJTU, Shanghai, China, 2018.11.15–18 73 / 80

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Graph Products and Concepts of Independence

4.5. Restricted Lattices — Counting Walks (cont)

1

L = L{(x, y) ∈ Z2 ; x ≥ y}. Since L ∼ = (Z+ ×K Z)O, Wm((0, 0); L) = Wm((0, 0); Z+ ×K Z) = Wm(0; Z+)Wm(0; Z) = Cm ( 2m m ) = 1 m + 1 ( 2m m )2 .

2

L = L{(x, y) ∈ Z2 ; x ≥ y ≥ −x}. Since L ∼ = (Z+ ×K Z+)O, Wm((0, 0); L) = Wm((0, 0); Z+ ×K Z+) = Wm(0; Z+)Wm(0; Z+) = C2

m =

1 (m + 1)2 ( 2m m )2 .

Nobuaki Obata (Tohoku University) Asymptotic Spectral Analysis SJTU, Shanghai, China, 2018.11.15–18 74 / 80

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4.5. Restricted Lattices — Spectral Distributions Theorem

For m ≥ 0 we have Wm(0; Z) = Mm(α), Wm(0; Z+) = Mm(w).

1

Arcsine distribution: α(x) = 1 π√4 − x2 1(−2,2)(x), x ∈ R, The moments of even orders are given by M2m(α) = ∫ +∞

−∞

x2mα(x) dx = ( 2m m ) , m ≥ 0.

2

Semi-circle distribution: w(x) = 1 2π √ 4 − x2 1[−2,2](x), x ∈ R, The moments of even orders are given by M2m(w) = ∫ +∞

−∞

x2mw(x) dx = Cm = 1 m + 1 ( 2m m ) , m ≥ 0.

Nobuaki Obata (Tohoku University) Asymptotic Spectral Analysis SJTU, Shanghai, China, 2018.11.15–18 75 / 80

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4.5. Restricted Lattices — Spectral Distributions (cont)

Domain D W2m(L[D], O) spectral distribution Z (2m

m

) α Z+ Cm w Z2 (2m

m

)2 α ∗ α = α ∗M α {x ≥ y} Cm (2m

m

) w ∗M α {x ≥ y ≥ −x} C2

m

w ∗M w {x ≥ 0, y ≥ 0} (A) w ∗ w {x ≥ y ≥ x − (n − 1)} (B) πn ∗M α { 0 ≤ x + y ≤ k − 1, 0 ≤ x − y ≤ l − 1 } (C) πk ∗M πl

(A) =

m

∑

k=0

(2m 2k ) CkCm−k, (B) = W2m(Pn, 0) (2m m ) , (C) = W2m(Pk, 0)W2m(Pl, 0).

Nobuaki Obata (Tohoku University) Asymptotic Spectral Analysis SJTU, Shanghai, China, 2018.11.15–18 76 / 80

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4.5. Restricted Lattices — Density Functions

Elliptic integrals For k2 < 1, the elliptic integrals are defined by K(k) = ∫ π/2 dθ √ 1 − k2 sin2 θ = ∫ 1 dx √ (1 − x2)(1 − k2x2) , E(k) = ∫ π/2 √ 1 − k2 sin2 θ dθ = ∫ 1 √ 1 − k2x2 1 − x2 dx.

1

The density function of w ∗M α is given by 1 π2 {K(ξ(x)) − E(ξ(x))}, ξ(x) = √ 1 − x2 16 , −4 ≤ x ≤ 4.

2

The density function of α ∗M α = α ∗ α is given by 1 2π2 K(ξ(x)), − 4 ≤ x ≤ 4.

3

The density function of w ∗M w is given by 2 π2 {( 1 + x2 16 ) K(ξ(x)) − 2E(ξ(x)) } , − 4 ≤ x ≤ 4.

Nobuaki Obata (Tohoku University) Asymptotic Spectral Analysis SJTU, Shanghai, China, 2018.11.15–18 77 / 80

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4.5. Restricted Lattices — Density Functions (cont)

w ∗M α

2 4 0.5 1.0 − 2 − 4

α ∗M α w ∗M w

0.2 0.4 2 4 − 2 − 4 0.5 1.0 2 4 − 2 − 4 Nobuaki Obata (Tohoku University) Asymptotic Spectral Analysis SJTU, Shanghai, China, 2018.11.15–18 78 / 80

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Summary and Perspectives

Summary

adjacency matrix combinatorics quantum decomposition quantum CLT interacting Fock space

Quantum Classical

quantum CLT use of independence

rthogonal polynomials

spectral distribution scaling limit asymptotic spectral distribution µ µν Aν Aν Aν Aν Aν Aν Aν Zν

ν ν

X X B

ε ε

Pn

n n

B B α ω α

Γ (

n

α

) B

B linear algebra

Nobuaki Obata (Tohoku University) Asymptotic Spectral Analysis SJTU, Shanghai, China, 2018.11.15–18 79 / 80

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Summary and Perspectives

Perspectives

1

Explore a bivariate extension of the method of quantum decomposition. — Closely related to orthogonal polynomials in two variables. — Trials: Strongly regular graphs, Association schemes [Morales–Obata–Tanaka arXiv:1809.03761]

2

Formulate {A(n)} as a discrete-time algebraic stochastic process such as A(n+1) = Φ(A(n), B(n))

r

A(n+1) = A(n)“+”B(n), where B(n) has special relation to A(n), i.e., a kind of independence (independent increment). And derive asymptotic spectral distribution from associated CLT.

Nobuaki Obata (Tohoku University) Asymptotic Spectral Analysis SJTU, Shanghai, China, 2018.11.15–18 80 / 80