Rigidity of determinantal point processes on the unit disc with sub-Bergman kernels
aa r X i v : . [ m a t h . P R ] J a n RIGIDITY OF DETERMINANTAL POINT PROCESSES ON THEUNIT DISC WITH SUB-BERGMAN KERNELS
YANQI QIU AND KAI WANG
Abstract.
We give natural constructions of number rigid determinantal point processeson the unit disc D with sub-Bergman kernels of the form K Λ ( z, w ) = X n ∈ Λ ( n + 1) z n ¯ w n , z, w ∈ D , with Λ an infinite subset of the set of non-negative integers. Our constructions are givenboth in a deterministic method and a probabilistic method. In the deterministic method,our proofs involve the classical Bloch functions. Introduction
The present paper is devoted to investigate the existence of some non-trivial and natural number rigid determinantal point processes over bounded domains in the complex plane.1.1.
Determinantal point processes.
Let us first recall some notations and concepts ofdeterminantal point processes. Let M be a locally compact and complete separable metricspace equipped with a σ -finite non-negative measure µ . Denote by Conf( M ) the space ofconfigurations over M which consists of non-negative integer-valued Radon measures on M . The topology of vague convergence on the set of Radon measures makes Conf( M ) aPolish space. We call any Borel probability measure P on Conf( M ) a point process on M .A point process P on M is called determinantal if it admits a reproducing kernel function K : M × M → C such that E P h n Y i =1 ( B i )!( B i − n i )! i = Z B n ×···× B nmm det h K ( x i , x j ) i ni,j =1 dµ ( x ) · · · dµ ( x n )(1.1)for any disjoint bounded Borel sets B , · · · , B m , m ≥ , n i ≥ , n + · · · + n m = n. Herethe counting function B : Conf( M ) → N = { , , , · · · } is defined by B ( X ) = R B dX for any X ∈ Conf( M ).Determinantal point processes were introduced by Macchi in the early seventies to de-scribe random fermion fields in quantum theory [15] and later developed in Soshnikov[19, 20], Shairai-Takahashi [18] and many other authors. Determinantal point processesappear in many branches of mathematics, such as eigenvalues of unitarily invariant ran-dom matrices as well as the zeros of random analytic functions on the unit disc [14]. Inmost interesting cases, the kernel function K ( x, y ) yields an integral operator on L ( M, dµ )of locally trace class. In the case that K is a locally trace class operator, a characterization Mathematics Subject Classification.
Primary 60G55; Secondary 30B20, 30H20.
Key words and phrases. determinantal point processes, sub-Bergman kernels, Bloch functions, lacu-nary sequences.Y. Qiu is supported by grants NSFC Y7116335K1, NSFC 11801547 and NSFC 11688101 of NationalNatural Science Foundation of China. K. Wang is supported by grants NSFC 11722102. for a point process P to be determinantal and induced by K is given by the following: forany pairwise disjoint bounded Borel sets B , · · · , B m and any z , · · · , z m ∈ C , we have E P h m Y i =1 z B i i i = det( Id + m X i =1 ( z i − χ B i Kχ B i ) . We refer the reader to [2, 14, 18, 19, 20] for further background and details of determinantalpoint processes.We now recall the definition of the number rigidity of point processes. For a given Borelsubset C ⊂ M , let F C be the σ -algebra on Conf( M ) generated by all random variables B with all Borel subsets B ⊂ C . For any point process P on M , we denote by F P C the completion of the σ -algebra F C with respect to P . A point process P on M is callednumber rigid if for any bounded Borel set B ⊂ M , the random variable B is F P M \ B measurable. This definition of number rigidity is due to Ghosh [8] where he shows thatthe sine-process is number rigid and Ghosh-Peres [12] where they show that the Ginibreprocess and the zero set of Gaussian analytic function on the plane are number rigid.Bufetov[3] shows that determinantal point processes with the Airy, the Bessel and theGamma kernels are rigid. He indeed establishes a general theorem that rigidity holds forthe kernel on the real axis R with a mild condition of growth. For more results on thenumber rigidity of point processes, we refer the reader to [4, 5, 9, 10, 11, 17, 16].However, Holroyd and Soo showed that the determinantal point process on the unit disc D with the standard Bergman kernel (with respect to the normalized Lebesgue measureon the unit disc D ): K D ( z, w ) = 1(1 − z ¯ w ) = ∞ X n =0 ( n + 1) z n ¯ w n is not number rigid [13]. See also [7] for an alternative proof of this result. More generally,among many other things, Bufetov, Fan and Qiu [6] showed that for any domain U in the d -dimensional complex Euclidean space C d without Liouville property (that is, there existsa non-constant bounded holomorphic function f : U → C ) and any weight ω : U → R + locally away from zero, the determinantal point process associated with the reproducingkernel of the weighted Bergman space L a ( U ; ω ) is not number rigid.These negative results lead us to ask whether there exist natural number rigid determi-nantal point processes on a bounded domain of the complex plane (of course, any finiterank orthogonal projection yields a number rigid determinantal point process, so here weare only interested in infinite rank orthogonal projections). In this paper, we answer af-firmatively this question with a deterministic and a probabilistic method. It also inspiresus to construct a series of examples involving lacunary series in the Bloch space.1.2. Statements of main results.
From now on, we focus on the case of unit disc D equipped with the normalized Lebesgue measure dm . We shall consider determinantalpoint processes induced by the orthogonal projection kernels (which we call sub-Bergmankernels) of the form K Λ ( z, w ) = X n ∈ Λ ( n + 1) z n ¯ w n , PP WITH SUB-BERGMAN KERNELS 3 where Λ ⊂ N = { , , , · · · } is an infinite subset of N . Note that K Λ is the orthogonalprojection onto the following subspaces of the Bergman space L a ( D ) = L ( D ) ∩ Hol ( D ):span L ( D ) n z n (cid:12)(cid:12)(cid:12) n ∈ Λ o ⊂ L a ( D ) . To indicate the idea of our proofs, in what follows, given any f = P ∞ n =0 a n z n , we write K f ( z, w ) = ∞ X n =0 a n ( n + 1) z n ¯ w n . In particular, for a subset Λ ⊂ N , we denote f Λ ( z ) = X n ∈ Λ z n . (1.2)Recall the definition of Bloch space on the unit disc B := n f ∈ Hol ( D ) (cid:12)(cid:12)(cid:12) k f k B := sup z ∈ D (1 − | z | ) | f ′ ( z ) | < ∞ o . Theorem 1.1.
Let Λ ⊂ N be an infinite subset. Suppose that the function f Λ defined in (1.2) satisfies f Λ ∈ B . Then the determinantal point process on D induced by the kernel K Λ ( z, w ) = K f Λ ( z, w ) is number rigid. We have a criterion when f Λ is included in the Bloch space involving lacunary series asfollows. Let Λ = { λ , λ , · · · } be a subset of N with λ < λ < · · · . We say Λ lacunary ifit satisfies the gap condition ρ Λ := lim inf k ∈ N λ k +1 λ k > . (1.3)The following characterization was already hinted in the proof of [1, Lemma 10] by An-derson and Shields. We remark a short proof for completeness. Proposition 1.2.
Let Λ be a subset of N . We have that f Λ ∈ B if and only if Λ is afinite union of some lacunary subsets of N . Now we turn to the probabilistic method. Throughout the paper, suppose that ( ξ n ) ∞ n =0 is a sequence of independent Bernoulli random variables with ξ n = ( n +1 − n +1 . (1.4)We shall consider the random analytic function on the unit disc D : f ξ = ∞ X n =0 ξ n z n . (1.5)By Kolmogorov Three Series Theorem, almost surely, we have P ∞ n =0 ξ n = ∞ . Therefore,for almost every realization ( ξ n ) ∞ n =0 , the kernel K f ξ ( z, w ) = ∞ X n =0 ξ n ( n + 1) z n ¯ w n (1.6)is an orthogonal projection onto the following infinite dimensional subspacespan L ( D ) n z n (cid:12)(cid:12)(cid:12) n ∈ N such that ξ n = 1 o ⊂ L a ( D ) YANQI QIU AND KAI WANG and yields a determinantal point process on the unit disc D . Theorem 1.3.
For almost every realization ξ , the determinantal point process inducedby the kernel K f ξ ( z, w ) is number rigid. Our probabilistic method yields indeed different construction of number rigid determi-nantal point processes on D by the following Proposition 1.4.
Almost surely, the function f ξ is not included in the Bloch space B .Or equivalently, almost surely, the subset Λ ξ := { n ∈ N | ξ n = 1 } is not a finite union of lacunary subsets of N . Rigidity of DPP with sub-Bergman kernels
This section is devoted to establish the existence of number rigid determinantal pointprocesses on the unit disc D with sub-Bergman kernels.For a bounded measurable compactly supported function φ on D , we denote by S φ theadditive functional on the configuration space Conf( D ) defined by the formula S φ ( X ) = Z D φdX. The following sufficient condition for number rigidity of a point process is showed byGhosh [8] and Ghosh, Peres [12].
Proposition 2.1 (Ghosh and Peres) . Let P be a Borel probability measure on Conf( M ) .Assume that for any ǫ > , and any bounded subset B ⊆ M , there exists a boundedmeasurable function φ : M → C of compact support such that φ ≡ on B , and Var S φ ≤ ǫ .Then P is number rigid. Recall that for a point process P with an orthogonal projection kernel K ( x, y ) on theunit disc D of locally trace class, we haveVar S φ = 12 Z D Z D | φ ( x ) − φ ( y ) | · | K ( x, y ) | dm ( x ) dm ( y ) . (2.7)The following lemma is our key estimation. Lemma 2.2.
For any ǫ > and < r < , there exists a bounded measurable function h : [0 , → R of compact support on [0 , such that h ≡ on [0 , r ] and Z [0 , Z [0 , | h ( t ) − h ( s ) | − st ) dsdt < ǫ. We will postpone the proof for the lemma until the next section.2.1.
Rigid kernel via the Bloch functions.
Proof of Theorem . Assume that f Λ ∈ B and consider the determinantal point process P on D induced by the orthogonal projection kernel K f Λ ( z, w ) = K Λ ( z, w ). Since f Λ ∈ B ,by [21, Thm 5.13], there exists C >
0, such that(1 − | z | ) | f ′ Λ ( z ) | ≤ k f Λ k B and (1 − | z | ) | f ′′ Λ ( z ) | ≤ C k f Λ k B . (2.8) PP WITH SUB-BERGMAN KERNELS 5
Let φ : D → R + be any compactly supported bounded radial function. Then by (2.7), wehave2Var S φ = Z z ∈ D Z w ∈ D | φ ( z ) − φ ( w ) | | K Λ ( z, w ) | dm ( z ) dm ( w )= Z t ∈ [0 , Z s ∈ [0 , | φ ( t ) − φ ( s ) | h Z α ∈ [0 , π ] Z β ∈ [0 , π ] | K Λ ( te iα , se iβ ) | dαπ dβπ i tsdtds. Note that Z α ∈ [0 , π ] | K Λ ( te iα , se iβ ) | dα π = Z α ∈ [0 , π ] (cid:12)(cid:12)(cid:12) X n ∈ Λ ( n + 1) t n s n e in ( α − β ) (cid:12)(cid:12)(cid:12) dα π = X n ∈ Λ ( n + 1) ( ts ) n . Moreover, we have X n ∈ Λ ( n + 1) ( ts ) n = X n ∈ Λ [ n ( n − ts ) n − t s + 3 n ( ts ) n − t s + ( ts ) n ]= t s f ′′ Λ ( t s ) + 3 t s f ′ Λ ( t s ) + f Λ ( t s ) . Therefore, by (2.8), there exists C ′ > S φ = 4 Z [0 , Z [0 , | φ ( t ) − φ ( s ) | X n ∈ Λ ( n + 1) ( ts ) n tsdtds = 4 Z [0 , Z [0 , | φ ( t ) − φ ( s ) | h t s f ′′ Λ ( t s ) + 3 t s f ′ Λ ( t s ) + f Λ ( t s ) i tsdtds ≤ C ′ Z [0 , Z [0 , | φ ( t ) − φ ( s ) | − t s ) dtds ≤ C ′ Z [0 , Z [0 , | φ ( t ) − φ ( s ) | − ts ) dtds. For any ǫ > < r <
1, if we take h r ,ǫ to be the function appeared in Lemma2.2 and set φ r ,ǫ ( z ) = h r ,ǫ ( | z | ) , then φ r ,ǫ : D → R is a bounded measurable function of compact support such that φ r ,ǫ ≡ { z ∈ D : | z | ≤ r } and Var S φ r ,ǫ ≤ ǫ. Since any compact subset B ⊂ D is included in { z ∈ D : | z | ≤ r } for some r ∈ (0 , (cid:3) Proof of Proposition . Suppose that f Λ ( z ) = P n ∈ Λ z n ∈ B . We have that X k ∈ Λ kr k = rf ′ ( r ) ≤ k f k B − r . YANQI QIU AND KAI WANG
For any fixed integer M ≥
2, set r = 1 − M . Note that there exists c > r k > c for any k ≤ M . It follows that for the constant c ′ = c > X ≤ k ≤ M, k ∈ Λ k ≤ X ≤ k ≤ M, k ∈ Λ k r k c ≤ c ′ X k ∈ Λ kr k ≤ c ′ k f k B − r = c ′ k f k B M. This implies that X n ≤ k< n +1 , k ∈ Λ ≤ X n ≤ k< n +1 , k ∈ Λ k n ≤ c ′ k f k B n +1 n ≤ c ′ k f k B . To ease the notations, write q := [2 c ′ k f k B ] + 1 andΛ even := Λ ∩ [ m ∈ N I m , Λ odd := Λ ∩ [ m ∈ N I m +1 , where I n = { k ∈ N : 2 n ≤ k < n +1 } . Then there exist subsets { Λ eveni } qi =1 such thatΛ even = ∪ qi =1 Λ eveni , and each Λ eveni has at most one element inside I m and no elementincluded in I m +1 for any m ∈ N . That is, each Λ eveni is either a finite subset or a subsetwhich satisfies the gap condition (1.3) with the gap ratio not less than 2. This impliesΛ even is the union of at most q many lacunary subsets. The same argument also holds forΛ odd . Therefore, Λ is the union of at most 2 q many lacunary subsets.On the other hand, without loss of generality, suppose that Λ = { λ , λ , · · · } is alacunary subset with λ < λ < · · · and inf k ∈ N λ k +1 λ k ≥
2. This implies that X n ≤ k< n +1 , k ∈ Λ ≤ , ∀ n ∈ N and hence for any r ∈ (0 ,
1) and any integer n ≥
1, we have X n ≤ k< n +1 , k ∈ Λ kr k ≤ sup n ≤ k< n +1 kr k ≤ n +1 r n . Therefore, by noting X n − ≤ k< n r k ≥ r n (2 n − n − ) = 2 n +1 r n , for any r ∈ (0 , X k ∈ Λ ,k ≥ kr k = ∞ X n =1 X n ≤ k< n +1 ,k ∈ Λ kr k ≤ ∞ X n =1 n +1 r n ≤ ∞ X n =1 X n − ≤ k< n r k ≤ − r . This implies that f Λ ∈ B . (cid:3) Rigid kernel via probabilistic methods.
Proof of Theorem . By the definition (1.6) of the kernel K f ξ ( z, w ), we have that Z [0 , π ] | K f ξ ( te iα , se iβ ) | dα π = Z [0 , π ] (cid:12)(cid:12)(cid:12) ∞ X n =0 ξ n ( n + 1) t n s n e in ( α − β ) (cid:12)(cid:12)(cid:12) dα π = ∞ X n =0 ξ n ( n + 1) t n s n . (2.9) PP WITH SUB-BERGMAN KERNELS 7
For any compact subset B in the unit disc, there exists 0 < r < B ⊂ { z ∈ C : | z | ≤ r } . For such real number r ∈ (0 ,
1) and any ǫ >
0, let h r ,ǫ be the function appeared inLemma 2.2 and set φ B,ǫ ( z ) = h r ,ǫ ( | z | ) . (2.10)By (2.9), the definition (1.4) of the random variables ξ n and the following elementaryidentity ∞ X n =0 ( n + 1) x n = 1(1 − x ) , we obtain E (cid:20)Z z ∈ D Z w ∈ D | φ B,ǫ ( z ) − φ B,ǫ ( w ) | | K f ξ ( z, w ) | dm ( z ) dm ( w ) (cid:21) = E (cid:20)Z t ∈ [0 , Z s ∈ [0 , ts | h r ,ǫ ( t ) − h r ,ǫ ( s ) | dtds Z α ∈ [0 , π ] Z β ∈ [0 , π ] | K f ξ ( te iα , se iβ ) | dαπ dβπ (cid:21) = 4 Z [0 , Z [0 , ts | h r ,ǫ ( t ) − h r ,ǫ ( s ) | ∞ X n =0 ( n + 1) ( ts ) n E ξ n dtds = 4 Z [0 , Z [0 , | h r ,ǫ ( t ) − h r ,ǫ ( s ) | ∞ X n =0 ( n + 1)( ts ) n tsdtds = 4 Z [0 , Z [0 , | h r ,ǫ ( t ) − h r ,ǫ ( s ) | − s t ) tsdtds ≤ Z [0 , Z [0 , | h r ,ǫ ( t ) − h r ,ǫ ( s ) | − st ) dtds ≤ ǫ. Now for any integer n ≥
1, set φ n ( z ) := φ B,n − ( z ) . (2.11)By the above computation, for each integer n ≥
1, we have E (cid:20)Z z ∈ D Z w ∈ D | φ n ( z ) − φ n ( w ) | | K f ξ ( z, w ) | dm ( z ) dm ( w ) (cid:21) ≤ n and hence ∞ X n =1 E (cid:20)Z z ∈ D Z w ∈ D | φ n ( z ) − φ n ( w ) | | K f ξ ( z, w ) | dm ( z ) dm ( w ) (cid:21) < ∞ . Levi lemma implies that ∞ X n =1 Z z ∈ D Z w ∈ D | φ n ( z ) − φ n ( w ) | | K f ξ ( z, w ) | dm ( z ) dm ( w ) < ∞ , a.s. It follows that lim n →∞ Z z ∈ D Z w ∈ D | φ n ( z ) − φ n ( w ) | | K f ξ ( z, w ) | dm ( z ) dm ( w ) = 0 , a.s. (2.12) YANQI QIU AND KAI WANG
Note that by (2.10), (2.11) and the property of h r ,ǫ , we know that for each n ≥
1, thefunction φ n : D → R is a bounded measurable function of compact support such that φ n ≡ B . Therefore, by Proposition 2.1 and the equality (2.7), the limit relation(2.12) implies that, for almost every realization of ξ , the determinantal point processinduced by the orthogonal projection kernel K f ξ ( z, w ) is number rigid. (cid:3) Proof of Proposition . Let I n = (2 n , n +1 ] ∩ N and N n = P k ∈ I n ξ k . We claim that forany integer C ≥
1, lim sup n N n ≥ C, a.s.
Note that for a lacunary set Λ = { λ , λ , · · · } with the gap ratio ρ = lim inf k λ k +1 λ k > ∩ I n has most [ log 2log( ρ +1) / ] + 1 elements when n is sufficiently large. Moregenerally, for an integer set Λ = ∪ pi =1 Λ i with each lacunary set Λ i having the gap ratio ρ i , one has that the set Λ ∩ I n contains at most P pi =1 [ log 2log( ρ i +1) / ] + p elements when n issufficiently large. Combining this with the claim, it follows that almost surely, the subsetΛ ξ = { k ∈ N | ξ k = 1 } is not a finite union of lacunary sets.We next prove the claim. Noting that for k ∈ I n ,Prob[ ξ k = 1] = 1 k + 1 ≥ n +1 + 1and Prob[ ξ k = 0] = 1 − k + 1 ≥ − n , we have that for a fixed integer C ≥ N n = C ] ≥ X A ⊆ I n , | A | = C (cid:18)
11 + 2 n +1 (cid:19) C (cid:18) − n (cid:19) n − C = (cid:18) n C (cid:19) n +1 ) C (cid:18) − n (cid:19) n − C . Note that lim n →∞ (cid:18) n C (cid:19) n +1 ) C = lim n →∞ n (2 n − · · · (2 n − C + 1) C !(1 + 2 n +1 ) C = 12 C C !and lim n →∞ (cid:18) − n (cid:19) n − C = lim n →∞ { [1 − n ] n } n − C n = 1 e . Therefore, there exist an integer M and β > n > M , (cid:18) n C (cid:19) n +1 ) C (cid:18) − n (cid:19) n − C > β. This implies that, for any integer n > M ,Prob[ N n ≥ C ] ≥ Prob[ N n = C ] > β PP WITH SUB-BERGMAN KERNELS 9 and hence ∞ X n =0 Prob[ N n ≥ C ] = ∞ . Noting that the random variables N n are independent, by Borel-Cantelli lemma, we havelim sup n →∞ N n ≥ C a.s.
This completes the proof. (cid:3) Proof of Lemma h such that the integral in Lemma 2.2 issmall enough. Comparing with the trace formulatr( H ∗ ¯ g H ¯ g ) = Z D Z D | g ( z ) − g ( w ) | | − z ¯ w | dm ( z ) dm ( w )for g ∈ B in the classical Hankel operator theory in Bergman space [21], one may guess thata function h on [0 ,
1] would satisfies the requirement in Lemma 2.2 if z g ( z ) = h ( | z | )has a small VMO norm, or the growth of such g is slow than the Poincar´e metric.Recall that the Poincar´e metric on D is defined by ρ ( z , z ) = log 1 + ϕ z ( z )1 − ϕ z ( z ) = log | z − z | + | − ¯ z z || z − z | − | − ¯ z z | , where ϕ z ( z ) = z − z − ¯ z z is the Mobius transformation on the unit disc D . For 0 < r 1, we write(3.13) h ( t ) = h ( r ,r ) ( t ) = t ≤ r ρ ( t,r ) ρ ( r ,r ) r ≤ t ≤ r t ≥ r . The following theorem implies our technique lemma in the above section. Theorem 3.1. For fixed < r < , we have that the integral Z [0 , Z [0 , | h ( t ) − h ( s ) | − st ) dsdt tends to when ϕ r ( r ) → .Proof. When 0 < t, s < r or r < t, s < 1, then the expression of the integral is equal tozero. So we shall estimate our integral over the following three domains0 < t < r < r < s ; r < t < s < r ; r < t < r < s. The first case: the integral ( I ) over the domain 0 < t < r < r < s can be calculatedexplicitly. ( I ) = Z r dt Z r ds − st ) = Z r ds (cid:20) s (1 − st ) (cid:12)(cid:12)(cid:12) r (cid:21) = Z r r − r s ds = log 1 − r r − r = log 1 + r r ϕ r ( r ) . Therefore, for fixed r , the integral ( I ) tends to 0 when ϕ r ( r ) → II ) over the domain r < t < s < r :( II ) = Z rr dt Z rt ds ρ ( s, t ) ρ ( r , r ) 1(1 − st ) . By the substitutions t → ϕ r ( t ) , s → ϕ r ( s ), we obtain Z rr dt Z rt ds ρ ( s, t ) ρ ( r , r ) 1(1 − st ) = Z ϕ r ( r )0 − r (1 + r t ) dt Z ϕ r ( r ) t − r (1 + r s ) ds ρ ( s, t ) ρ (0 , ϕ r ( r )) 1(1 − s + r r s t + r r t ) = Z ϕ r ( r )0 dt Z ϕ r ( r ) t ds ρ ( s, t ) ρ (0 , ϕ r ( r )) 1(1 − st ) . Now make the substitution s → ϕ t ( s ), we get( II ) = Z ϕ r ( r )0 dt Z ϕ r ( r ) t ds ρ ( s, r ) ρ (0 , ϕ r ( r )) 1(1 − st ) = Z ϕ r ( r )0 dt Z ϕ t ( ϕ r ( r ))0 ds ρ ( s, ρ (0 , ϕ r ( r )) 11 − t ≤ ρ (0 , ϕ r ( r )) Z ϕ r ( r )0 − t dt Z ρ ( s, ds = C ρ (0 , ϕ r ( r )) , where C := R ρ ( s, ds < ∞ . Therefore, integral ( II ) tends to 0 when ϕ r ( r ) → . The third case: We now estimate the integral ( III )( III ) = Z rr dt Z r ds ρ ( r, t ) ρ ( r , r ) 1(1 − st ) . PP WITH SUB-BERGMAN KERNELS 11 With the substitutions t → ϕ r ( t ) , s → ϕ r ( s ), the change of variables formula yields that( III ) = Z rr dt Z r ds ρ ( r, t ) ρ ( r , r ) 1(1 − st ) = Z − ϕ r ( r ) dt Z ds ρ (0 , t ) ρ (0 , ϕ r ( r )) 1(1 − st ) ≤ ρ (0 , ϕ r ( r )) Z − ρ ( t, dt Z ds = C ρ (0 , ϕ r ( r )) . Therefore, integral ( III ) tends to 0 when ϕ r ( r ) → , which completes the proof. (cid:3) References [1] J. M. Anderson, A. L. Shields, Coefficient multipliers of Bloch functions , Trans. Amer. Math. 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Soshnikov, Gaussian limit for determinantal random point fields , Annals of probability 30(2001),no. 1, 1–17.[21] K. Zhu, Operator theory in function spaces, Mathematical Surveys and Monographs, 138, AmericanMathematical Society, Providence, RI, 2007. Yanqi QIU: Institute of Mathematics and Hua Loo-Keng Key Laboratory of Mathe-matics, AMSS, Chinese Academy of Sciences, Beijing 100190, China. E-mail address : [email protected]; [email protected] Kai WANG: School of Mathematical Sciences, Fudan University, Shanghai, 200433,China. E-mail address ::