Limit theorems for determinantal point processes Tomoyuki Shirai 1 2 Kyushu University May. 8, 2019 1 Joint work with Makoto Katori (Chuo University) 2 A Probability Conference “Random Matrices and Related Topics” @KIAS, Seoul, Korea, May 6–10, 2019. Tomoyuki Shirai (Kyushu University) Limit theorems for determinantal point processes May. 8, 2019 1 / 37
Content of this talk 1 Brief review on determinantal point processes (DPPs) 2 L 1 -limit for generalized accumulated spectrograms 3 Circular Unitary Ensemble (CUE) 4 Two DPPs on the 2-dimensional sphere and limit theorems 5 An extension to the d -dimensional sphere 6 An extension to compact Riemannian manifolds Tomoyuki Shirai (Kyushu University) Limit theorems for determinantal point processes May. 8, 2019 2 / 37
Reproducing kernel Hilbert space (RKHS) Let S be a set and H a Hilbert space of C -valued functions on S . H is said to be a reproducing kernel Hilbert space (RKHS) if, for every y ∈ S , the point evaluation map L y : H → C L y ( f ) = f ( y ) ( f ∈ H ) is bounded (continuous). Since L y is a bounded linear functional, by Riesz’s theorem, we have K y ∈ H such that L y ( f ) = ( f , K y ) H . K ( x , y ) := K y ( x ) is called a reproducing kernel in the sense that f ( y ) = ( f , K ( · , y )) H ∀ f ∈ H , ∀ y ∈ S . Theorem (Moore-Aronszain) Let K be a Hermitian positive definite kernel K : S × S → C . Then, there exists a unique Hilbert space H K of C -valued functions on S for which K is a reproducing kernel. Tomoyuki Shirai (Kyushu University) Limit theorems for determinantal point processes May. 8, 2019 3 / 37
Example: Paley-Wiener space Band limited functions: PW a = { f ∈ L 2 ( R ) : supp � f ⊂ [ − a , a ] } , ∫ where � R f ( x ) e − i ω x dx . f ( ω ) = √ a � � ∫ a � � | f ( x ) | ≤ 1 � f ( ω ) e i ω x d ω � � � ≤ π ∥ f ∥ . � 2 π − a Reproducing kernel: K a ( x , y ) = sin a ( x − y ) → δ y ( x ) ( a → ∞ ) . π ( x − y ) RKHS ( PW a , K a ) is called the Paley-Wiener space. Tomoyuki Shirai (Kyushu University) Limit theorems for determinantal point processes May. 8, 2019 4 / 37
Determinantal point processes (DPPs) We recall determinantal point processes (DPPs) on S . S : a base space (locally compact Polish space) λ ( ds ): Radon measure on S Conf ( S ) = { ξ = ∑ i δ x i : x i ∈ S , ξ ( K ) < ∞ for all bounded set K } : the set of Z ≥ 0 -valued Radon measures H K ⊂ L 2 ( S , λ ): reproducing kernel Hilbert space (RKHS) with kernel K ( · , · ) : S × S → C . Theorem (Determinantal point process with ( K , λ ) or H K ) There exists a point process ξ = ξ ( ω ) on S , i.e., a Conf ( S )-valued random variable such that the n th correlation function w.r.t. λ ⊗ n is given by ρ n ( s 1 , . . . , s n ) = det( K ( s i , s j )) n i , j =1 . Tomoyuki Shirai (Kyushu University) Limit theorems for determinantal point processes May. 8, 2019 5 / 37
DPP and Gaussian process – RKHS RKHS Moore-Aronszain Linear structure correlation kernel Determinantal point process Positive definite kernel K covariance kernel Gaussian process Tomoyuki Shirai (Kyushu University) Limit theorems for determinantal point processes May. 8, 2019 6 / 37
Determinantal point processes (DPPs) II Example. (Paley-Wiener Space): S = R , λ ( dx ) = dx and K ( x , y ) = sin a ( x − y ) . x − y The RHKS H K is PW a , then the corresponding DPP is the limiting CUE (also GUE) eigenvalues process. Later we will discuss a generalization of this procss. Example (Bargmann-Fock space): S = C and λ ( dz ) = π − 1 e −| z | 2 dz and K ( z , w ) = e zw . The RKHS H K is the Bargmann-Fock space, i.e., H K := { f ∈ L 2 ( C , λ ) : f is entire } The DPP in this case is the Ginibre point process. Tomoyuki Shirai (Kyushu University) Limit theorems for determinantal point processes May. 8, 2019 7 / 37
Determinantal point processes (DPPs) III Number of points: If K is of rank N , i.e., dim H K is N , then the number of points is N a.s. Density of points w.r.t. λ ( dx ) and negative correlation: ρ 1 ( x ) = K ( x , x ) ρ 2 ( x , y ) = K ( x , x ) K ( y , y ) − | K ( x , y ) | 2 ≤ ρ 1 ( x ) ρ 1 ( y ) Gauge invariance: For u : S → U (1), a gauge transformation K ( s , t ) �→ ˜ K ( s , t ) := u ( s ) K ( s , t ) u ( t ) does not change the law of DPP. Scaling property: When S = R d and λ ( dx ) = dx , for a configuration ξ = ∑ i δ x i , we define ∑ S c ( ξ ) = δ cx i . i If ξ ( ω ) is DPP with K , then S c ( ξ ( ω )) is also DPP with K c ( x , y ) = c − d K ( c − 1 x , c − 1 y ) Tomoyuki Shirai (Kyushu University) Limit theorems for determinantal point processes May. 8, 2019 8 / 37
DPP associated with partial isometry We say that W : L 2 ( S 1 , λ 1 ) → L 2 ( S 2 , λ 2 ) is partial isometry if for all f ∈ (ker W ) ⊥ ∥W f ∥ L 2 ( S 2 ,λ 2 ) = ∥ f ∥ L 2 ( S 1 ,λ 1 ) Let W : L 2 ( S 1 , λ 1 ) → L 2 ( S 2 , λ 2 ) and its dual W ∗ : L 2 ( S 2 , λ 2 ) → L 2 ( S 1 , λ 1 ) be partial isometries, or equivalently, K 2 := WW ∗ (orthogonal projections) K 1 = W ∗ W , Suppose that both K 1 and K 2 are of locally trace class, i.e., P Λ 1 K 1 P Λ 1 , P Λ 2 K 2 P Λ 2 are of trace class for any bounded set Λ i ⊂ S i ( i = 1 , 2). Then K 1 and K 2 admit kernel K 1 ( x , x ′ ) and K 2 ( y , y ′ ), which are reproducing kernels of (ker W ) ⊥ and (ker W ∗ ) ⊥ , respectively. Let Ξ i ( i = 1 , 2) be the DPPs associated with ( K i , λ i ) ( i = 1 , 2), respectively. M.Katori-T.Shirai, Partial Isometry, Duality, and Determinantal Point Processes, available at https://arxiv.org/abs/1903.04945 Tomoyuki Shirai (Kyushu University) Limit theorems for determinantal point processes May. 8, 2019 9 / 37
Orthogonal polynomial ensemble (1) Orthogonal polynomial ensemble . W : L 2 ( R , λ ) → ℓ 2 ( Z ≥ 0 ) defined by the kernel ∫ ( W f )( n ) = φ n ( x ) f ( x ) λ ( dx ) R where { φ n ( x ) } n ∈ Z ≥ 0 are orthonormal polynomials for L 2 ( R , λ ). N − 1 ∑ K { 0 , 1 ,..., N − 1 } ( x , y ) = φ j ( x ) φ j ( y ) = ⇒ DPP Ξ 1 on R . 1 j =0 ∫ ∞ K [ r , ∞ ) ( n , m ) = φ n ( x ) φ m ( x ) λ ( dx ) = ⇒ DPP Ξ 2 on Z ≥ 0 . 2 r Duality relation: for any m = 0 , 1 , . . . , ( ) ( ) P Ξ 1 ([ r , ∞ )) = m = P Ξ 2 ( { 0 , 1 , . . . , N − 1 } ) = m Tomoyuki Shirai (Kyushu University) Limit theorems for determinantal point processes May. 8, 2019 10 / 37
Weyl-Heisenberg ensemble (2) Weyl-Heisenberg ensemble (Abreu-Pereira-Romero-Torquato(’17)) : W : L 2 ( R d ) → L 2 ( R d × R d ) is the short-time Fourier transform defined by ∫ z := ( x , ξ ) ∈ R d × R d , R d f ( t ) g ( t − x ) e 2 π i ξ t dt , W f ( z ) = where g is a window function such that ∥ g ∥ L 2 ( R d ) = 1. It is easy to see that K = WW ∗ (orthogonal proj. on L 2 ( R d × R d )) . W ∗ W = I L 2 ( R d ) , DPP on R d × R d associated with K is called Weyl-Heisenberg ensemble. Example: When d = 1, g ( t ) = 2 1 / 4 e − π t 2 , by identifying R × R with C , we have K 2 ( z , w ) = e i π Re z Im z e i π Re w Im w e π { zw − 1 2 ( | z | 2 + | w | 2 ) } ( z , w ∈ C ) . The corresponding Weyl-Heisenberg ensemble is the Ginibre point process. Tomoyuki Shirai (Kyushu University) Limit theorems for determinantal point processes May. 8, 2019 11 / 37
DPP associated with partial isometry We focus on a generalized framework of Weyl-Heisenberg ensembles. Let W : L 2 ( S 1 , λ 1 ) → L 2 ( S 2 , λ 2 ) be an isometry and its dual W ∗ : L 2 ( S 2 , λ 2 ) → L 2 ( S 1 , λ 1 ) be a partial isometry, i.e., W ∗ W = I L 2 ( S 1 ,λ 1 ) , WW ∗ =: K 2 (orthogonal projection on (ker W ∗ ) ⊥ Suppose that K 2 is of locally trace class, i.e., K 2 admits a kernel K 2 ( y , y ′ ). Let Ξ 2 the DPP on S 2 associated with ( K 2 , λ 2 ). Tomoyuki Shirai (Kyushu University) Limit theorems for determinantal point processes May. 8, 2019 12 / 37
Generalized accumulated spectrogram Ξ 2 is the DPP on S 2 associated with ( K 2 , λ 2 ). For Λ ⊂ S 2 such that E [Ξ 2 (Λ)] < ∞ , we define the restriction ( K 2 ) Λ := P Λ K 2 P Λ (trace class) and consider the eigenvalue problem ( K 2 ) Λ Φ (Λ) = µ (Λ) Φ (Λ) ( j = 1 , 2 , . . . ) j j j such that 1 ≥ µ (Λ) ≥ µ (Λ) ≥ · · · ≥ 0 1 2 and Φ (Λ) is the normalized eigenfunction for µ (Λ) . j j Set N Λ = ⌈ E [Ξ 2 (Λ)] ⌉ and define a generalized accumulated spectrogram N Λ ∑ | Φ (Λ) ( y ) | 2 ρ Λ ( y ) := ( y ∈ S 2 ) . j j =1 Tomoyuki Shirai (Kyushu University) Limit theorems for determinantal point processes May. 8, 2019 13 / 37
Example 1 Weyl-Heisenberg case (Ginibre case): For Λ ⊂ R × R ≃ C , we set N Λ = ⌈ E [Ξ(Λ)] ⌉ and define N Λ ∑ ( π z ) j | z | 2 j e − π | z | 2 ρ Λ ( z ) := (accumulated spectrogram) , j ! j =1 where N Λ = ⌈ E [Ξ(Λ)] ⌉ . (Corresponding to Circular law for Ginibre) Let D 1 = { ( x , ξ ) ∈ R 2 : x 2 + ξ 2 ≤ 1 } ⊂ C . As R → ∞ , in L 1 ( C ) , ρ R D 1 ( R · ) → 1 D 1 where N R D 1 ≈ π R 2 . Tomoyuki Shirai (Kyushu University) Limit theorems for determinantal point processes May. 8, 2019 14 / 37
Example 2 Weyl-Heisenberg case (Ginibre case): For Λ = star, we have the following figure. In the talk, I used here the figure 3 in the following paper. L. D. Abreu, K. Gr¨ ochenig, and J. L. Romero, On accumulated spectrograms, Trans. Amer. Math. Soc 368 (2016), 3629-3649. Tomoyuki Shirai (Kyushu University) Limit theorems for determinantal point processes May. 8, 2019 15 / 37
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