Quantum Probability and Asymptotic Spectral Analysis of Growing Roma, GraphsNovember 14, / 47

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1 Quantum Probability and Asymptotic Spectral Analysis of Growing Graphs Nobuaki Obata GSIS, Tohoku University Roma, November 14, 2013 Quantum Probability and Asymptotic Spectral Analysis of Growing Roma, GraphsNovember 14, / 47

2 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 Quantum Probability and Asymptotic Spectral Analysis of Growing Roma, GraphsNovember 14, / 47

3 1 Asymptotic Spectral Analysis of Graphs Quantum Probability and Asymptotic Spectral Analysis of Growing Roma, GraphsNovember 14, / 47

4 11 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 K 5 2-dim lattice homogeneous tree star graph T 4 Definition (adjacency matrix) The adjacency matrix of a graph G = (V, E) is defined by 1, x y, A = [A xy ] x,y V A xy = 0, otherwise The adjacency matrix possesses all the information of a graph Quantum Probability and Asymptotic Spectral Analysis of Growing Roma, GraphsNovember 14, / 47

5 12 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 ( ) λ i Spec (G) = m i The spectral (eigenvalue) distribution of G is defined by µ G = 1 V m iδ λi 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 Cvetković, M Doob and H Sachs: Spectra of Graphs, Academic Press, 1979 i Quantum Probability and Asymptotic Spectral Analysis of Growing Roma, GraphsNovember 14, / 47

6 13 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 A xyf(y), f C 0(V ) l 2 (V ); 2 a random variable in a suitable framework of probability theory Quantum Probability and Asymptotic Spectral Analysis of Growing Roma, GraphsNovember 14, / 47

7 14 Illustration: Path Graphs P n ad n A = n 1 kπ 2 cos Spec (P n ) = n n 1 k n π 4 x Quantum Probability and Asymptotic Spectral Analysis of Growing Roma, GraphsNovember 14, / 47

8 2 Quantum Probability = Non-Commutative Probability K R Parthasarathy: An Introduction to Quantum Stochastic Calculus, Birkhäuser, 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 Quantum Probability and Asymptotic Spectral Analysis of Growing Roma, GraphsNovember 14, / 47

9 21 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: µ X = 1 2 δ δ+1 Quantum Probability and Asymptotic Spectral Analysis of Growing Roma, GraphsNovember 14, / 47

10 21 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: µ X = 1 2 δ δ+1 Moment sequence is one of the most fundamental characteristics Moment sequence {M m } Probability distributions µ For a coin-toss we have M m(µ X) = Up to determinate moment problem + x m 1, if m is even, µ X(dx) = 0, otherwise Quantum Probability and Asymptotic Spectral Analysis of Growing Roma, GraphsNovember 14, / 47

11 21 Quantum Probabilistic Model for Coin-toss (cont) Let us observe: [ ] [ ] [ ] A =, e 0 =, e 1 = It is straightforward to see that { } e 0, A m 1, if m is even, e 0 = = M m (µ X ) = 0, otherwise, + x m µ X (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) = e 0, ae 0, a A, We call A an algebraic realization of the random variable X Quantum Probability and Asymptotic Spectral Analysis of GrowingRoma, Graphs November 14, / 47

12 22 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 1 A, and a state φ : A C, ie, (i) φ is linear; (ii) φ(a a) 0; (iii) φ(1 A ) = 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(a m ) = + x m µ(dx) M m(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) Quantum Probability and Asymptotic Spectral Analysis of GrowingRoma, Graphs November 14, / 47

13 23 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] = Quantum Probability and Asymptotic Spectral Analysis of GrowingRoma, Graphs November 14, / 47

14 23 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[X 4 Y 3 ] = E[X 4 ]E[Y 3 ] Quantum Probability and Asymptotic Spectral Analysis of GrowingRoma, Graphs November 14, / 47

15 23 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[X 4 Y 3 ] = E[X 4 ]E[Y 3 ] Classical central limit theorem (classical CLT) If X 1, X 2, are independent, identically distributed, normalized (mean zero, variance one) random variables, then distribution) in the limit: ( lim P N 1 N 1 N N n=1 N n=1 X n obeys the standard normal law (Gaussian X n a ) = 1 2π a e x2 /2 dx, or equivalently, for all bounded continuous functions f, [ ( N )] lim E 1 f X n = 1 + f(x)e x2 /2 dx N N n=1 2π Quantum Probability and Asymptotic Spectral Analysis of GrowingRoma, Graphs November 14, / 47

16 Classical CLT in moment form If X 1, X 2, are independent, identically distributed, normalized (mean zero, variance one) random variables having moments of all orders, we have lim E N [( 1 N N n=1 X n ) m ] = 1 + x m e x2 /2 dx, m = 1, 2, 2π 1 For the proof the factorization rule is essential E[X 1 X 3 X 1 X 2 X 1 X 3 X 1 ] = E[X 4 1 X 2X 2 3 ] = E[X 4 1 ]E[X 2]E[X 2 3 ] 2 If X 1, X 2, X 2, are non-commutative, what can we do for E[X 1 X 3 X 1 X 2 X 1 X 3 X 1 ]? = E[X 4 1 X 2 X 2 3 ]? = E[X 4 1 ]E[X 2 ]E[X 2 3 ] 3 As a result, there are many different factorization rules various concepts of independence Quantum Probability and Asymptotic Spectral Analysis of GrowingRoma, Graphs November 14, / 47

17 24 Four Concepts of Independence and Quantum CLTs Consider an algebraic probability space and (A, φ) and a A 1, b A 2 commutative free Boolean monotone φ(aba) φ(a 2 )φ(b) φ(a 2 )φ(b) φ(a) 2 φ(b) φ(a 2 )φ(b) φ(bab) φ(a)φ(b 2 ) φ(a)φ(b 2 ) φ(a)φ(b) 2 φ(a)φ(b) 2 φ(a) 2 φ(b 2 ) φ(abab) φ(a 2 )φ(b 2 ) +φ(a 2 )φ(b) 2 φ(a) 2 φ(b) 2 φ(a 2 )φ(b) 2 φ(a) 2 φ(b) 2 CLM Gaussian Wigner Bernoulli arcsine Quantum Probability and Asymptotic Spectral Analysis of GrowingRoma, Graphs November 14, / 47

18 24 Four Concepts of Independence and Quantum CLTs Consider an algebraic probability space and (A, φ) and a A 1, b A 2 commutative free Boolean monotone φ(aba) φ(a 2 )φ(b) φ(a 2 )φ(b) φ(a) 2 φ(b) φ(a 2 )φ(b) φ(bab) φ(a)φ(b 2 ) φ(a)φ(b 2 ) φ(a)φ(b) 2 φ(a)φ(b) 2 φ(a) 2 φ(b 2 ) φ(abab) φ(a 2 )φ(b 2 ) +φ(a 2 )φ(b) 2 φ(a) 2 φ(b) 2 φ(a 2 )φ(b) 2 φ(a) 2 φ(b) 2 CLM Gaussian Wigner Bernoulli arcsine CLM (Central Limit Measure) µ is obtained from a sequence of normalized random variables {a n = a n} which are {commutative/free/boolean/monotone} independent: [( 1 lim φ N N N n=1 a n ) m ] = + x m µ(dx), m = 1, 2, Axiomatization: Ben Ghorbal Schürmann (2002), Muraki (2002), Franz (2003) Quantum Probability and Asymptotic Spectral Analysis of GrowingRoma, Graphs November 14, / 47

19 25 Monotone Independence Definition (monotone independence) Let (A, φ) be an algebraic probability space and {A n } a family of -subalgebras We say that {A n} is monotone independent if for a 1 A n1,, a m A nm we have φ(a 1 a m ) = φ(a i )φ(a 1 ǎ i a m ) (ǎ i stands for omission) holds when n i 1 < n i and n i > n i+1 happen for i {1, 2,, n} Computing φ(a 1a 2a 3 a m) where a 1 A 2, a 2 A 1, a 3 A 4, φ( ) = φ(4)φ(4)φ(66)φ( ) = 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) Quantum Probability and Asymptotic Spectral Analysis of GrowingRoma, Graphs November 14, / 47

20 Monotone Central Limit Theorem (Muraki 2001) Let (A, φ) be an algebraic probability space and let {a n } n=1 be a sequence of algebraic random variables satisfying the following conditions: (i) a n are real, ie, a n = a n; (ii) a n are normalized, ie, φ(a n ) = 0, φ(a 2 n) = 1; (iii) a n have uniformly bounded mixed moments, ie, for each m 1 there exists C m 0 such that φ(a n1 a nm ) C m for any choice of n 1,, n m (iv) {a n} are monotone independent Then, lim φ N ({ 1 N N i=1 a i } m ) = 1 π x m dx, m = 1, 2, 2 x 2 The probability measure in the right hand side is the normalized arcsine law Quantum Probability and Asymptotic Spectral Analysis of GrowingRoma, Graphs November 14, / 47

21 Monotone Central Limit Theorem (Muraki 2001) Let (A, φ) be an algebraic probability space and let {a n } n=1 be a sequence of algebraic random variables satisfying the following conditions: (i) a n are real, ie, a n = a n; (ii) a n are normalized, ie, φ(a n ) = 0, φ(a 2 n) = 1; (iii) a n have uniformly bounded mixed moments, ie, for each m 1 there exists C m 0 such that φ(a n1 a nm ) C m for any choice of n 1,, n m (iv) {a n} are monotone independent Then, lim φ N ({ 1 N N i=1 a i } m ) = 1 π x m dx, m = 1, 2, 2 x 2 The probability measure in the right hand side is the normalized arcsine law Proof is to show { lim φ 1 N N N i=1 } 2m a i = (2m)! 2 m m!m! = π 2 x m 2 x 2 dx Quantum Probability and Asymptotic Spectral Analysis of GrowingRoma, Graphs November 14, / 47

22 Arcsine Law as a Central Limit Measure for Monotone Independence Arcsine law vs normal (Gaussian) law 1 π 1 e x2 /2 2 x 2 2π Quantum Probability and Asymptotic Spectral Analysis of GrowingRoma, Graphs November 14, / 47

23 3 Quantum Probabilistic Approach to Spectral Graph Theory A Hora and N Obata: Quantum Probability and Spectral Analysis of Graphs, Springer, 2007 Quantum Probability and Asymptotic Spectral Analysis of GrowingRoma, Graphs November 14, / 47

24 31 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 R satisfying A m = + x m µ(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 + x m µ(dx), m = 1, 2, µ is called the asymptotic spectral distribution of G (ν) in the state ν Quantum Probability and Asymptotic Spectral Analysis of GrowingRoma, Graphs November 14, / 47

25 32 States on A(G) Quantum Probability and Asymptotic Spectral Analysis of GrowingRoma, Graphs November 14, / 47

26 32 States on A(G) (I) Trace For a finite graph G we set a tr = 1 1 Tr (a) = V V δ x, aδ x, a A(G) x V Then (A(G), tr) becomes an algebraic probability space The spectral distribution µ of A, determined (uniquely) by µ = 1 V A m tr = + x m µ(dx), m = 1, 2,, coincides with the eigenvalue distribution of A: ( m i δ λi, Spec (G) = Spec (A) = i λ i m i ) Quantum Probability and Asymptotic Spectral Analysis of GrowingRoma, Graphs November 14, / 47

27 (II) Vacuum State The vacuum state at o V is the vector state defined by a o = δ o, aδ o, a A(G) For the adjacency matrix A we have A m o = δ o, A m δ o = {m-step walks from o to o} = (III) Deformed Vacuum State The Q-matrix of G is defined by Q = Q q = [q (x,y) ] x,y V 1 q 1 Then the deformed vacuum state is defined by + x m µ(dx) a q = Qδ o, aδ o, a A(G) 1 q gives rise to a one-parameter deformation of vacuum states 2 In general, q is not necessarily positive (ie, 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 Q q is positive definite Quantum Probability and Asymptotic Spectral Analysis of GrowingRoma, Graphs November 14, / 47

28 4 Graph Products and Concepts of Independence Quantum Probability and Asymptotic Spectral Analysis of GrowingRoma, Graphs November 14, / 47

29 41 Independence and Graph Structures (I) Cartesian product Definition Let G 1 = (V 1, E 1) and G 2 = (V 2, E 2) be two graphs For (x, y), (x, y ) V 1 V 2 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 V 1 V 2 becomes a graph in such a way that (x, y), (x, y ) V 1 V 2 are adjacent if (x, y) (x, y ) This graph is called the Cartesian product or direct product of G 1 and G 2, and is denoted by G 1 G 2 Example (C 4 C 3 ) (1,3 (1, (1,1 (4,1 C C C C (2,1 (3,1 Quantum Probability and Asymptotic Spectral Analysis of GrowingRoma, Graphs November 14, / 47

30 Theorem Let G = G 1 G 2 be the cartesian product of two graphs G 1 and G 2 Then the adjacency matrix of G admits a decomposition: A = A 1 I 2 + I 1 A 2 as an operator acting on l 2 (V ) = l 2 (V 1 V 2 ) = l 2 (V 1 ) l 2 (V 2 ) 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) = K N K N A d,n = n {}}{{}}{ I I B I I i=1 i 1 (d-fold cartesian power of complete graphs) where B is the adjacency matrix of K N Hora (1998) obtained the limit distribution by a direct calculation of eigenvalues n i Quantum Probability and Asymptotic Spectral Analysis of GrowingRoma, Graphs November 14, / 47

31 42 Independence and Graph Structures (II) Comb product Definition Let G 1 = (V 1, E 1) and G 2 = (V 2, E 2) be two graphs We fix a vertix o 2 V 2 For (x, y), (x, y ) V 1 V 2 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 = o 2 Then V 1 V 2 becomes a graph, denoted by G 1 o2 G 2, and is called the comb product or the hierarchical product Example (C 4 C 3 ) (1,3 (1,2 (4, (1,1 C C C C (2,1 (3,1 Quantum Probability and Asymptotic Spectral Analysis of GrowingRoma, Graphs November 14, / 47

32 Theorem As an operator on C 0(V 1) C 0(V 2) the adjacency matrix of G 1 o2 G 2 is given by A = A 1 P 2 + I 1 A 2 where P 2 : C 0(V 2) C 0(V 2) is the projection onto the space spanned by δ o2 and I 1 is the identity matrix acting on C 0 (V 1 ) 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 (i 1) A (i) P (i+1) P (n), i=1 i 1 where P (i) the projection from l 2 (V (i) ) onto the one-dimensional subspace spanned by δ oi Moreover, the right-hand side is a sum of monotone independent random variables with respect to ψ δ o2 δ on, where ψ is an arbitrary state on B(l 2 (V (1) )) n i Quantum Probability and Asymptotic Spectral Analysis of GrowingRoma, Graphs November 14, / 47

33 43 Independence and Graph Structures (III) Star product Definition Let G 1 = (V 1, E 1) and G 2 = (V 2, E 2) be two graphs with distinguished vertices o 1 V 1 and o 2 V 2 Define a subset of V 1 V 2 by V 1 V 2 = {(x, o 2) ; x V 1} {(o 1, y) ; y V 2} The induced subgraph of G 1 G 2 spanned by V 1 V 2 is called the star product of G 1 and G 2 (with contact vertices o 1 and o 2), and is denoted by G 1 G 2 = G 1 o1 o2 G 2 Example (C 4 C 3 ) (1,3 (1,2 (4, (1,1 C C C C (2,1 (3,1 Quantum Probability and Asymptotic Spectral Analysis of GrowingRoma, Graphs November 14, / 47

34 Theorem (O IIS(2004)) As an operator on C 0 (V 1 ) C 0 (V 2 ) the adjacency matrix of G 1 G 2 is given by A = A (1) P (2) + P (1) A (2), P (i) = Φ (i) 0 Φ (i) 0 where P (i) : C 0 (V i ) C 0 (V i ) is the projection onto the space spanned by δ oi 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 {}}{{}}{ P (1) P (i 1) A (i) P (i+1) P (n), i=1 i 1 where P (i) the projection from l 2 (V (i) ) onto the one-dimensional subspace spanned by δ oi Moreover, the right-hand side is a sum of Boolean independent random variables with respect to δ o1 δ o2 δ on n i Quantum Probability and Asymptotic Spectral Analysis of GrowingRoma, Graphs November 14, / 47

35 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 Quantum Probability and Asymptotic Spectral Analysis of GrowingRoma, Graphs November 14, / 47

36 5 Method of Quantum Decomposition Quantum Probability and Asymptotic Spectral Analysis of GrowingRoma, Graphs November 14, / 47

37 51 A Simple Prototype: Coin-toss [ ] [ ] Recall that the coin-toss is modelled by A = and e 0 = A prototype of quantum decomposition [ ] [ ] A = A + + A = e 0, A m e 0 = e 0, (A + + A ) m e 0 = ϵ 1,,ϵ m {±} e 0, A ϵm A ϵ 1 e 0 = Relations among non-commutative quantum components A ± are useful e 1 e 0 G A + A m e 0, A m 1, if m is even, e 0 = 0, otherwise Quantum Probability and Asymptotic Spectral Analysis of GrowingRoma, Graphs November 14, / 47

38 52 Quantum Decomposition of the Adjacency Matrix Fix an origin o V of G = (V, E) V n+1 Φ n+1 V n Φ n V n-1 Φ n-1 Γ(G) V 1 V 0 Φ Φ 1 0 Quantum Probability and Asymptotic Spectral Analysis of GrowingRoma, Graphs November 14, / 47

39 52 Quantum Decomposition of the Adjacency Matrix Fix an origin o V of G = (V, E) Stratification (Distance Partition) V n+1 Φ n+1 V = n=0 V n V n = {x V ; (o, x) = n} V n V n-1 Φ Φ n n-1 Γ(G) V 1 Φ 1 V 0 Φ 0 Quantum Probability and Asymptotic Spectral Analysis of GrowingRoma, Graphs November 14, / 47

40 52 Quantum Decomposition of the Adjacency Matrix Fix an origin o V of G = (V, E) Stratification (Distance Partition) V n+1 Φ n+1 V = n=0 V n V n Φ n V n = {x V ; (o, x) = n} V n-1 Φ n-1 Γ(G) Associated Hilbert space Γ(G) l 2 (V ) Γ(G) = CΦ n n=0 Φ n = V n 1/2 x V n δ x V 1 V 0 Φ Φ 1 0 Quantum Probability and Asymptotic Spectral Analysis of GrowingRoma, Graphs November 14, / 47

41 52 Quantum Decomposition of the Adjacency Matrix (cont) (A + ) yx =1 y V n+1 (A ) =1 yx x y V n A = A + + A + A (A + ) = A, (A ) = A (A ) yx =1 y V n 1 Quantum Probability and Asymptotic Spectral Analysis of GrowingRoma, Graphs November 14, / 47

42 52 Quantum Decomposition of the Adjacency Matrix (cont) (A + ) yx =1 y V n+1 (A ) =1 yx x y V n A = A + + A + A (A + ) = A, (A ) = A (A ) yx =1 y V n 1 In general, Γ(G) = CΦ n is not invariant under the actions of A ϵ n=0 Cases so far studied in detail 1 Γ(G) is invariant under A ϵ distance-regular graphs 2 Γ(G) is asymptotically invariant under A ϵ Quantum Probability and Asymptotic Spectral Analysis of GrowingRoma, Graphs November 14, / 47

43 53 When Γ(G) is Invariant Under A ϵ For x V n we define ω ϵ (x) = {y V n+ϵ ; y x}, Then, Γ(G) is invariant under A ϵ if and only if ( ) ω ϵ(x) is constant on each V n Typical examples: distance-regular graphs (in this case the constant in ( ) is independent of the choice of o V ) eg, homogeneous trees, Hamming graphs, Johnson graphs, odd graphs, ϵ = +,, Vn+1 Vn Vn 1 { { { ω (x) ω (x) x ω (x) Quantum Probability and Asymptotic Spectral Analysis of GrowingRoma, Graphs November 14, / 47

44 53 When Γ(G) is Invariant Under A ϵ For x V n we define ω ϵ (x) = {y V n+ϵ ; y x}, Then, Γ(G) is invariant under A ϵ if and only if ( ) ω ϵ(x) is constant on each V n Typical examples: distance-regular graphs (in this case the constant in ( ) is independent of the choice of o V ) eg, homogeneous trees, Hamming graphs, Johnson graphs, odd graphs, ϵ = +,, Vn+1 Vn Vn 1 { x { Theorem If Γ(G) is invariant under A +, A, A, there exists a pair of sequences {α n } and {ω n} such that A + Φ n = ω n+1 Φ n+1, A Φ n = ω n Φ n 1, A Φ n = α n+1 Φ n In other words, Γ {ωn },{α n } = (Γ(G), {Φ n}, A +, A, A ) is an interacting Fock space associated with Jacobi parameters ({ω n }, {α n }) { ω (x) ω (x) ω (x) Quantum Probability and Asymptotic Spectral Analysis of GrowingRoma, Graphs November 14, / 47

45 54 Computing the Spectral Distribution 1 By the interacting Fock space structure (Γ(G), A +, A, A ): AΦ n = (A + + A + A )Φ n = ω n+1 Φ n+1 + α n+1 Φ n + ω n Φ n 1 2 Associated with a probability distribution µ on R, the orthogonal polynomials {P 0(x) = 1,, P n(x) = x n + } verify the three term recurrence relation: xp n = P n+1 + α n+1p n + ω np n 1, P 0 = 1, P 1 = x α 1, where ({ω n}, {α n}) is called the Jacobi parameters of µ Quantum Probability and Asymptotic Spectral Analysis of GrowingRoma, Graphs November 14, / 47

46 54 Computing the Spectral Distribution 1 By the interacting Fock space structure (Γ(G), A +, A, A ): AΦ n = (A + + A + A )Φ n = ω n+1 Φ n+1 + α n+1 Φ n + ω n Φ n 1 2 Associated with a probability distribution µ on R, the orthogonal polynomials {P 0(x) = 1,, P n(x) = x n + } verify the three term recurrence relation: xp n = P n+1 + α n+1p n + ω np n 1, P 0 = 1, P 1 = x α 1, where ({ω n}, {α n}) is called the Jacobi parameters of µ 3 An isometry U : Γ(G) L 2 (R, µ) is defined by Φ n P n 1 P n 4 Then UAU = x (multiplication operator) and Φ 0, A m Φ 0 = UΦ 0, UA m U UΦ 0 = P 0, x m P 0 µ = + x m µ(dx) Quantum Probability and Asymptotic Spectral Analysis of GrowingRoma, Graphs November 14, / 47

47 54 Computing the Spectral Distribution 1 By the interacting Fock space structure (Γ(G), A +, A, A ): AΦ n = (A + + A + A )Φ n = ω n+1 Φ n+1 + α n+1 Φ n + ω n Φ n 1 2 Associated with a probability distribution µ on R, the orthogonal polynomials {P 0(x) = 1,, P n(x) = x n + } verify the three term recurrence relation: xp n = P n+1 + α n+1p n + ω np n 1, P 0 = 1, P 1 = x α 1, where ({ω n}, {α n}) is called the Jacobi parameters of µ 3 An isometry U : Γ(G) L 2 (R, µ) is defined by Φ n P n 1 P n 4 Then UAU = x (multiplication operator) and Φ 0, A m Φ 0 = UΦ 0, UA m U UΦ 0 = P 0, x m P 0 µ = + x m µ(dx) Theorem (Graph structure = ({ω n }, {α n }) = spectral distribution) If Γ(G) is invariant under A +, A, A, the vacuum spectral distribution µ defined by Φ 0, A m Φ 0 = + x m µ(dx), m = 1, 2,, is a probability distribution on R that has the Jacobi parameters ({ω n }, {α n }) Quantum Probability and Asymptotic Spectral Analysis of GrowingRoma, Graphs November 14, / 47

48 55 How to know µ from the Jacobi Parameters ({ω n }, {α n }) We need to find a probability distribution µ for which the orthogonal polynomials satisfy xp n = P n+1 + α n+1p n + ω np n 1, P 0 = 1, P 1 = x α 1, Quantum Probability and Asymptotic Spectral Analysis of GrowingRoma, Graphs November 14, / 47

49 55 How to know µ from the Jacobi Parameters ({ω n }, {α n }) We need to find a probability distribution µ for which the orthogonal polynomials satisfy xp n = P n+1 + α n+1p n + ω np n 1, P 0 = 1, P 1 = x α 1, Cauchy Stieltjes transform G µ (z) = + µ(dx) z x = 1 z α 1 where the right-hand side is convergent in {Im z 0} if the moment problem is determinate, eg, if ω n = O((n log n) 2 ) (Carleman s test) ω 1 ω 2 ω 3 z α 2 z α 3 z α 4 1 = ω 1 z α 1 ω 2 z α 2 ω 3 z α 3 z α 4 Quantum Probability and Asymptotic Spectral Analysis of GrowingRoma, Graphs November 14, / 47

50 55 How to know µ from the Jacobi Parameters (cont) Cauchy Stieltjes transform G µ (z) = + µ(dx) z x = 1 z α 1 ω 1 ω 2 ω 3 z α 2 z α 3 z α 4 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 {F (s) + F (s 0)} 2 ρ(x) = 1 π = 1 π lim y +0 t lim Im G µ(x + iy) y +0 s Im G µ (x + iy)dx, s < t, Quantum Probability and Asymptotic Spectral Analysis of GrowingRoma, Graphs November 14, / 47

51 55 Illustration: Homogeneous tree T κ (κ 2) Stratification of T 4 Quantum Probability and Asymptotic Spectral Analysis of GrowingRoma, Graphs November 14, / 47

52 55 Illustration: Homogeneous tree T κ (κ 2) Stratification of T 4 (1) Quantum decomposition: A = A + + A A + Φ 0 = κ Φ 1, A + Φ n = κ 1 Φ n+1 (n 1) A Φ 0 = 0, A Φ 1 = κ Φ 0, A Φ n = κ 1 Φ n 1 (n 2) Quantum Probability and Asymptotic Spectral Analysis of GrowingRoma, Graphs November 14, / 47

53 55 Illustration: Homogeneous tree T κ (κ 2) Stratification of T 4 (1) Quantum decomposition: A = A + + A A + Φ 0 = κ Φ 1, A + Φ n = κ 1 Φ n+1 (n 1) A Φ 0 = 0, A Φ 1 = κ Φ 0, A Φ n = κ 1 Φ n 1 (n 2) (2) Jacobi parameters: {ω 1 = κ, ω 2 = ω 3 = = κ 1}, {α n 0} Quantum Probability and Asymptotic Spectral Analysis of GrowingRoma, Graphs November 14, / 47

54 (3) Cauchy Stieltjes transform: (ω 1 = κ, ω 2 = ω 3 = = κ 1) + µ(dx) z x = G µ(z) = 1 ω 1 ω 2 ω 3 ω 4 ω 5 z z z z z z = (κ 2)z κ z 2 4(κ 1) 2(κ 2 z 2 ) Quantum Probability and Asymptotic Spectral Analysis of GrowingRoma, Graphs November 14, / 47

55 (3) Cauchy Stieltjes transform: (ω 1 = κ, ω 2 = ω 3 = = κ 1) + µ(dx) z x = G µ(z) = 1 ω 1 ω 2 ω 3 ω 4 ω 5 z z z z z z = (κ 2)z κ z 2 4(κ 1) 2(κ 2 z 2 ) (4) Spectral distribution: µ(dx) = ρ κ(x)dx ρ κ(x) = x 2 κ 1 κ 4(κ 1) x 2 2π κ 2 x 2 Kesten Measures (1959) ρ 4, ρ 8, ρ 12 Quantum Probability and Asymptotic Spectral Analysis of GrowingRoma, Graphs November 14, / 47

56 6 Method of Quantum Decomposition for Growing Graphs G G G Quantum Probability and Asymptotic Spectral Analysis of GrowingRoma, Graphs November 14, / 47

57 61 Growing Regular Graphs G (ν) = (V (ν), A (ν) ) Here Γ(G (ν) ) is not necessarily invariant but asymptotically invariant under A ϵ ν Statistics of ω ϵ (x) M(ω ϵ V n ) = 1 V n Σ 2 (ω ϵ V n) = 1 V n x V n ω ϵ (x) x V n { ωϵ(x) M(ω ϵ V n) } 2 Vn+1 Vn { ω (x) ω (x) x { L(ω ϵ V n) = max{ ω ϵ(x) ; x V n} Vn 1 { ω (x) Quantum Probability and Asymptotic Spectral Analysis of GrowingRoma, Graphs November 14, / 47

58 61 Growing Regular Graphs G (ν) = (V (ν), A (ν) ) Here Γ(G (ν) ) is not necessarily invariant but asymptotically invariant under A ϵ ν Statistics of ω ϵ (x) M(ω ϵ V n ) = 1 V n Σ 2 (ω ϵ V n) = 1 V n x V n ω ϵ (x) x V n { ωϵ(x) M(ω ϵ V n) } 2 Vn+1 Vn { ω (x) ω (x) x { L(ω ϵ V n) = max{ ω ϵ(x) ; x V n} Conditions for a growing regular graph G (ν) = (V (ν), E (ν) ) (A1) lim ν deg(g (ν) ) = (A2) for each n = 1, 2,, lim ν Vn 1 M(ω V (ν) n ) = ω n <, lim Σ 2 (ω V (ν) ν n ) = 0, sup L(ω V (ν) n ) < ν (A3) for each n = 0, 1, 2,, lim ν M(ω V (ν) n ) Σ 2 (ω V (ν) n = α n+1 <, lim ) κ(ν) ν κ(ν) = 0, sup ν { ω (x) L(ω V (ν) n ) κ(ν) < Quantum Probability and Asymptotic Spectral Analysis of GrowingRoma, Graphs November 14, / 47

59 Theorem (QCLT in a general form [Hora-Obata: TAMS (2008)]) Let {G (ν) = (V (ν), E (ν) )} be a growing regular graph satisfying (A1) lim ν κ(ν) =, where κ(ν) = deg(g (ν) ) (A2) for each n = 1, 2,, lim ν M(ω V (ν) n ) = ω n <, lim Σ 2 (ω V (ν) ν n ) = 0, sup L(ω V (ν) n ) < ν (A3) for each n = 0, 1, 2,, lim ν M(ω V (ν) n ) Σ 2 (ω V (ν) n = α n+1 <, lim ) κ(ν) ν κ(ν) = 0, sup ν L(ω V (ν) n ) κ(ν) < Then the asymptotic spectral distribution of the normalized A ν in the vacuum state is a probability distribution µ determined by ({ω n }, {α n }): lim ν ( Aν κ(ν) ) m = + x m µ(dx), m = 1, 2, Proof is to show that lim ν A ϵ ν κ(ν) = B ϵ (stochastically) where (Γ, {Ψ n }, B +, B, B ) is the interacting Fock space associated with the Jacobi parameters ({ω n}, {α n}) Quantum Probability and Asymptotic Spectral Analysis of GrowingRoma, Graphs November 14, / 47

60 Outline of Proof (1) A ϵ Φ n = γn+ϵ ϵ Φ n+ϵ + Sn+ϵ ϵ, ϵ {+,, }, n = 0, 1, 2, κ γ + n = M(ω V n) ( Vn κ V n 1 ) 1/2 (, γ n = M(ω Vn + V n) κ V n+1 ) 1/2, γ n = M(ω V n ) κ { n 1 (2) V n = M(ω V k )} κ n + O(κ n 1 ) k=1 (3) lim γ + ν n = ω n, lim γ ν n = ω n+1, lim γ ν n = α n+1 (4) A ϵm κ Aϵ 1 κ Φ n = γ ϵ 1 n+ϵ 1 γ ϵ 2 n+ϵ 1 +ϵ 2 γ ϵ m n+ϵ1 + +ϵ m Φ n+ϵ1 + +ϵ m + m k=1 γ ϵ 1 n+ϵ 1 γ ϵ k 1 n+ϵ 1 + +ϵ k 1 } {{ } (k 1) times A ϵm κ Aϵ k+1 κ } {{ } (m k) times (5) Estimate the error terms and show that lim Φ (ν) A ϵ m ν j, Aϵ k+1 S ϵ k n+ϵ1 + +ϵ = 0 κ(ν) κ(ν) k S ϵ k n+ϵ1 + +ϵ k Quantum Probability and Asymptotic Spectral Analysis of GrowingRoma, Graphs November 14, / 47

61 62 Illustration: Z N as N Quantum Probability and Asymptotic Spectral Analysis of GrowingRoma, Graphs November 14, / 47

62 62 Illustration: Z N as N 1 Asymptotic invariance of Γ(Z N ) under A ϵ : 2 Normalized adjacency matrices: A + Φ n = 2N n + 1 Φ n+1 + O(1), A Φ n = 2N n Φ n 1 + O(N 1/2 ) A ϵ κ(n) = Aϵ 2N 3 The interacting Fock space in the limit as N is the Boson Fock space: B + Ψ n = n + 1 Ψ n+1, B Φ n = n Ψ n 1, B = 0 4 The asymptotic spectral distribution is the standard Gaussian distribution: ( ) m AN lim Φ 0, Φ 0 = Ψ 0, (B + + B ) m Ψ 0 N 2N = 1 + x m e x2 /2 dx 2π Quantum Probability and Asymptotic Spectral Analysis of GrowingRoma, Graphs November 14, / 47

63 63 Some Results: Asymptotic Spectral Distributions graphs IFS vacuum state deformed vacuum state Hamming graphs ω n = n Gaussian (N/d 0) Gaussian H(d, N) (Boson) Poisson (N/d λ 1 > 0) or Poisson Johnson graphs ω n = n 2 exponential (2d/v 1) Poissonization of J(v, d) geometric (2d/v p (0, 1)) exponential distribution odd graphs ω 2n 1 = n two-sided Rayleigh? O k ω 2n = n homogeneous ω n = 1 Wigner semicircle free Poisson trees T κ (free) integer lattices ω n = n Gaussian Gaussian Z N (Boson) symmetric groups ω n = n Gaussian Gaussian S n (Coxeter) (Boson) Coxeter groups ω n = 1 Wigner semicircle free Poisson (Fendler) (free) Spidernets ω 1 = 1 free Meixner law? S(a, b, c) ω 2 = = r Quantum Probability and Asymptotic Spectral Analysis of GrowingRoma, Graphs November 14, / 47

64 Quantum Probabilistic Approach to Spectral Analysis of Graphs adjacency matrix A ν combinatorics linear algebra spectral distribution µ ν Classical scaling limit quantum decomposition A ν Aν A ν A ν A ν quantum CLT ε A ν ε B ν Z ν use of independence X X ν Quantum B quantum CLT B α asymptotic spectral distribution µ interacting Fock space Γ ( α n ) B B orthogonal polynomials P n ω n α n Quantum Probability and Asymptotic Spectral Analysis of GrowingRoma, Graphs November 14, / 47

65 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 A 2) 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, eg, coupled oscillators, random walks (or more generally Markov chains), quantum walks, etc Quantum Probability and Asymptotic Spectral Analysis of GrowingRoma, Graphs November 14, / 47