S PECTRAL DYNAMICS OF RANDOM REGULAR GRAPHS AND THE P OISSON FREE FIELD Soumik Pal The Pitman conference June 21, 2014
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.
R ANDOM M ATRIX T HEORY A GOE is a square random matrix with
R ANDOM M ATRIX T HEORY A GOE is a square random matrix with − 0 . 6 0 . 7 0 . 1 0 . 3 2 . 1 2 . 5 − 0 . 1 upper triangular entries − 2 . 2 1 . 1 chosen iid N ( 0 , 1 ) ; 0 . 4 A sample of a 4 × 4 GOE matrix and its 3 × 3 minor.
R ANDOM M ATRIX T HEORY A GOE is a square random matrix with − 0 . 6 0 . 7 0 . 1 0 . 3 0 . 7 2 . 1 2 . 5 − 0 . 1 upper triangular entries 0 . 1 2 . 5 − 2 . 2 1 . 1 chosen iid N ( 0 , 1 ) ; 0 . 3 − 0 . 1 1 . 1 0 . 4 symmetric. A sample of a 4 × 4 GOE matrix and its 3 × 3 minor.
R ANDOM M ATRIX T HEORY A GOE is a square random matrix with − 0 . 6 0 . 7 0 . 1 0 . 3 0 . 7 2 . 1 2 . 5 − 0 . 1 upper triangular entries 0 . 1 2 . 5 − 2 . 2 1 . 1 chosen iid N ( 0 , 1 ) ; 0 . 3 − 0 . 1 1 . 1 0 . 4 symmetric. Minor=principal submatrix, A sample of a 4 × 4 GOE matrix also GOE. and its 3 × 3 minor.
GOE VS . RANDOM GRAPHS Adjacency matrices are not GOE (or, Wigner). Rows are sparse; no independence.
GOE VS . RANDOM GRAPHS Adjacency matrices are not GOE (or, Wigner). Rows are sparse; no independence. However, for large d , approximately GOE. Eigenvalue distribution (McKay ’81, Dumitriu-P . ’10, Tran-Vu-Wang ’10) Linear eigenvalue statistics (Dumitriu-Johnson-P .-Paquette ’11) Simulations. Not Erd˝ os-Rényi, e.g. connected.
E IGENVALUE FLUCTUATIONS W ∞ - GOE array. W n - n × n minor. E-values { λ n i } . Linear eigenvalue statistics � λ n n � � i 2 √ n tr f ( W 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
D YNAMICS OF EIGENVALUE FLUCTUATIONS (A. Borodin ’10) GOE array W ∞ ( s ) in time with entries as Brownian motions. Choose ( t i , s i , f i , i = 1 , . . . , k ) . Polynomial f i ’s. � � � � lim tr f i W ⌊ nt i ⌋ ( s i ) − E tr f i ( · ) , i ∈ [ k ] = Gaussian . n →∞ Mean zero. Covariance kernel?
D YNAMICS OF EIGENVALUE FLUCTUATIONS (A. Borodin ’10) GOE array W ∞ ( s ) in time with entries as Brownian motions. Choose ( t i , s i , f i , i = 1 , . . . , k ) . Polynomial f i ’s. � � � � lim tr f i W ⌊ nt i ⌋ ( s i ) − E tr f i ( · ) , i ∈ [ k ] = Gaussian . n →∞ Mean zero. Covariance kernel? Fix s . Limiting Height Function is the Gaussian Free Field. Nontrivial correlation across s .
M AIN QUESTION What dynamics on random regular graphs leads to similar eigenvalue fluctuations in dimension × time?
Description of the dynamics
D YNAMICS IN DIMENSION (Dubins-Pitman) Chinese restaurant process on d permutations. i th customers arrive simultaneously. Sits independently. Let T i = Exp( i ), i ∈ N , � m � � n t = max m : T i ≤ t . i = 1 G ( t , 0 ) := G ( n t , 2 d ) , for 0 ≤ t ≤ T . dimension t ; time 0.
D YNAMICS IN TIME Fix T large. d permutations on n labels. Run random transposition MC simultaneously. � n � Any transposition selected at rate 1 / n . 2 Successive product on left. Superimpose - G ( T , s ) for s ≥ 0. Delete labels successively: G ( T + t , s ) , t ∈ [ − T , 0 ] , s ≥ 0 .
C YCLES AND EIGENVALUES N k - # k -cycles in the graph G ( n , 2 d ) . As n → ∞ , ( N k , k ∈ N ) - linear eigenvalue statistics. In fact 2 kN k ≈ tr ( T k ( G ( n , 2 d ))) . ( T k , k ∈ N ) - Chebyshev polynomials of first kind.
Dynamics of cycles in dimension
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 ( T ) ( t ) = # k -cycles in G ( T + t , 0 ) , t ∈ [ − T , 0 ] . k Non-Markovian process in t , with T fixed.
C YCLE COUNTS C ( T ) ( t ) = # k -cycles in G ( T + t , 0 ) , t ∈ [ − T , 0 ] . k Non-Markovian process in t , with T fixed. ( C ( T ) ( t ) , k ∈ N , t < 0 ) converges as T → ∞ . k Limiting process ( N k ( t ) , k ∈ N , t ≤ 0 ) 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 .
P OISSON FIELD OF Y ULE PROCESSES k x x x Poisson point process χ on N × ( −∞ , ∞ ) . Intensity µ ⊗ Leb.
P OISSON FIELD OF Y ULE PROCESSES k x x x Poisson point process χ on N × ( −∞ , ∞ ) . Intensity µ ⊗ Leb. For ( k , y ) ∈ χ , start indep Yule processes ( X k , y ( t ) , t ≥ 0 ) .
P OISSON FIELD OF Y ULE PROCESSES k x x x Poisson point process χ on N × ( −∞ , ∞ ) . Intensity µ ⊗ Leb. For ( k , y ) ∈ χ , start indep Yule processes ( X k , y ( t ) , t ≥ 0 ) . Define � N k ( t ) := 1 { X j , y ( t − y ) = k } . ( j , y ) ∈ χ ∩{ [ k ] × ( −∞ , t ] }
I NVARIANT DISTRIBUTION � � C ( T ) ( t ) , k ∈ N , t ∈ ( −∞ , 0 ] − → ( N k ( t ) , k ∈ N , t ∈ ( −∞ , 0 ]) . k Marginal distribution: � a ( d , k ) � ( N k ( t ) , k ∈ N ) ∼ ⊗ Poi . 2 k
I NVARIANT DISTRIBUTION � � C ( T ) ( t ) , k ∈ N , t ∈ ( −∞ , 0 ] − → ( N k ( t ) , k ∈ N , t ∈ ( −∞ , 0 ]) . k Marginal distribution: � a ( d , k ) � ( N k ( t ) , k ∈ N ) ∼ ⊗ Poi . 2 k Dumitriu-Johnson-P .-Paquette ’11 Bollobás ’80, Wormald ’81.
C YCLES IN TIME j 1 1 6 2 6 2 5 3 5 3 4 4 F IGURE : A cycle that vanishes due to transposition ( 1 , j ) , j > 6. Random transpositions make short cycles vanish or appear at random. Other effects are of negligible probability.
T HE JOINT LIMITING PROCESS (Ganguly-P . ’14) Take limit as T → ∞ . Fix t < 0. Consider in s ≥ 0. ( N k ( t , · ) , k ∈ N ) - independent birth-and-death chains. Joint convergence to a Poisson surface: � � C ( T ) ( t , s ) , k ∈ N , t ≤ 0 , s ≥ 0 − → ( N k ( t , s )) . k Yule process in dimension, birth-and-death chains in time. Markov field. Stationary along axis. Joint law by intertwining.
Diffusion limit
L ARGE DIMENSION , SMALL TIME Take centered+scaling limit as d → ∞ and s = ve − T 0 , t = − T 0 + u , T 0 → ∞ , u ≥ 0 , v ≥ 0 . Large dimension; very small time. Imagine observing random transposition chain acting on infinite symmetric group.
O RNSTEIN -U HLENBECK T HEOREM (J OHNSON -P. ’12, G ANGULY -P. ’14) Joint convergence to Gaussian field: ( 2 d − 1 ) − k / 2 � 2 kN k ( − T 0 + u , ve − T 0 ) − E ( · ) � − → ( U k ( u , v )) . U k ( · , · ) - continuous Gaussian surfaces, independent among k. Infinite-dimensional O-U surface. Marginally N ( 0 , k / 2 ) . In dimension and time ( U k ) time-changed stationary O-U: dU k ( t , · ) = − kU k ( t , · ) dt + kdW k ( t ) , t ≥ 0 .
C OMPARISON WITH W IGNER Recall 2 kN k ≈ tr ( T k ( · )) . Allows to compute covariances of polynomials linear eigenvalue statistics. Same as GOE. A diffusion dynamics on the Gaussian Free Field.
Thank you Jim for all the beautiful math and happy birthday.
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