T HE PERMUTATION MODEL G ( n , 2 d ) 1 , . . . , d iid uniform - - PowerPoint PPT Presentation

EIGENVALUES OF SPARSE

RANDOM REGULAR GRAPHS

Soumik Pal University of Washington Seattle Seminar on Stochastic Processes, 2012

GRAPHS AND ADJACENCY MATRICES

Undirected graphs on n labeled vertices. Regular: degree d. Adjacency matrix = n × n symmetric matrix. Sparse - d ≪ n.

1 2 3 4 5 6        1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1       

MODELS OF RANDOM REGULAR GRAPHS

The permutation model: G(n, 2). π - random permutation on [n]. 2-regular graph:

1 2 4 3 8 5 10 6 7 9

THE PERMUTATION MODEL G(n, 2d)

π1, . . . , πd iid uniform permutations. Superimpose.

THE PERMUTATION MODEL G(n, 2d)

π1, . . . , πd iid uniform permutations. Superimpose. 1 2 3 4 5 π1 = π2 =

THE PERMUTATION MODEL G(n, 2d)

π1, . . . , πd iid uniform permutations. Superimpose. 1 2 3 4 5 π1 = (1 3 2)(4 5) π2 =

THE PERMUTATION MODEL G(n, 2d)

π1, . . . , πd iid uniform permutations. Superimpose. 1 2 3 4 5 π1 = (1 3 2)(4 5) π2 = (1 4 2)

THE PERMUTATION MODEL G(n, 2d)

π1, . . . , πd iid uniform permutations. Superimpose. 1 2 3 4 5 π1 = (1 3 2)(4 5) π2 = (1 4 2) Multiple edges, loops OK.

SPECTRAL GRAPH THEORY

Eigenvalues of the adjacency matrix. Why care? Test RMT universality: McKay ’81, Dumitriu-P . ’10, Tran-Vu-Wang’10 Ben Arous-Dang ’11, Erdos-Knowles-Yau ’11 Expansion properties: Broder-Shamir ’87, Friedman ’91, ’08 Quantum chaos: Smilansky ’10, Elon-Smilansky ’10 Simulations: Jacobson et al. ’99, Miller et al. ’08. and many more . . .

RANDOM MATRIX THEORY

A Wigner matrix is a square random matrix with

RANDOM MATRIX THEORY

A Wigner matrix is a square random matrix with upper triangular entries chosen independently;        −0.6 0.7 0.1 0.6 2.1 2.5 − 0.1 −2.2 1.1 − 0.6        A sample of a 4 × 4 Wigner matrix.

RANDOM MATRIX THEORY

A Wigner matrix is a square random matrix with upper triangular entries chosen independently; symmetric.        −0.6 0.7 0.1 0.6 0.7 2.1 2.5 − 0.1 0.1 2.5 −2.2 1.1 0.6 − 0.1 1.1 − 0.6        A sample of a 4 × 4 Wigner matrix.

RANDOM MATRIX THEORY

A Wigner matrix is a square random matrix with upper triangular entries chosen independently; symmetric. Minor=principal submatrix, also Wigner.        −0.6 0.7 0.1 0.6 0.7 2.1 2.5 − 0.1 0.1 2.5 −2.2 1.1 0.6 − 0.1 1.1 − 0.6        A sample of a 4 × 4 Wigner matrix.

EIGENVALUE FLUCTUATIONS

W∞ - Wigner array. Wn - n × n minor. E-values {λn

i }.

Linear eigenvalue statistics tr f (Wn) :=

n

• i=1

f λn

i

2√n

• .

(Classical Theorem) If f is analytic lim

n→∞ [tr f (Wn) − E tr f (Wn)] = N

• 0, σ2

f

• .

EIGENVALUE FLUCTUATIONS

W∞ - Wigner array. Wn - n × n minor. E-values {λn

i }.

Linear eigenvalue statistics tr f (Wn) :=

n

• i=1

f λn

i

2√n

• .

(Classical Theorem) If f is analytic lim

n→∞ [tr f (Wn) − E tr f (Wn)] = N

• 0, σ2

f

• .

Eigenvalue repulsion.

JOINT CONVERGENCE ACROSS MINORS

Results by Borodin ’10. Choose 0 < t1 ≤ t2 ≤ . . . ≤ tk and f1, f2, . . . , fk. lim

n→∞

• tr fi
• W⌊nti⌋
• − E tr fi (·) , i ∈ [k]
• = Gaussian.

Mean zero. Covariance kernel?

JOINT CONVERGENCE ACROSS MINORS

Results by Borodin ’10. Choose 0 < t1 ≤ t2 ≤ . . . ≤ tk and f1, f2, . . . , fk. lim

n→∞

• tr fi
• W⌊nti⌋
• − E tr fi (·) , i ∈ [k]
• = Gaussian.

Mean zero. Covariance kernel?

‘Limiting fluctuation of the Height Function is the Gaussian Free Field.’

CHEBYSHEV POLYNOMIALS - FIRST KIND

Tn(x), n ≥ 0 - Poly of degree n. Orthogonal polynomials on [−1, 1]. Weight measure: Arc-Sine law. Fourier expansion: f =

n cnTn.

1.0 0.5 0.5 1.0 1.0 0.5 0.5 1.0

FIGURE: T1, T3, T8

CHEBYSHEV POLYNOMIALS - FIRST KIND

Tn(x), n ≥ 0 - Poly of degree n. Orthogonal polynomials on [−1, 1]. Weight measure: Arc-Sine law. Fourier expansion: f =

n cnTn.

1.0 0.5 0.5 1.0 1.0 0.5 0.5 1.0

FIGURE: T1, T3, T8

(Borodin ’10) Fix s ≤ t and i, j ∈ N. lim

n→∞ Cov

• tr Ti
• W⌊nt⌋
• , tr Tj
• W⌊ns⌋
• = δij

j 2 s t j/2 .

Main Results - (with Toby Johnson ’12)

CHINESE RESTAURANT PROCESS

(Dubins-Pitman) Randomly grows a permutation. Every point adds neighbor to left at rate one. New fixed points at rate one. σ = (1)

CHINESE RESTAURANT PROCESS

(Dubins-Pitman) Randomly grows a permutation. Every point adds neighbor to left at rate one. New fixed points at rate one. σ = (1)(2)

CHINESE RESTAURANT PROCESS

(Dubins-Pitman) Randomly grows a permutation. Every point adds neighbor to left at rate one. New fixed points at rate one. σ = (1)(2 3)

CHINESE RESTAURANT PROCESS

(Dubins-Pitman) Randomly grows a permutation. Every point adds neighbor to left at rate one. New fixed points at rate one. σ = (1)(2 4 3)

CHINESE RESTAURANT PROCESS

(Dubins-Pitman) Randomly grows a permutation. Every point adds neighbor to left at rate one. New fixed points at rate one. σ = (1)(2 4 3)(5)

CHINESE RESTAURANT PROCESS

(Dubins-Pitman) Randomly grows a permutation. Every point adds neighbor to left at rate one. New fixed points at rate one. σ = (1)(2 4 3)(5)(6)

CHINESE RESTAURANT PROCESS

(Dubins-Pitman) Randomly grows a permutation. Every point adds neighbor to left at rate one. New fixed points at rate one. σ = (1)(2 4 3)(5 7)(6)

CHINESE RESTAURANT PROCESS

(Dubins-Pitman) Randomly grows a permutation. Every point adds neighbor to left at rate one. New fixed points at rate one. σ = (1)(2 4 3)(5 8 7)(6)

CHINESE RESTAURANT PROCESS

(Dubins-Pitman) Randomly grows a permutation. Every point adds neighbor to left at rate one. New fixed points at rate one. σ = (1)(2 4 3 9)(5 8 7)(6)

CHINESE RESTAURANT PROCESS

(Dubins-Pitman) Randomly grows a permutation. Every point adds neighbor to left at rate one. New fixed points at rate one. σ = (1)(2 4 3 9)(5 10 8 7)(6)

CYCLES AND EIGENVALUES

Case of G(n, 2). Adjacency matrix - blocks of cycles. E-values of cycle of size k:

• cos

2πj k

• , j ∈ [k]
• .

CYCLES AND EIGENVALUES

Case of G(n, 2). Adjacency matrix - blocks of cycles. E-values of cycle of size k:

• cos

2πj k

• , j ∈ [k]
• .

Under the CRP: Cycle of size k − → size (k + 1) at rate k. New cycle size 1 at rate 1. Stochastic process of eigenvalues. Kerov-Olshanki-Vershik ’93 Olshanki-Vershik ’96, Bourgade-Najnudel-Nikeghbali ’11

THE CASE OF GENERAL G(n, 2d)

Simultaneous insertion in (π1, π2, . . . , πd) by CRP . Let Ti = Exp(i), i ∈ N, nt = max

• m :

m

• i=1

Ti ≤ t

• .

G(t) := G(nt, 2d). No cyclic decomposition.

GROWTH OF A CYCLE

(Johnson-P . ’12) Existing cycles grow in size. 1 2 3 4 5 π1 π1 π2 π1 π2 π1 = (1 2 3)(4 5) π2 = (1 5)(4 3)(2) 1 2 3 4 5 6 π1 π1 π1 π2 π1 π2 π1 = (1 2 6 3)(4 5) π2 = (1 5)(4 3)(2 6)

FIGURE: Vertex 6 is inserted between 2 and 3 in π1.

BIRTH OF A CYCLE

1 2 3 4 5 π2 π1 π2 π1 π1 = (2 3 1)(4 5) π2 = (2 1 3 4 5) 1 2 3 4 5 6 π2 π3 π2 π1 π1 π2 π1 = (2 3 1 6)(4 5) π2 = (2 1 3 4 6 5)

FIGURE: A cycle forms “spontaneously”.

CYCLE COUNTS

C(s)

k (t) = # k-cycles in G(s + t).

Non-Markovian process in t, with s fixed.

CYCLE COUNTS

C(s)

k (t) = # k-cycles in G(s + t).

Non-Markovian process in t, with s fixed. (C(s)

k (t), k ∈ N, −∞ < t < ∞) converges as s → ∞.

Limiting process (Nk(t), k ∈ N, −∞ < t < ∞) is Markov. Running in stationarity.

THE LIMITING PROCESS

(Johnson-P . ’12) In the limit: Existing k-cycles grows to (k + 1) at rate k. New k-cycles created at rate µ(k) ⊗ Leb. Here: µ(k) = 1 2 [a(d, k) − a(d, k − 1)] , k ∈ N, a(d, 0) := 0, where a(d, k) =

• (2d − 1)k − 1 + 2d,

k even, (2d − 1)k + 1, k odd.

MORE FORMALLY ...

(Johnson-P . ’12) Poisson point process χ on N × (−∞, ∞). Intensity µ ⊗ Leb. Yule process: Lf(k) = k (f(k + 1) − f(k)) , k ∈ N.

MORE FORMALLY ...

(Johnson-P . ’12) Poisson point process χ on N × (−∞, ∞). Intensity µ ⊗ Leb. Yule process: Lf(k) = k (f(k + 1) − f(k)) , k ∈ N. For (k, y) ∈ χ, start indep (Xk,y(t), t ≥ 0). Yule process from k.

MORE FORMALLY ...

(Johnson-P . ’12) Poisson point process χ on N × (−∞, ∞). Intensity µ ⊗ Leb. Yule process: Lf(k) = k (f(k + 1) − f(k)) , k ∈ N. For (k, y) ∈ χ, start indep (Xk,y(t), t ≥ 0). Yule process from k. Define Nk(t) :=

• (j,y)∈χ∩{[k]×(−∞,t]}

1 {Xj,y(t − y) = k} .

INVARIANT DISTRIBUTION

(C(s)

k (t), k ∈ N, t ∈ R) s

− → (Nk(t), k ∈ N, t ∈ R). Marginal distribution: (Nk(t), k ∈ N) ∼ ⊗Poi a(d, k) 2k

• .

INVARIANT DISTRIBUTION

(C(s)

k (t), k ∈ N, t ∈ R) s

− → (Nk(t), k ∈ N, t ∈ R). Marginal distribution: (Nk(t), k ∈ N) ∼ ⊗Poi a(d, k) 2k

• .

Dumitriu-Johnson-P .-Paquette ’11 Bollobás ’80, Wormald ’81.

EIGENVALUES AND CYCLES

EIGENVALUES AND CYCLES

What do the eigenvalues count?

EIGENVALUES AND CYCLES

What do the eigenvalues count? Cyclically non-backtracking walks (CNBW).

EIGENVALUES AND CYCLES

What do the eigenvalues count? Cyclically non-backtracking walks (CNBW). backtracking

EIGENVALUES AND CYCLES

What do the eigenvalues count? Cyclically non-backtracking walks (CNBW). backtracking

EIGENVALUES AND CYCLES

What do the eigenvalues count? Cyclically non-backtracking walks (CNBW). backtracking

EIGENVALUES AND CYCLES

What do the eigenvalues count? Cyclically non-backtracking walks (CNBW). backtracking cyclically non-backtracking

EIGENVALUES AND CYCLES

What do the eigenvalues count? Cyclically non-backtracking walks (CNBW). backtracking cyclically non-backtracking

EIGENVALUES AND CYCLES

What do the eigenvalues count? Cyclically non-backtracking walks (CNBW). backtracking cyclically non-backtracking

EIGENVALUES AND CYCLES

What do the eigenvalues count? Cyclically non-backtracking walks (CNBW). backtracking cyclically non-backtracking

EIGENVALUES AND CYCLES

What do the eigenvalues count? Cyclically non-backtracking walks (CNBW). backtracking cyclically non-backtracking

EIGENVALUES AND CYCLES

What do the eigenvalues count? Cyclically non-backtracking walks (CNBW). backtracking cyclically non-backtracking

EIGENVALUES AND CYCLES

What do the eigenvalues count? Cyclically non-backtracking walks (CNBW). backtracking cyclically non-backtracking

EIGENVALUES AND CYCLES

What do the eigenvalues count? Cyclically non-backtracking walks (CNBW). backtracking cyclically non-backtracking

COUNTING CNBW

(Friedman ’08, DJPP ’11) Nn

k = # CNBW’s of size k. Then

(2d − 1)−k/2Nn

k = n

• i=1

Γk(λi), polynomial Γk. Here {λi, i ∈ [n]} are e-values of G(n, 2d)/2 √ 2d − 1.

COUNTING CNBW

(Friedman ’08, DJPP ’11) Nn

k = # CNBW’s of size k. Then

(2d − 1)−k/2Nn

k = n

• i=1

Γk(λi), polynomial Γk. Here {λi, i ∈ [n]} are e-values of G(n, 2d)/2 √ 2d − 1. Moreover w.h.p. all CNBW’s are repeated cycles.

PROCESS OF EIGENVALUES

THEOREM (JOHNSON-P. ’12)

For any polynomial f,

• tr f(G(s + t)), t ∈ R
• converges to a linear combination of (Nk(t), k ∈ N).

PROCESS OF EIGENVALUES

THEOREM (JOHNSON-P. ’12)

For any polynomial f,

• tr f(G(s + t)), t ∈ R
• converges to a linear combination of (Nk(t), k ∈ N).

Note the unusual Markov property.

BACK TO CHEBYSHEV POLYNOMIALS

THEOREM (JOHNSON-P. ’12)

{Tk, k ∈ N} Chebyshev polys. lim

d→∞ lim s→∞ (tr Tk (G(s + t)) − E (·) , t ∈ R, k ∈ N)

= (Uk(t), t ∈ R, k ∈ N) . (Uk) indep stationary O-U: dUk(t) = −kUk(t)dt + kdWk(t), t ≥ 0.

BACK TO CHEBYSHEV POLYNOMIALS

THEOREM (JOHNSON-P. ’12)

{Tk, k ∈ N} Chebyshev polys. lim

d→∞ lim s→∞ (tr Tk (G(s + t)) − E (·) , t ∈ R, k ∈ N)

= (Uk(t), t ∈ R, k ∈ N) . (Uk) indep stationary O-U: dUk(t) = −kUk(t)dt + kdWk(t), t ≥ 0. Moreover, for u < t, lim

d→∞ lim s→∞ Cov (tr Ti (G(s + t)) , tr Tj (G(s + u))) = δij

j 2ej(u−t).

COMPARISON WITH WIGNER

(Borodin) Process of Wigner minors: Ti(t), Tj(s) = δij j 2 s t j/2 , s, t > 0. (Johnson-P .) Random regular graphs with CRP: Ti(t), Tj(s) = δij j 2ej(s−t), s, t ∈ R.

COMPARISON WITH WIGNER

(Borodin) Process of Wigner minors: Ti(t), Tj(s) = δij j 2 s t j/2 , s, t > 0. (Johnson-P .) Random regular graphs with CRP: Ti(t), Tj(s) = δij j 2ej(s−t), s, t ∈ R. Deterministic time-change.

COMPARISON WITH WIGNER

(Borodin) Process of Wigner minors: Ti(t), Tj(s) = δij j 2 s t j/2 , s, t > 0. (Johnson-P .) Random regular graphs with CRP: Ti(t), Tj(s) = δij j 2ej(s−t), s, t ∈ R. Deterministic time-change. Unfortunately, no GFF convergence yet ...

An easy example

AN EASY EXAMPLE

G(n, 2d). Take T1(x) = x. tr T1(G) =

n

• i=1

λi = 2

d

• i=1

mi, where mi = # fixed points in πi. |mi − Poi(1)| = ε(n),

• i

mi ≈ Poi(d). The process of fixed point count ⇒ OU as d → ∞.

SEVERAL OPEN PROBLEMS ...

Dynamic case: GFF convergence. Thank You

SEVERAL OPEN PROBLEMS ...

Dynamic case: GFF convergence. Static case: How to capture eigenvalue distribution functions? No information about long walks/cycles known. Properties of eigenvectors. Gaussian law? Generalization to graphs with fixed-degree distribution. Thank You