E IGENVALUES OF SPARSE RANDOM REGULAR GRAPHS Soumik Pal University of Washington Seattle Seminar on Stochastic Processes, 2012
G RAPHS AND ADJACENCY MATRICES 3 2 Undirected graphs on n labeled vertices. 4 1 Regular: degree d . 5 6 Adjacency matrix = n × n 0 1 0 0 1 1 symmetric matrix. 1 0 0 1 1 0 0 0 0 1 1 1 Sparse - d ≪ n . 0 1 1 0 0 1 1 1 1 0 0 0 1 0 1 1 0 0
M ODELS OF RANDOM REGULAR GRAPHS The permutation model: G ( n , 2 ) . π - random permutation on [ n ] . 2-regular graph: 3 9 4 8 2 1 7 10 6 5
T HE PERMUTATION MODEL G ( n , 2 d ) π 1 , . . . , π d iid uniform permutations. Superimpose.
T HE PERMUTATION MODEL G ( n , 2 d ) π 1 , . . . , π d iid uniform permutations. Superimpose. 4 3 π 1 = 5 2 π 2 = 1
T HE PERMUTATION MODEL G ( n , 2 d ) π 1 , . . . , π d iid uniform permutations. Superimpose. 4 3 π 1 = ( 1 3 2 )( 4 5 ) 5 2 π 2 = 1
T HE PERMUTATION MODEL G ( n , 2 d ) π 1 , . . . , π d iid uniform permutations. Superimpose. 4 3 π 1 = ( 1 3 2 )( 4 5 ) 5 2 π 2 = ( 1 4 2 ) 1
T HE PERMUTATION MODEL G ( n , 2 d ) π 1 , . . . , π d iid uniform permutations. Superimpose. 4 3 π 1 = ( 1 3 2 )( 4 5 ) 5 2 π 2 = ( 1 4 2 ) 1 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 . . .
R ANDOM M ATRIX T HEORY A Wigner matrix is a square random matrix with
R ANDOM M ATRIX T HEORY A Wigner matrix is a square random matrix with − 0 . 6 0 . 7 0 . 1 0 . 6 upper triangular entries 2 . 1 2 . 5 − 0 . 1 chosen independently; − 2 . 2 1 . 1 − 0 . 6 A sample of a 4 × 4 Wigner matrix.
R ANDOM M ATRIX T HEORY A Wigner matrix is a square random matrix with − 0 . 6 0 . 7 0 . 1 0 . 6 upper triangular entries 0 . 7 2 . 1 2 . 5 − 0 . 1 chosen independently; 0 . 1 2 . 5 − 2 . 2 1 . 1 symmetric. 0 . 6 − 0 . 1 1 . 1 − 0 . 6 A sample of a 4 × 4 Wigner matrix.
R ANDOM M ATRIX T HEORY A Wigner matrix is a square random matrix with − 0 . 6 0 . 7 0 . 1 0 . 6 upper triangular entries 0 . 7 2 . 1 2 . 5 − 0 . 1 chosen independently; 0 . 1 2 . 5 − 2 . 2 1 . 1 symmetric. 0 . 6 − 0 . 1 1 . 1 − 0 . 6 Minor=principal submatrix, also Wigner. A sample of a 4 × 4 Wigner matrix.
E IGENVALUE FLUCTUATIONS W ∞ - Wigner array. W n - n × n minor. E-values { λ n i } . Linear eigenvalue statistics � λ n n � � i tr f ( W n ) := 2 √ n . f i = 1 (Classical Theorem) If f is analytic 0 , σ 2 � � n →∞ [ tr f ( W n ) − E tr f ( W n )] = N lim . f
E IGENVALUE FLUCTUATIONS W ∞ - Wigner array. W n - n × n minor. E-values { λ n i } . Linear eigenvalue statistics � λ n n � � i tr f ( W n ) := 2 √ n . f i = 1 (Classical Theorem) If f is analytic 0 , σ 2 � � n →∞ [ tr f ( W n ) − E tr f ( W n )] = N lim . f Eigenvalue repulsion.
J OINT CONVERGENCE ACROSS MINORS Results by Borodin ’10. Choose 0 < t 1 ≤ t 2 ≤ . . . ≤ t k and f 1 , f 2 , . . . , f k . � � � � lim tr f i W ⌊ nt i ⌋ − E tr f i ( · ) , i ∈ [ k ] = Gaussian . n →∞ Mean zero. Covariance kernel?
J OINT CONVERGENCE ACROSS MINORS Results by Borodin ’10. Choose 0 < t 1 ≤ t 2 ≤ . . . ≤ t k and f 1 , f 2 , . . . , f k . � � � � lim tr f i W ⌊ nt i ⌋ − E tr f i ( · ) , i ∈ [ k ] = Gaussian . n →∞ Mean zero. Covariance kernel? ‘Limiting fluctuation of the Height Function is the Gaussian Free Field.’
C HEBYSHEV POLYNOMIALS - FIRST KIND 1.0 T n ( x ) , n ≥ 0 - Poly of degree n . 0.5 Orthogonal polynomials on [ − 1 , 1 ] . � 1.0 � 0.5 0.5 1.0 Weight measure: Arc-Sine law. � 0.5 Fourier expansion: f = � n c n T n . � 1.0 F IGURE : T 1 , T 3 , T 8
C HEBYSHEV POLYNOMIALS - FIRST KIND 1.0 T n ( x ) , n ≥ 0 - Poly of degree n . 0.5 Orthogonal polynomials on [ − 1 , 1 ] . � 1.0 � 0.5 0.5 1.0 Weight measure: Arc-Sine law. � 0.5 Fourier expansion: f = � n c n T n . � 1.0 F IGURE : T 1 , T 3 , T 8 (Borodin ’10) Fix s ≤ t and i , j ∈ N . j � s � j / 2 � � � � �� n →∞ Cov lim tr T i W ⌊ nt ⌋ , tr T j W ⌊ ns ⌋ = δ ij . 2 t
Main Results - (with Toby Johnson ’12)
C HINESE R ESTAURANT P ROCESS (Dubins-Pitman) Randomly grows a permutation. Every point adds neighbor to left at rate one. New fixed points at rate one. σ = ( 1 )
C HINESE R ESTAURANT P ROCESS (Dubins-Pitman) Randomly grows a permutation. Every point adds neighbor to left at rate one. New fixed points at rate one. σ = ( 1 )( 2 )
C HINESE R ESTAURANT P ROCESS (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 )
C HINESE R ESTAURANT P ROCESS (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 )
C HINESE R ESTAURANT P ROCESS (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 )
C HINESE R ESTAURANT P ROCESS (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 )
C HINESE R ESTAURANT P ROCESS (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 )
C HINESE R ESTAURANT P ROCESS (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 )
C HINESE R ESTAURANT P ROCESS (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 )
C HINESE R ESTAURANT P ROCESS (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 )
C YCLES AND EIGENVALUES Case of G ( n , 2 ) . Adjacency matrix - blocks of cycles. E-values of cycle of size k : � � 2 π j � � cos , j ∈ [ k ] . k
C YCLES AND EIGENVALUES Case of G ( n , 2 ) . Adjacency matrix - blocks of cycles. E-values of cycle of size k : � � 2 π j � � cos , j ∈ [ k ] . 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
T HE CASE OF GENERAL G ( n , 2 d ) Simultaneous insertion in ( π 1 , π 2 , . . . , π d ) by CRP . Let T i = Exp( i ), i ∈ N , � m � � n t = max m : T i ≤ t . i = 1 G ( t ) := G ( n t , 2 d ) . No cyclic decomposition.
G ROWTH OF A CYCLE (Johnson-P . ’12) Existing cycles grow in size. 4 π 2 3 4 π 2 3 π 1 6 π 1 π 1 π 1 π 1 5 2 5 2 π 2 π 1 π 2 π 1 1 1 π 1 = ( 1 2 3 )( 4 5 ) π 1 = ( 1 2 6 3 )( 4 5 ) π 2 = ( 1 5 )( 4 3 )( 2 ) π 2 = ( 1 5 )( 4 3 )( 2 6 ) F IGURE : Vertex 6 is inserted between 2 and 3 in π 1 .
B IRTH OF A CYCLE 6 π 1 π 2 π 2 π 1 π 2 π 1 π 2 π 3 π 2 π 1 1 2 3 4 5 1 2 3 4 5 π 1 = ( 2 3 1 )( 4 5 ) π 1 = ( 2 3 1 6 )( 4 5 ) π 2 = ( 2 1 3 4 5 ) π 2 = ( 2 1 3 4 6 5 ) F IGURE : A cycle forms “spontaneously”.
C YCLE COUNTS C ( s ) k ( t ) = # k -cycles in G ( s + t ) . Non-Markovian process in t , with s fixed.
C YCLE 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 ( N k ( t ) , k ∈ N , −∞ < t < ∞ ) is Markov. Running in stationarity.
T HE 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 � ( 2 d − 1 ) k − 1 + 2 d , k even , a ( d , k ) = ( 2 d − 1 ) k + 1 , k odd .
M ORE 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 .
M ORE 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 ( X k , y ( t ) , t ≥ 0 ) . Yule process from k .
Recommend
More recommend
