On Number Rigidity for Pfaffian Point Processes
aa r X i v : . [ m a t h . P R ] O c t ON NUMBER RIGIDITY FOR PFAFFIAN POINT PROCESSES
ALEXANDER I. BUFETOV, PAVEL P. NIKITIN, AND YANQI QIU
Abstract.
Our first result states that the orthogonal and symplectic Bessel processesare rigid in the sense of Ghosh and Peres. Our argument in the Bessel case proceedsby an estimate of the variance of additive statistics in the spirit of Ghosh and Peres.Second, a sufficient condition for number rigidity of stationary Pfaffian processes, relyingon the Kolmogorov criterion for interpolation of stationary processes and applicable, inparticular, to pfaffian sine-processes, is given in terms of the asymptotics of the spectralmeasure for additive statistics. Introduction
Point processes and number rigidity.
Let Conf( R ) be the set of non-negativeinteger-valued Radon measures on the real line R . Elements of Conf( R ) are called (locallyfinite) configurations on R . The space Conf( R ) is a Polish space equipped with the vaguetopology generated by the maps: ξ Z R f dξ for compactly supported continuous functions f : R → C .By definition, a point process on R is a Borel probability measure on Conf( R ).A configuration ξ ∈ Conf( R ) is called simple if ξ ( { x } ) ∈ { , } for all x ∈ R . A pointprocess P is called simple, if P -almost every configuration is simple.Given an element ξ ∈ Conf( R ) and any Borel subset S ⊂ R , we denote ξ | S the restric-tion of the measure ξ on S . Definition 1.1 (Ghosh [7], Ghosh-Peres[8]) . A point process P on R is called number rigid if for any bounded Borel subset B ⊂ R , there exists a Borel function F B : Conf( R ) → Z such that ξ ( B ) = F B ( ξ | R \ B ) , for P -almost every ξ ∈ Conf( R ) . Pfaffian point processes.
Recall that for a simple point process P on R , the k -point correlation function ρ ( k ) P of P with respect to the Lebesgue measure, if it exists, is thenon-negative function ρ ( k ) P : R k → R such that for any continuous compactly supportedfunction ϕ : R k → C , we have Z Conf( R ) ∗ X x ,...,x k ∈X ϕ ( x , . . . , x k ) P ( d X ) = Z R k ϕ ( x , . . . , x k ) ρ ( k ) P ( x , . . . , x k ) dx · · · dx k , where ∗ P denotes the sum over all ordered k -tuples of distinct points ( x , . . . , x k ) ∈ X k . Mathematics Subject Classification.
Primary 60G55; Secondary 60G10.
Key words and phrases.
Pfaffian point process, stationary point process, number rigidity.
A simple point process P on R is said to be a Pfaffian point process if there existsa matrix kernel K : R × R → C × such that for all positive integers k , the k -pointcorrelation functions of P exist and have the form ρ ( k ) P ( x , · · · , x k ) = Pf[ K ( x i , x j ) J ] ≤ i,j ≤ k . Here Pf( A ) is the Pfaffian of an antisymmetric matrix and the matrix kernel K mustsatisfy the condition ( K ( x, y ) J ) t = − K ( y, x ) J, where J = (cid:20) − (cid:21) (1.1)to ensure that the 2 k × k matrix [ K ( x i , x j ) J ] ≤ i,j ≤ k is antisymmetric. In this situation,we say that the point process P is the Pfaffian point process induced by the matrix kernel K and is denoted P K .We can write the matrix kernel K as K ( x, y ) = (cid:20) K ( x, y ) K ( x, y ) K ( x, y ) K ( x, y ) (cid:21) , (1.2)where the entries K ij : R × R → C are scalar functions and then the condition (1.1) saysthat(1.3) K ( x, y ) = K ( y, x ) , K ( x, y ) = − K ( y, x ) , K ( x, y ) = − K ( y, x ) . We recall the general structure of the Pfaffian kernels for β = 4 (symplectic) ensemblesand β = 1 (orthogonal) ensembles and their scaling limits: K ( x, y ) = 12 (cid:20) K ( x, y ) − R yx K ( x, t ) dt ∂∂x K ( x, y ) K ( y, x ) (cid:21) , (1.4) K ( x, y ) = (cid:20) K ( x, y ) − R yx K ( x, t ) dt − / x − y ) ∂∂x K ( x, y ) K ( y, x ) (cid:21) , (1.5)for some particular K ( x, y ), K ( x, y ). In the integrable case a kernel has the followingform K β ( x, y ) = A ( x ) B ( y ) − B ( x ) A ( y ) x − y + C ( x ) D ( y ) . Section 2 is devoted to the Pfaffian Bessel processes. Recall that a classical polynomial β -ensemble is defined by the probability density functionconst( β, w β ) N Y i =1 w β ( x i ) Y ≤ i The symplectic Bessel kernel K Bessel ,s ( x, y ) is given by the formula K Bessel ,s ( x, y ) = x / J s +1 ( x / ) J s ( y / ) − y / J s +1 ( y / ) J s ( x / )2( x − y ) ,K Bessels ( x, y ) = 2 (cid:18) xy (cid:19) / K Bessel , s − (4 x, y ) − J s − (2 y / )2 y / Z x / J s − (2 t ) dt, K Bessel ,s ( x, y ) = (cid:20) K Bessels ( x, y ) R xy K Bessels ( x, t ) dt ∂∂x K Bessels ( x, y ) K Bessels ( y, x ) (cid:21) , where s > 0. Regarding the formula for K Bessels ( x, y ), see Proposition 2.15.The orthogonal Bessel kernel K Bessel ,s ( x, y ) is given by the formula K ,s ( x, y ) = (cid:18) xy (cid:19) / K Bessel ,s +1 ( x, y ) + J s +1 ( y / )4 y / Z ∞ x / J s +1 ( t ) dt, K Bessel ,s ( x, y ) = (cid:20) K ,s ( x, y ) − R yx K ,s ( x, t ) dt − / x − y ) ∂∂x K ,s ( x, y ) K ,s ( y, x ) (cid:21) . Theorem 1.2. (i) The symplectic Bessel process is number rigid.(ii) The orthogonal Bessel process is number rigid. The number rigidity of the Pfaffian Bessel processes is proved in subsections 2.3, 2.4.We use the following sufficient condition for the number rigidity of a point process due toGhosh and Peres. Proposition 1.3 (Ghosh [7], Ghosh and Peres [8]) . Let M be a complete metric separablespace. Let P be a Borel probability measure on Conf( M ) . Assume that for any ε > and any bounded subset B ⊂ M there exists a bounded measurable function f of boundedsupport such that f ≡ on B and Var P S f < ε , where S f ( X ) = P x ∈ X f ( x ) , X ∈ Conf( M ) .Then the measure P is number rigid. We give an explicit formula for the variance of an additive functional of a Pfaffianpoint process in (2.11), subsection 2.1. Preliminary integral estimates are discussed insubsection 2.2. A difference with the determinantal case can be seen as follows. Considera point process on R with the first correlation function ρ (1) ( x ), second corrleation function ρ (2) ( x, y ) and truncated second correlation function ρ (2 ,T ) ( x, y ) = ρ (2) ( x, y ) − ρ (1) ( x ) ρ (1) ( y ).For a determinantal process governed by an orthogonal projection, we have(1.6) Z R ρ (2 ,T ) ( x, y ) dy = − ρ (1) ( x ) , In a discussion of perfect screening in [6], 14.1, p.660, Forrester writes that property (1.6)should “remain valid in the thermodynamic limit”. We show that (1.6) is not valid forthe Pfaffian Bessel processes, see (2.33), (2.39). Nonetheless, a weaker integral propertyholds (see Proposition 2.16 and Proposition 2.19) and suffices for our purposes.We next give a general sufficient condition for the number rigidity of a stationary pointprocess convenient for working with stationary Pfaffian processes. Let P be a stationarypoint process on R admitting the first and the second correlation functions ρ (1) P and ρ (2) P .The first correlation function is a constant, and we set ρ = ρ (1) P ( x ) . ALEXANDER I. BUFETOV, PAVEL P. NIKITIN, AND YANQI QIU Denote also F ( x ) = ρ (2) P ( x, − ρ . Proposition 1.4. Assume that there exists C > such that the Fourier transform b F of F satisfies ≤ b F ( λ ) + ρ ≤ C | λ | for all λ ∈ R . (1.7) Then the point process P is number rigid. For example, Pfaffian sine processes arise as bulk scaling limits of the Pfaffian Gaussianensembles, see [6], 7.6.1. and 7.8.1. Let K sine , ( x, y ) = S ( x − y ) , S ( x ) := sin( πx ) πx be the standard sine-kernel. Then, the orthogonal sine process or Sine -process, is thePfaffian point process on R with a matrix correlation kernel K sine , ( x, y ) = (cid:20) S ( x − y ) IS ( x − y ) − ε ( x − y ) S ′ ( x − y ) S ( x − y ) (cid:21) , where IS ( x ) := Z x S ( t ) dt and ε ( x ) := 12 sgn( x ) , (1.8)and the symplectic sine process, the Sine -process, is the Pfaffian point process on R witha matrix correlation kernel K sine , ( x, y ) = 12 (cid:20) S ( x − y ) IS ( x − y ) S ′ ( x − y ) S ( x − y ) (cid:21) . In these cases, the quantity b F ( λ ) − ρ = b F ( λ ) − b F (0) is computed in Forrester [6]. For theorthogonal sine process, Forrester [6, formula (7.136)] gives b F ( λ ) − b F (0) = ( | λ | − | λ | log(1 + 2 | λ | ) if | λ | ≤ − | λ | log | λ | +12 | λ |− if | λ | ≥ . For the symplectic sine process, Forrester [6, formula (7.95)] gives b F ( λ ) − b F (0) = | λ | − | λ | log (cid:12)(cid:12)(cid:12) − | λ | (cid:12)(cid:12)(cid:12) if | λ | ≤ if | λ | ≥ . In both the orthogonal and symplectic case, we have0 ≤ b F ( λ ) − b F (0) ≤ C | λ | , and Proposition 1.4 yields Proposition 1.5. The orthogonal sine process is number rigid. Proposition 1.6. The symplectic sine process is number rigid. Remark. For the general Sine β processes, rigidity is due to Chhaibi and Najnudel [5] , andPropositions , of course follow from their result. Their argument is quite different.It would be interesting to obtain a spectral asymptotics at zero for general Sine β processes. Remark. The soft edge scaling limit yields Pfaffian Airy kernels, and it would be inter-esting to prove the rigidity of the corresponding point processes. N NUMBER RIGIDITY FOR PFAFFIAN POINT PROCESSES 5 Pfaffian Bessel point processes Variance of an additive functional. For a general point process we have thefollowing formula for the variance of S f ( X ) = P x ∈ X f ( x ), X ∈ Conf( M ).Var( S f ) = E P K ( | S f | ) − | E P K ( S f ) | == E P K (cid:18)X x ∈ X | f ( x ) | (cid:19) + E P K (cid:18) X x,y ∈ X,x = y f ( x ) f ( y ) (cid:19) − (cid:12)(cid:12)(cid:12)(cid:12) E P K (cid:18)X x ∈ X f ( x ) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) == Z R | f ( x ) | ρ (1) P K ( x ) dx + Z R f ( x ) f ( y ) ρ (2) P K ( x, y ) dxdy − (cid:12)(cid:12)(cid:12)(cid:12)Z R f ( x ) ρ (1) P K ( x ) dx (cid:12)(cid:12)(cid:12)(cid:12) == Z R | f ( x ) | ρ (1) P K ( x ) dx + Z R f ( x ) f ( y ) ρ (2 ,T ) P K ( x, y ) dxdy == Z R | f ( x ) | (cid:18) ρ (1) P K ( x ) + Z R ρ (2 ,T ) P K ( x, y ) dy (cid:19) dx − Z R | f ( x ) − f ( y ) | ρ (2 ,T ) P K ( x, y ) dxdy, where ρ (2 ,T ) P K ( x, y ) = ρ (2) P K ( x, y ) − ρ (1) P K ( x ) ρ (1) P K ( y ) is the truncated second correlation function of P K . If we additionally have (1.6), then we finally obtain the formula(2.9) Var( S f ) = − Z R | f ( x ) − f ( y ) | ρ (2 ,T ) P K ( x, y ) dxdy. For a Pfaffian point process induced by a matrix kernel K , by definition, we have ρ (1) P K ( x ) = K , ( x, x ) , ρ (2 ,T ) P K ( x, y ) = − det K ( x, y ) . Thus using (2.9) we obtain that if the condition (1.6) holds, then(2.10) Var( S f ) = 12 Z R | f ( x ) − f ( y ) | det K ( x, y ) dxdy and in general case, we have(2.11) Var( S f ) = Z R | f ( x ) | (cid:18) K , ( x, x ) − Z R det K ( x, y ) dy (cid:19) dx ++ 12 Z R | f ( x ) − f ( y ) | det K ( x, y ) dxdy. We recall once again the formulas for the kernels for β = 4 (symplectic ensembles) and β = 1 (orthogonal ensembles): K ( x, y ) = 12 (cid:20) K ( x, y ) − R yx K ( x, t ) dt ∂∂x K ( x, y ) K ( y, x ) (cid:21) , K ( x, y ) = (cid:20) K ( x, y ) − R yx K ( x, t ) dt − sgn( x − y ) ∂∂x K ( x, y ) K ( y, x ) (cid:21) . Remark 2.1. We will consider Pfaffian sine and Bessel processes, they arise as limitsof the Pfaffian polynomial ensembles. Kernels of these polynomial ensembles are skew-symmetric by construction, therefore the limit kernels are also skew-symmetric (it is notobvious from the definition of the Pfaffian Bessel kernel). We note also that det K ( x, y ) is a symmetric function if K ( x, y ) J is skew-symmetric. ALEXANDER I. BUFETOV, PAVEL P. NIKITIN, AND YANQI QIU Proposition 2.2. If K ( x, y ) is a projection, ∂∂x K ( x, y ) is skew-symmetric function, andfor any x ∈ R we have lim y →±∞ K ( y, x ) = 0 , lim x →±∞ K ( y, x ) Z yx K ( x, t ) dt = 0 , then the condition (1.6) holds for the kernels K ( x, y ) and K ( x, y ) .Proof. For K ( x, y ) we havedet K ( x, y ) = 14 (cid:18) K ( x, y ) K ( y, x ) + ∂∂x K ( x, y ) Z yx K ( x, t ) dt (cid:19) , and we use the equality ∂∂x K ( x, y ) = − ∂∂y K ( y, x ) to write Z R (cid:18) ∂∂x K ( x, y ) Z yx K ( x, t ) dt (cid:19) dy = − Z R (cid:18) ∂∂y K ( y, x ) Z yx K ( x, t ) dt (cid:19) dy = − K ( y, x ) Z yx K ( x, t ) dt (cid:12)(cid:12)(cid:12)(cid:12) ∞−∞ + Z R K ( y, x ) K ( x, y ) dy = Z R K ( x, y ) K ( y, x ) dy, and finally Z R det K ( x, y ) dy = 12 Z R K ( x, y ) K ( y, x ) dy = 12 K ( x, x ) . For K ( x, y ) we havedet K ( x, y ) = (cid:18) K ( x, y ) K ( y, x ) + ∂∂x K ( x, y ) Z yx K ( x, t ) dt (cid:19) ++ 12 sgn( x − y ) ∂∂x K ( x, y ) = 4 det K ( x, y ) − 12 sgn( x − y ) ∂∂y K ( y, x ) . Integrating the last term we obtain − Z R sgn( x − y ) ∂∂y K ( y, x ) dy = − Z x −∞ ∂∂y K ( y, x ) dy ++ 12 Z ∞ x ∂∂y K ( y, x ) dy = − K ( x, x ) , thus Z R det K ( x, y ) dy = 2 K ( x, x ) − K ( x, x ) = K ( x, x ) . (cid:3) For a Pfaffian process with a kernel of the form K ( x, y ) or K ( x, y ), let us define thedefect of the process as the differenceDef K ( x ) := Z R K ( x, y ) K ( y, x ) dy − K ( x, x ) . We have the following corollary from the proof of the previous proposition; it will beuseful for the analysis of the Bessel processes later. N NUMBER RIGIDITY FOR PFAFFIAN POINT PROCESSES 7 Corollary 2.3. For K = K we have − Z R ρ (2 ,T ) P K ( x, y ) dy − ρ (1) P K ( x ) = 14 h K ( x ) − K ( y, x ) Z yx K ( x, t ) dt (cid:12)(cid:12)(cid:12)(cid:12) y = ∞ y = −∞ i . For K = K , if for any x ∈ R we have lim y →±∞ K ( y, x ) = 0 , then − Z R ρ (2 ,T ) P K ( x, y ) dy − ρ (1) P K ( x ) = 2 Def K ( x ) − K ( y, x ) Z yx K ( x, t ) dt (cid:12)(cid:12)(cid:12)(cid:12) y = ∞ y = −∞ . Preliminary propositions. We plan to estimate the variance (2.11) term by term.From the formulas in the previous section we see that it leads to the estimates of theintegrals of the type(2.12) Z R | f ( x ) − f ( y ) | Π( x, y ) dxdy, where for most of the summands we will have Π( x, y ) = A ( x, y ) / ( x − y ) , or Π( x, y ) = B ( x, y ) / ( x − y ) or Π( x, y ) = C ( x ) D ( y ) for some reasonable functions A ( x, y ), B ( x, y ), C ( x ) , D ( y ). The corresponding integrals usually will be not absolutely convergent, and wewill use the oscillation of Π( x, y ) to obtain a required estimate. In this subsection sufficientconditions on a kernel Π( x, y ) in these three cases are stated in Propositions 2.5, 2.9 and2.12 respectively. Remark 2.4. We have noted that det K ( x, y ) is symmetric for a skew-symmetric kernel,therefore in our case it is sufficient to estimate all the integrals only for y ≥ x . Forthe general case one can split the domain of integration into two parts and change thevariables.Moreover, in the present paper we consider the case of R only for the sine process; theproof of the rigidity in this case is much simpler than for Bessel process, and we can alsouse the stationarity to give an alternative proof as presented in section . Thus we decidedto present the detailed proof for the case of the processes on R + . Analogous propositionsare true when we consider R instead of R + , with almost the same arguments. We will use the family of functions that were used in [2] for the determinantal pointprocesses. Take R > T > R and set ϕ ( R,T ) ( x ) = , x ≤ R ;1 − log( x − R + 1)log( T − R + 1) if R ≤ x ≤ T ;0 , T ≤ x. We split the domain of integration in the following way: D = { ( x, y ) | y ≥ x ≥ } ⊂ R + × R + , D = D >R ⊔ D Proposition 2.5. Let Π( x, y ) be a function on D .(i) Assume that there exists α ∈ (0 , / and, for any R > , a constant const( R ) > such that for any y ≥ x ≥ R we have | Π( x, y ) | ≤ const( R ) · ( x/y ) α + ( y/x ) α ( x − y ) . Then we have Z D >R | ϕ ( R,T ) ( x ) − ϕ ( R,T ) ( y ) | | Π( x, y ) | dxdy −−−→ T →∞ . (ii) Assume that there exists ε > , and, for any R > , a constant const( R ) > suchthat for any y ≥ R we have Z R − R | Π( x, y ) | dx ≤ const( R ) y ε . Then we have Z D The result of Proposition is also valid for Π( x, y ) = Π ( x, y ) / ( x − y ) ,once Π ( x, y ) is bounded. The assumption (ii) of Proposition doesn’t hold, but wecan see from the proof of the Proposition that what we actually need is the decreasingbound for the integral (cid:12)(cid:12)(cid:12)(cid:12) ∞ Z R R Z ( ϕ ( R,T ) ( x ) − Π( x, y ) dydx (cid:12)(cid:12)(cid:12)(cid:12) ≤ const( R )(log T ) ∞ Z R R Z log ( x − R + 1)( x − y ) dydx ≤≤ const( R )(log T ) ∞ Z R log ( x − R + 1) x ( x − R ) dx ≤ const( R )(log T ) . Recall the following well-known lemma (Abel-Dirichlet’s test) Lemma 2.7. Let f ( x ) , g ( x ) be two functions on [ a, b ) , where g ( x ) is monotonic, differ-entiable, lim x → b g ( x ) = 0 , and there exists a constant M > such that for any c ∈ [ a, b ) we have (cid:12)(cid:12)(cid:12)(cid:12)Z ca f ( x ) dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ M. Then (2.13) (cid:12)(cid:12)(cid:12)(cid:12)Z ba f ( x ) g ( x ) dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ M · | g ( a ) | . We will also use the following simple analog of Riemann-Lebesgue lemma: Lemma 2.8. Let f ( x ) be a differentiable function on [ a, b ] , then for every α ∈ R we have (cid:12)(cid:12)(cid:12)(cid:12)Z ba f ( x ) sin( mx α + n ) x α − dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ const m (cid:18) max a ≤ x ≤ b | f ( x ) | + Z ba | f ′ ( x ) | dx (cid:19) . Proposition 2.9. Let Π( x, y ) be a function on D . We set λ = y/x and ¯Π( x, λ ) =Π( x, λx ) . N NUMBER RIGIDITY FOR PFAFFIAN POINT PROCESSES 9 (i) Assume that there exist constants ≥ ε > ε ≥ , > ε > ε ≥ , ε > ε ≥ , λ > , λ < , a positive function ψ , ψ ( T ) = o (log ( T )) when T → ∞ , and, forany R > , a constant const( R ) > such that the following holds: max a,b>R (cid:12)(cid:12)(cid:12)(cid:12)Z ba ¯Π( x, λ ) dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ const( R ) ψ (max( a, b )) (cid:18) λ ε | λ − | ε + λ ε | λ − λ | ε + λ ε | λ − λ | ε (cid:19) , (2.14) | Π( x, y ) | ≤ const( R ) for y > x > R. (2.15) Then we have (2.16) Z D >R | ϕ ( R,T ) ( x ) − ϕ ( R,T ) ( y ) | Π( x, y ) y − x dxdy −−−→ T →∞ . (ii) If ε = 1 and ψ ( T ) = o (log( T )) when T → ∞ , then (2.16) also holds.(iii) Assume that there exist constants ε > − , ε > , and, for any R > , a constant const( R ) > such that the following holds: (2.17) | Π( x, y ) | ≤ const( R ) x ε y − ε for x < R, y > R. Then we have (2.18) Z D We start with the proof of (i). We split the domain D >R into three parts: D >R = { x, y ∈ D : R ≤ x ≤ y < T } ⊔ { x, y ∈ D : R ≤ x < T ≤ y } ⊔⊔ { x, y ∈ D : T ≤ x ≤ y } . Note that the integral is zero on { x, y ∈ D : T ≤ x ≤ y } . The First Case : R ≤ x ≤ y < T .(2.19) constlog ( T − R + 1) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) T Z R T Z x (cid:18) log( x − R + 1) − log( y − R + 1) (cid:19) Π( x, y ) x − y dydx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ const( R )(log T ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) T Z R T/x Z log ( λ ) ¯Π( x, λ ) λ − dλdx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ++ const( R )(log T ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) T Z R T/x Z log( λ ) (cid:18) log(1 − ( R − x − ) − log(1 − ( R − λx ) − ) (cid:19) ¯Π( x, λ ) λ − dλdx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ++ const( R )(log T ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) T Z R T Z x (cid:18) log(1 − ( R − x − ) − log(1 − ( R − y ) − ) (cid:19) Π( x, y ) y − x dydx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . where we have used a simple estimate log( T − R + 1) ≥ const log( T ) for T sufficientlylarge. For the first term we change the order of integration to obtain(2.20)const( R )(log T ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) T Z R T/x Z log ( λ ) ¯Π( x, λ ) λ − dλdx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = const( R )(log T ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) T/R Z log ( λ ) λ − (cid:18) T/λ Z R ¯Π( x, λ ) dx (cid:19) dλ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ const( R ) ψ ( T )(log T ) T/R Z log ( λ ) λ − (cid:18) λ ε | λ − | ε + λ ε | λ − λ | ε + λ ε | λ − λ | ε (cid:19) dλ −−−→ T →∞ . We have | log(1 − ( R − y − ) − const( R ) y − | ≤ const( R ) y − , therefore for the secondterm in (2.19) we obtainconst( R )(log T ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) T Z R T/x Z log( λ ) (cid:18) log(1 − ( R − x − ) − log(1 − ( R − λx ) − ) (cid:19) ¯Π( x, λ ) λ − dλdx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ const( R )(log T ) T/R Z log( λ ) λ − (cid:12)(cid:12)(cid:12)(cid:12) T/λ Z R log(1 − ( R − x − ) ¯Π( x, λ ) dx (cid:12)(cid:12)(cid:12)(cid:12) dλ +const( R )(log T ) T/R Z log( λ ) λ ( λ − (cid:12)(cid:12)(cid:12)(cid:12) T/λ Z R ¯Π( x, λ ) x dx (cid:12)(cid:12)(cid:12)(cid:12) dλ + const( R )(log T ) T/R Z T/λ Z R log( λ ) x λ ( λ − dxdλ ≤ const( R ) ψ ( T )(log T ) T/R Z log( λ ) λ − (cid:18) λ ε | λ − | ε + λ ε | λ − λ | ε + λ ε | λ − λ | ε (cid:19) dλ + const( R )(log T ) −−−→ T →∞ , where we have used (2.13) and (2.14) to estimate the first two terms, and (2.15) to bound | Π( x, y ) | in the last term.We have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) log (cid:18) − ( R − x − − ( R − y − (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ const( R )( y − x ) xy , for y > x > R, therefore for the last term in (2.19) we obtainconst( R )(log T ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) T Z R T Z x (cid:18) log(1 − ( R − x − ) − log(1 − ( R − y ) − ) (cid:19) Π( x, y ) y − x dydx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤≤ const( R )(log T ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) T Z R T Z x ( y − x ) | Π( x, y ) | x y dydx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) −−−→ T →∞ , where we have used (2.15) once again. The Second Case : R ≤ x < T ≤ y . N NUMBER RIGIDITY FOR PFAFFIAN POINT PROCESSES 11 We need to estimate the integralconst( R ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ Z T T Z R (cid:18) − log( x − R + 1)log( T − R + 1) (cid:19) Π( x, y ) y − x dxdy (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤≤ const( R )log ( T ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) T/R Z T Z T/λ log (cid:18) x − R + 1 T − R + 1 (cid:19) ¯Π( x, λ ) dx dλλ − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ++ const( R )log ( T ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ Z T/R T Z R log (cid:18) x − R + 1 T − R + 1 (cid:19) ¯Π( x, λ ) dx dλλ − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . We use (2.13) to bound both summands from above:const( R )log ( T ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) T/R Z T Z T/λ log (cid:18) x − R + 1 T − R + 1 (cid:19) ¯Π( x, λ ) dx dλλ − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ const( R ) ψ ( T )log ( T ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) T/R Z log (cid:18) T /λ − R + 1 T − R + 1 (cid:19) · (cid:18) λ ε | λ − | ε + λ ε | λ − λ | ε + λ ε | λ − λ | ε (cid:19) dλλ − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤≤ const( R ) ψ ( T )log ( T ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) T/R Z log λ · (cid:18) λ ε | λ − | ε + λ ε | λ − λ | ε + λ ε | λ − λ | ε (cid:19) dλλ − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) −−−→ T →∞ . andconst( R )log ( T ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ Z T/R T Z R log (cid:18) x − R + 1 T − R + 1 (cid:19) ¯Π( x, λ ) dx dλλ − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ const( R ) ψ ( T )log ( T ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ Z T/R log ( T − R + 1) · (cid:18) λ ε | λ − | ε + λ ε | λ − λ | ε + λ ε | λ − λ | ε (cid:19) dλλ − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) −−−→ T →∞ . If ε = 1 and ψ ( T ) = o (log( T )) then we split the interval [1 , T /R ] into two parts: P = [1 , λ − T − ] ∪ [ λ + T − , T /R ] and P = [ λ − T − , λ + T − ]. For the first part wehave(2.21) Z P log j ( λ ) λ − (cid:18) λ ε | λ − | ε + λ ε | λ − λ | ε + λ ε | λ − λ | ε (cid:19) dλ ≤ const log( T ) , j ∈ Z + , and for the second part we combine the estimate (2.15) with an obvious estimate(2.22) T Z R Z P F ( x, λ ) dλdx ≤ const( R ) , where F ( x, λ ) is bounded on P , | F ( x, λ ) | ≤ const( R ). The cases (i) and (ii) are fullyproved.Now we prove (iii). We split the domain D Remark 2.10. The assumptions of Proposition obviously hold if (cid:12)(cid:12) ¯Π( x, λ ) (cid:12)(cid:12) ≤ const( R ) x (cid:18) λ ε | λ − | ε + λ ε | λ − λ | ε + λ ε | λ − λ | ε (cid:19) , | Π( x, y ) | ≤ const( R ) , y > x > R. We use this fact many times below. In some of the applications below the condition (2.17) doesn’t hold. In this case wewill apply the following obvious corollary from the proof of the previous theorem. N NUMBER RIGIDITY FOR PFAFFIAN POINT PROCESSES 13 Corollary 2.11. Let Π( x, y ) be a function on D . If we have T ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) T Z R R Z log ( y − R + 1) Π( x, y ) y − x dxdy (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) −−−→ T →∞ , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ Z T R Z Π( x, y ) y − x dxdy (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) −−−→ T →∞ , then the convergence (2.18) holds. We use similar but simpler arguments to show the convergence to zero of the requiredintegrals in case when the variables are split: Proposition 2.12. Let Π( x, y ) = Π ( x ) · Π ( y ) be a function on D .(i) Assume that there exists a positive function ˜ ψ , ˜ ψ ( T ) = o (log( T )) when T → ∞ ,and, for any R > , a constant const( R ) > such that for m ∈ { , , } , we have max RR | ϕ ( R,T ) ( x ) − ϕ ( R,T ) ( y ) | Π( x, y ) dxdy −−−→ T →∞ . (ii) Assume additionally that Π ( x ) is integrable on [0 , R ] for any R > . Then wehave Z D For R ≤ x ≤ y < T we should estimate the integralconstlog ( T − R + 1) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) T Z R T Z x (cid:18) log( x − R + 1) − log( y − R + 1) (cid:19) Π( x, y ) dydx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ const( R )(log T ) X m =0 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z TR log m ( x − R + 1)Π ( x ) dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ×× max R
Assume that Π ( x ) = f ( x ) g ( x ) + h ( x ) , Π ( y ) = f ( y ) g ( y ) + h ( y ) ,and there exist constants R > and ε > − , ε > such that • max a,b>R (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)R ba f i ( x ) dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ const( R ) , for i ∈ { , } ; • g i ( x ) log ( x − R + 1) are decreasing to zero for x sufficiently large, i ∈ { , } ; • | h i ( x ) | ≤ const( R ) x − ε for x > R , i ∈ { , } ; • | Π ( x ) | ≤ const( R ) , x ≥ R , | Π ( y ) | ≤ const( R ) , y ≥ R ; • | Π ( x ) | ≤ const( R ) x ε for x < R .Then we have Z D | ϕ ( R,T ) ( x ) − ϕ ( R,T ) ( y ) | Π ( x )Π ( y ) dxdy −−−→ T →∞ . Proof. Let g ( y ) log m ( t − R + 1) be decreasing for y > R > R , m ∈ { , , } . We havemax R
We also have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z ∞ T Π ( y ) dy (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ const( R ) | g ( T ) | + T − ε −−−→ T →∞ , and (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z R Π ( x ) dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ const( R ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z R x ε dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ const( R ) . Thus we can apply Proposition 2.12 to obtain the required estimate. (cid:3) From the proofs of Proposition 2.12 and Corollary 2.13 we obtain also the correspondingone-dimensional result. Corollary 2.14. Assume that Π ( x ) = f ( x ) g ( x ) + h ( x ) and there exist constants R > and ε > such that • max a,b>R (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)R ba f ( x ) dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ const( R ) ; • g ( t ) log ( t − R + 1) is decreasing to zero for t sufficiently large; • | h ( x ) | ≤ const( R ) x − ε for x ≥ R ; • | Π ( x ) | ≤ const( R ) for x ≥ R .Then we have Z ∞ (cid:18) | ϕ ( R,T ) ( x ) | − (cid:19) Π ( x ) dx −−−→ T →∞ . Symplectic Bessel process. We will use the following estimate for the Besselfunction for small x :(2.23) J s ( x ) = ( x/ s Γ( s + 1) + O ( x s +2 )(cf. e.g. 9.1.10 in Abramowitz and Stegun [1]) and the asymptotic expansion(2.24) J s ( x ) = r πx cos( x − sπ/ − π/ 4) + O ( x − / )of the Bessel function of a large argument (cf. e.g. 9.2.1 in Abramowitz and Stegun [1]).From the relation(2.25) J ′ s ( x ) = ± sx J s ( x ) ∓ J s ± ( x ) , (cf. e.g. 9.1.27 in Abramowitz and Stegun [1]) we obtain(2.26) J ′ s ( x ) = s ( x/ s − s + 1) + O ( x s +2 )for small x and(2.27) J ′ s ( x ) = − r πx sin( x − sπ/ − π/ 4) + O ( x − / )for x → ∞ . Also integrating the asymptotic expansion we have(2.28) Z ∞ x J s ( t ) dt = r πx sin( x − sπ/ − π/ 4) + O ( x − / ) . Forrester [6, p. 312, (7.109)] gives the following definition of the hard edge scaling limit(scaling limit of the Laguerre symplectic ensemble in our case):(2.29) ˜ S ( x, y ) = 12 (cid:18) xy (cid:19) / K ( L )2 N ( x, y )++ (2 N )! y ( s − / e − y/ L s N ( y )4Γ( s + 2 N ) Z ∞ x t ( s − / e − t/ L s N − ( t ) dt,K hard edges ( X, Y ) = lim N →∞ N S (cid:18) X N , Y N (cid:19) , S ( x, y ) = 2 ˜ S (2 x, y ) (cid:12)(cid:12)(cid:12)(cid:12) s → s − , K hard edge ,s ( x, y ) = (cid:20) K hard edges ( x, y ) R xy K hard edges ( x, t ) dt ∂∂x K hard edges ( x, y ) K hard edges ( y, x ) (cid:21) , where L sN ( x ) is the N -th Laguerre polynomial and K ( L )2 N ( x, y ) is the corresponding Christoffel–Darboux kernel. Proposition 2.15. The (hard edge) scaling limit of the Laguerre symplectic ensemble isdefined by the following kernel: K Bessels ( x, y ) = 2 (cid:18) xy (cid:19) / K Bessel , s − (4 x, y ) − J s − (2 y / )2 y / Z x / J s − (2 t ) dt, K Bessel ,s ( x, y ) = (cid:20) K Bessels ( x, y ) R xy K Bessels ( x, t ) dt ∂∂x K Bessels ( x, y ) K Bessels ( y, x ) (cid:21) . First proof. We have e − x/ x s/ L sN ( x ) ∼ N s/ J s (cid:0) N x ) / (cid:1) for N → ∞ and(2.30) Z ∞ t ( s − / e − t/ L s N − ( t ) dt = 0 , and the required formula follows. (cid:3) Second proof. Forrester [6, p. 312, (7.111)] gives K hard edges ( x, y ) = 2 K Bessel , s (4 x, y ) − J s − (2 y / )2 y / Z x / J s +1 (2 t ) dt. Therefore it is sufficient to check that2 K Bessel , s (4 x, y ) − (cid:18) xy (cid:19) / K Bessel , s − (4 x, y ) == J s − (2 y / )2 y / Z x / J s +1 (2 t ) dt − Z x / J s − (2 t ) dt ! . From the relations (2.25), (2.24) we have(2.31) sx / J s (2 x / ) = J s − (2 x / ) + J s +1 (2 x / ) N NUMBER RIGIDITY FOR PFAFFIAN POINT PROCESSES 17 and Z x / J s +1 (2 t ) dt − Z x / J s − (2 t ) dt = − J s (2 x / ) . Regarding the left-hand side,2 K Bessel , s (4 x, y ) − (cid:18) xy (cid:19) / K Bessel , s − (4 x, y ) = 12( x − y ) − y / J s +1 (2 y / ) J s (2 x / )++ J s (2 y / ) (cid:18) x / J s +1 (2 x / ) + x / J s − (2 x / (cid:19) − xy − / J s (2 x / ) J s − (2 y / )) ! == J s ( x / )2( x − y ) − x J s − ( y / ) y / + y / (cid:18) sJ s (2 y / ) y / − J s +1 (2 y / ) (cid:19)! == − J s (2 x / ) J s − (2 y / )2 y / , where we have used (2.31) several times. (cid:3) Unfortunately the condition (1.6) doesn’t hold for the kernel K Bessel ,s ( x, y ). But a weakercondition does hold, and it will be sufficient for our purposes. Proposition 2.16. Z ∞ Z ∞ det K Bessel ,s ( x, y ) dy − K Bessels ( x, x ) ! dx = 0 . Proof. We plan to use Corollary 2.3, and we will first simplify the expressions for thedefect Def K Bessel ,s ( x ) and for the limits − K Bessels ( y, x ) Z yx K Bessels ( y, t ) dt (cid:12)(cid:12)(cid:12)(cid:12) ∞ y =0 . First, we see that 4 K Bessel , s − (4 x, y ) = 14 Z J s − ( √ ux ) J s − ( √ uy ) du is an orthogonal projection onto the subspace of functions f ( x ) such that has its Hankeltransform supported in [0 , J s − (2 √ x ) ∈ Ran(4 K Bessel , s − (4 x, y )), and it follows that Z ∞ J s − (2 y / ) · K Bessel , s − (4 x, y ) dy = J s − (2 x / ) . Therefore we have Z ∞ K Bessels ( x, y )4 (cid:16) yx (cid:17) / K Bessel , s − (4 x, y ) dy = Z ∞ (cid:18) (cid:18) xy (cid:19) / K Bessel , s − (4 x, y ) −− J s − (2 y / )2 y / Z x / J s − (2 t ) dt (cid:19) (cid:16) yx (cid:17) / K Bessel , s − (4 x, y ) dy == 12 Z ∞ K Bessel , s − (4 x, y ) · K Bessel , s − (4 x, y ) dy −− x / Z x / J s − (2 t ) dt · Z ∞ J s − (2 y / ) · K Bessel , s − (4 x, y ) dy = 2 K Bessel , s − (4 x, y ) − x / Z x / J s − (2 t ) dt · J s − (2 x / ) = K Bessels ( x, x )and Z ∞ (cid:18) xy (cid:19) / K Bessel , s − (4 x, y ) Z y / J s − (2 t ) dtdy = Z x / J s − (2 t ) dt. We also have2 Z ∞ J s − (2 y / )2 y / Z y / J s − (2 t ) dtdy = Z ∞ d (cid:18)Z y / J s − (2 t ) dt (cid:19) = 14 , where we have used that(2.32) Z ∞ J µ ( z ) dz = 1for ℜ µ > − Z ∞ K Bessels ( x, y ) K Bessels ( y, x ) dy = Z ∞ K Bessels ( x, y )4 (cid:16) yx (cid:17) / K Bessel , s − (4 x, y ) dy −− J s − (2 x / )2 x / Z ∞ (cid:18) xy (cid:19) / K Bessel , s − (4 x, y ) Z y / J s − (2 t ) dtdy ++ J s − (2 x / ) x / Z x / J s − (2 t ) dt Z ∞ J s − (2 y / )2 y / Z y / J s − (2 t ) dtdy = K Bessels ( x, x ) − J s − (2 x / ) x / Z x / J s − (2 t ) dt, and 2 Def K Bessel ,s ( x ) = − J s − (2 x / ) x / Z x / J s − (2 t ) dt. Now since K Bessels (0 , x ) = 0 and lim y →∞ K Bessels ( y, x ) = − J s − (2 x / )4 x / , N NUMBER RIGIDITY FOR PFAFFIAN POINT PROCESSES 19 we have − K Bessels ( y, x ) Z yx K Bessels ( y, t ) dt (cid:12)(cid:12)(cid:12)(cid:12) ∞ y =0 = J s − (2 x / )4 x / Z yx (cid:18)(cid:16) yt (cid:17) / K Bessel , s − (4 y, t ) − J s − (2 t / )2 t / Z y / J s − (2 p ) dp (cid:19) dt (cid:12)(cid:12)(cid:12)(cid:12) y = ∞ = J s − (2 x / )2 x / lim y →∞ Z yx (cid:16) yt (cid:17) / K Bessel , s − (4 y, t ) dt −− J s − (2 x / )8 x / Z ∞ x / J s − (2 p ) dp. We write Z yx (cid:16) yt (cid:17) / K Bessel , s − (4 y, t ) dt = Z ∞ (cid:16) yt (cid:17) / K Bessel , s − (4 y, t ) dt − Z x (cid:16) yt (cid:17) / K Bessel , s − (4 y, t ) dt − Z ∞ y (cid:16) yt (cid:17) / K Bessel , s − (4 y, t ) dt, where we have lim y →∞ (cid:12)(cid:12)(cid:12)(cid:12)Z x (cid:16) yt (cid:17) / K Bessel , s − (4 y, t ) dt (cid:12)(cid:12)(cid:12)(cid:12) ≤ lim y →∞ const( x ) y / y − x = 0 , and we use Lemma 2.8 to obtainlim y →∞ (cid:12)(cid:12)(cid:12)(cid:12)Z ∞ y (cid:16) yt (cid:17) / K Bessel , s − (4 y, t ) dt (cid:12)(cid:12)(cid:12)(cid:12) =const lim y →∞ (cid:12)(cid:12)(cid:12)(cid:12)Z ∞ cos( √ y − sπ − π/ v − / cos( √ yv − sπ + π/ v / ( v − −− cos( √ y − sπ + π/ v / cos( √ yv − sπ − π/ v / ( v − dv (cid:12)(cid:12)(cid:12)(cid:12) = 0 . We also have Z ∞ (cid:16) yt (cid:17) / K Bessel , s − (4 y, t ) dt = √ y Z ∞ (cid:18)Z J s − (2 √ uy ) J s − (2 √ ut ) du (cid:19) dt √ t = √ y Z J s − (2 √ uy ) Z ∞ J s − (2 √ up ) dpdu = √ y Z J s − (2 √ uy )2 √ u du = 12 Z y / J s − (2 p ) dp, therefore lim y →∞ Z yx (cid:16) yt (cid:17) / K Bessel , s − (4 y, t ) dt = 12 lim y →∞ Z y / J s − (2 p ) dp = 14 , and − K Bessels ( y, x ) Z yx K Bessels ( y, t ) dt (cid:12)(cid:12)(cid:12)(cid:12) ∞ y =0 = J s − (2 x / )8 x / − J s − (2 x / )8 x / Z ∞ x / J s − (2 p ) dp. Write Z ∞ det K Bessel ,s ( x, y ) dy − K Bessels ( x, x ) =2 Def K Bessel ,s ( x ) − K Bessels ( y, x ) Z yx K Bessels ( y, t ) dt (cid:12)(cid:12)(cid:12)(cid:12) ∞ y =0 == J s − (2 x / )8 x / (cid:18) − Z x / J s − (2 t ) dt + 1 − Z ∞ x / J s − (2 p ) dp (cid:19) == J s − (2 x / )16 x / − J s − (2 x / )4 x / Z x / J s − (2 t ) dt. We directly see that(2.33) J s − (2 x / )16 x / − J s − (2 x / )4 x / Z x / J s − (2 t ) dt = 0and therefore the relation (1.6) does not hold for the kernel K Bessel ,s . Nonetheless, we have Z ∞ (cid:18) J s − (2 x / )16 x / − J s − (2 x / )4 x / Z x / J s − (2 t ) dt (cid:19) dx == 116 Z ∞ J s − (2 x / ) x / dx − (cid:18)Z x / J s − (2 t ) dt (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) ∞ = 116 − 116 = 0 . (cid:3) Remark 2.17. The assumptions of Proposition do hold for the Pfaffian Laguerrekernel ˜ S ( x, y ) , for the finite N (see the definition of the kernel in (2.29) ), and thereforethe condition (1.6) holds also. In this case, first of all, S ( x, y ) = (cid:18) xy (cid:19) / K ( L )2 N ( x, y ) + f N ( x ) g N ( y ) has a reproducing property: Π( x, y ) = (cid:16) xy (cid:17) / K ( L )2 N ( x, y ) is a projection because K ( L )2 N ( x, y ) is a Christoffel–Darboux kernel, f N ( x ) = R ∞ x t ( s − / e − t/ L s N − ( t ) dt lies in the image of Π( x, y ) and g N ( y ) = (2 N )!2Γ( s +2 N ) y ( s − / e − y/ L s N ( y ) is orthogonal to the image of Π( x, y ) .And we also have ˜ S (0 , x ) = 0 , lim y →∞ ˜ S ( y, x ) = 0 , because the same is true for the kernel (cid:0) yx (cid:1) / K ( L )2 N ( y, x ) and because the integral (2.30) iszero.Neither property holds when we consider the limiting kernel K Bessels ( x, y ) . First, thereis no reproducing property: scaling limits for L s N − ( y ) and L s N ( y ) are the same, thereforethe scaling limit of g N ( y ) is not orthogonal to the image of the limiting projection. And,second, as we see from (2.32) , the integral (2.30) is not zero after the scaling limit, and lim y →∞ K Bessels ( y, x ) = 0 . N NUMBER RIGIDITY FOR PFAFFIAN POINT PROCESSES 21 Lemma 2.18. (i) The convergence to zero of the integrals (2.16) , (2.18) holds for Π( x, y ) = (cid:18) yx (cid:19) ε / J t (2 √ y ) J t + ε (2 √ y ) J v (2 √ x ) , where ε ∈ { , } and t > − , v > − .(ii) The convergence to zero of the integrals (2.16) , (2.18) holds for Π( x, y ) = (cid:18) yx (cid:19) ε / − / J t (2 √ y ) J t + ε (2 √ x ) J v (2 √ x ) , where ε ∈ { , } and t > − , v > − .Proof. (i). We set λ = y/x , we write ¯Π( x, λ ) = Π( x, λx ) as a sum(2.34) λ ε / − / J t (2 √ λx ) J t + ε (2 √ x ) J v (2 √ x ) = (cid:18) λ ε / J t (2 √ λ √ x ) J t + ε (2 √ λ √ x ) J v (2 √ x ) −− λ ε / − / π √ x cos(2 √ λ √ x − tπ/ − π/ 4) cos(2 √ λ √ x − ( t + ε ) π/ − π/ J v (2 √ x ) (cid:19) ++ λ ε / − / π √ x cos(2 √ λ √ x − tπ/ − π/ 4) cos(2 √ λ √ x − ( t + ε ) π/ − π/ ×× (cid:18) J v (2 √ x ) − √ πx / cos(2 √ x − vπ/ − π/ (cid:19)! ++ λ ε / − / π √ x cos(2 √ λ √ x − tπ/ − π/ 4) cos(2 √ λ √ x − ( t + ε ) π/ − π/ ×× √ πx / cos(2 √ x − vπ/ − π/ , and we check the convergence to zero of the integrals (2.16) and (2.18) term by term.We fix arbitrary R > (cid:12)(cid:12)(cid:12)(cid:12) λ ε / J t (2 √ λ √ x ) J t + ε (2 √ λ √ x ) J v (2 √ x ) −− λ ε / − / π √ x cos(2 √ λ √ x − tπ/ − π/ 4) cos(2 √ λ √ x − ( t + ε ) π/ − π/ J v (2 √ x ) (cid:12)(cid:12)(cid:12)(cid:12) ≤≤ const( R ) J v (2 √ x ) √ λx , for y = λx ≥ R , where we have used the estimate (2.24) and the fact that λ ≥ 1. Nowwe can combine this estimate with (2.23) to see that the assumptions of Proposition 2.9are satisfied for the difference (2.35). As a next step we need to estimate the intergal (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z ba λ ε / − / π √ x cos(2 √ λ √ x − tπ/ − π/ 4) cos(2 √ λ √ x − ( t + ε ) π/ − π/ ×× (cid:18) J v (2 √ x ) − √ πx / cos(2 √ x − vπ/ − π/ (cid:19) dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ λ ε / − / π (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z ba √ x cos( ε π/ (cid:18) J v (2 √ x ) − √ πx / cos(2 √ x − vπ/ − π/ (cid:19) dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ++ λ ε / − / π (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z ba √ x cos(4 √ λ √ x − (2 t + ε + 1) π/ ×× (cid:18) J v (2 √ x ) − √ πx / cos(2 √ x − vπ/ − π/ (cid:19) dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ const( R ) √ λ , where we have used estimates (2.24) and (2.27) and also (2.13) for non-zero first term,and Lemma 2.8 for the second term.Next, for a, b > R for the main part we can write (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z ba const λ ε / − / x / cos(2 √ λ √ x − tπ/ − π/ 4) cos(2 √ λ √ x − ( t + ε ) π/ − π/ ×× cos(2 √ x − vπ/ − π/ (cid:19) dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ const √ λ + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z ba const λ ε / − / x / ×× cos(4 √ λ √ x − (2 t + ε + 1) π/ 2) cos(2 √ x − vπ/ − π/ dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤≤ const( R ) (cid:18) √ λ + 12 √ λ + 1 + 12 √ λ − (cid:19) , where we have used (2.13) once again. Therefore we can apply Proposition 2.9(i) in thiscase.If ε = 0 then Proposition 2.9(iii) obviously holds, therefore we can now put ε = 1.We haveconst( R )(log T ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) R Z T Z R log ( y − R + 1) cos(4 √ y − tπ ) J v (2 √ x ) √ x ( y − x ) dydx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤≤ const( R )(log T ) R Z x ( v − / dx R ′ Z R log ( y − R + 1) y − R dy + √ R ′ log ( R ′ − R + 1) R ′ − R ! ≤ const( R )(log T ) , where the function √ y log ( y − R + 1) / ( y − R ) is decreasing for y ≥ R ′ . And (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) R Z ∞ Z T cos(4 √ y − tπ ) J v (2 √ x ) √ x ( y − x ) dydx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ const( R ) √ TT − R R Z x ( v − / dx −−−→ T →∞ , N NUMBER RIGIDITY FOR PFAFFIAN POINT PROCESSES 23 thus we can use Corollary 2.11.(ii). We fix again arbitrary R > (cid:12)(cid:12)(cid:12)(cid:12) λ ε / − / J t (2 √ λ √ x ) J t + ε (2 √ x ) J t (2 √ x ) −− λ ε / − / √ πx / cos(2 √ λ √ x − tπ/ − π/ J t + ε (2 √ x ) J v (2 √ x ) (cid:12)(cid:12)(cid:12)(cid:12) ≤≤ const( R ) J t + ε (2 √ x ) J v (2 √ x ) λ / x / , for y = λx ≥ R , where we have used the estimate (2.24) and the fact that λ ≥ 1. Nowwe can combine this estimate with (2.23) to see that the assumptions of Proposition 2.9are satisfied for the considered difference.As a next step we need to estimate the intergal (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z ba λ ε / − / √ π √ x cos(2 √ λ √ x − tπ/ − π/ (cid:18) x / J t + ε (2 √ x ) J v (2 √ x ) −− πx / cos(2 √ x − ( t + ε ) π/ − π/ 4) cos(2 √ x − vπ/ − π/ (cid:19) dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ const( R ) λ / , where we have used estimates (2.24) and (2.27) and then Lemma 2.8.Next, for a, b > R for the main part we can write (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z ba const λ ε / − / x / cos(2 √ λ √ x − tπ/ − π/ 4) cos(2 √ x − ( t + ε ) π/ − π/ ×× cos(2 √ x − vπ/ − π/ (cid:19) dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z ba const λ ε / − / x / cos(2 √ λ √ x − tπ/ − π/ ×× (cid:18) cos(4 √ x − ( t + v + ε + 1) π/ 2) + cos(( t − v + ε ) π/ (cid:19) dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤≤ const( R ) λ / (cid:18) √ λ + 1 √ λ + 2 + 1 √ λ − (cid:19) , where we have used (2.13) for the last estimate. Therefore we can apply Proposition 2.9(i)and (ii) in this case.Finally for x < R , y > R we have (cid:12)(cid:12)(cid:12)(cid:12) λ ε / − / √ πx / cos(2 √ λ √ x − tπ/ − π/ J t + ε (2 √ x ) J v (2 √ x ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ const( R ) x ( t + v + ε ) / y / , therefore the assumptions of Proposition 2.9(iii) hold for this term and Lemma is fullyproved. (cid:3) Proof of the Theorem . Our plan is to expand the determinant,det K Bessel ,s ( x, y ) = K Bessels ( x, y ) K Bessels ( y, x ) − ∂∂x K Bessels ( x, y ) Z xy K Bessels ( x, t ) dt and then to show that all the summands in the formula (2.11) for the variance, with f ( x ) = ϕ ( R,T ) ( x ), tend to zero term by term. The first part, Z R + | ϕ ( R,T ) ( x ) | (cid:18) K Bessels ( x, x ) − Z R det K Bessel ,s ( x, y ) dy (cid:19) dx. We setΠ( x ) = K Bessels ( x, x ) − Z ∞ det K Bessel ,s ( x, y ) dy == J s − (2 x / )4 x / Z x / J s − (2 t ) dt − J s − (2 x / )16 x / == J s − (2 x / )16 x / − J s − (2 x / )4 x / Z ∞ x / J s − (2 t ) dt, we use Proposition 2.16 to write Z ∞ | ϕ ( R,T ) ( x ) | Π( x ) dx = Z ∞ (cid:18) | ϕ ( R,T ) ( x ) | − (cid:19) Π( x ) dx and then we use estimates (2.24) and (2.28) and Corollary 2.14 to see that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z ∞ (cid:18) | ϕ ( R,T ) ( x ) | − (cid:19) Π( x ) dx (cid:12)(cid:12)(cid:12)(cid:12) −−−→ T →∞ . The second part, Z D | ϕ ( R,T ) ( x ) − ϕ ( R,T ) ( y ) | K Bessels ( x, y ) · K Bessels ( y, x ) dxdy. N NUMBER RIGIDITY FOR PFAFFIAN POINT PROCESSES 25 We have K Bessels ( x, y ) · K Bessels ( y, x ) = (cid:18) xy (cid:19) / K Bessel , s − (4 x, y ) + J s − (2 y / )2 y / (cid:18) − 12 + Z ∞ x / J s − (2 t ) dt (cid:19)! ×× (cid:16) yx (cid:17) / K Bessel , s − (4 y, x ) + J s − (2 x / )2 x / (cid:18) − 12 + Z ∞ y / J s − (2 t ) dt (cid:19)! =4 (cid:0) K Bessel , s − (4 x, y ) (cid:1) + (cid:18) xy (cid:19) / K Bessel , s − (4 x, y ) J s − (2 x / )2 x / ×× Z ∞ y / J s − (2 t ) dt + J s (2 x / ) J s − (2 y / )4( x − y ) Z ∞ x / J s − (2 t ) dt ! ++ y / x − / J s (2 y / ) J s − (2 x / ) J s − (2 y / )4( x − y ) Z ∞ x / J s − (2 t ) dt −− (cid:18) xy (cid:19) / K Bessel , s − (4 x, y ) J s − (2 x / )2 x / + (cid:16) yx (cid:17) / K Bessel , s − (4 y, x ) J s − (2 y / )2 y / ! ++ J s − (2 x / )2 x / Z x / J s − (2 t ) dt J s − (2 y / )2 y / Z y / J s − (2 t ) dt =:=: S ( x, y )( x − y ) + S ( x, y ) x − y + S ( x, y ) x − y + S ( x, y ) x − y + S ( x, y ) . The integral for the first term, S ( x, y ) / ( x − y ) = 4 (cid:18) K Bessel , s − (4 x, y ) (cid:19) , was estimatedin [2] by Proposition 2.5.We use (2.24) and (2.28) to obtain (cid:12)(cid:12)(cid:12)(cid:12) S ( x, y ) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:0) x / J s (2 x / ) J s − (2 y / ) − y / J s (2 y / ) J s − (2 x / ) (cid:1) × J s − (2 x / )4 y / Z ∞ y / J s − (2 t ) dt + J s (2 x / ) J s − (2 y / )4 Z ∞ x / J s − (2 t ) dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ const 1 λ / x , where we have set, as usual, λ = y/x . Thus 2.14 is satisfied for x, y > R and any R > (cid:12)(cid:12)(cid:12)(cid:12) S ( x, y ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ const(R , s) y − / for x < R, y > R, and we can apply Proposition 2.9. We also have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) S ( x, y ) − π √ x cos(2 y / − sπ − π/ 4) cos(2 y / − (2 s − π/ − π/ J s − (2 x / ) ×× Z ∞ x / J s − (2 t ) dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ const (cid:12)(cid:12) J s − (2 x / ) R ∞ x / J s − (2 t ) dt (cid:12)(cid:12) √ xy , thus we can combine this estimate with (2.23), (2.24), (2.28) to use Proposition 2.9 inthis case.For the main term of S ( x, y ) we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z ba π √ x cos(2 y / − sπ − π/ 4) cos(2 y / − (2 s − π/ − π/ J s − (2 x / ) ×× Z ∞ x / J s − (2 t ) dtdx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z ba π √ x cos(4 λ / x / − sπ ) J s − (2 x / ) ×× Z ∞ x / J s − (2 t ) dtdx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ const log( b ) √ λ by Lemma 2.8, and we can use Proposition 2.9(i). We also haveconst( R )(log T ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) R Z T Z R log ( y − R + 1) cos(4 y / − sπ ) J s − (2 x / ) R ∞ x / J s − (2 t ) dtx / ( y − x ) dydx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤≤ const( R )(log T ) R Z x s − dx R ′ Z R log ( y − R + 1) y − R dy + √ R ′ log ( R ′ − R + 1) R ′ − R ! ≤ const( R )(log T ) , where the function √ y log ( y − R + 1) / ( y − R ) is decreasing for y ≥ R ′ . And (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) R Z ∞ Z T cos(4 y / − sπ ) J s − (2 x / ) R ∞ x / J s − (2 t ) dtx / ( y − x ) dydx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ const( R ) √ TT − R R Z x s − dx −−−→ T →∞ , thus we can use Corollary 2.11.For the term S ( x, y ) we use Lemma 2.18 for all the four summands.In the last term S ( x, y ) the variables are split, and we have S ( x, y ) = J s − (2 x / )2 x / Z x / J s − (2 t ) dt × J s − (2 y / )2 y / Z y / J s − (2 t ) dt == J s − (2 x / )2 x / (cid:18) − Z ∞ x / J s − (2 t ) dt (cid:19) × J s − (2 y / )2 y / (cid:18) − Z ∞ y / J s − (2 t ) dt (cid:19) . N NUMBER RIGIDITY FOR PFAFFIAN POINT PROCESSES 27 We use the estimates (2.23), (2.24) to obtain (cid:12)(cid:12)(cid:12)(cid:12) J s − (2 x / )2 x / Z x / J s − (2 t ) dt (cid:12)(cid:12)(cid:12)(cid:12) ≤ const( R ) x s − for x < R. Moreover, from the estimates (2.24), (2.28) we see that(2.36) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) J s − (2 x / )2 x / (cid:18) − Z ∞ x / J s − (2 t ) dt (cid:19) − √ πx / cos(2 x / − (2 s − π/ − π/ ×× (cid:18) − √ πx / sin(2 x / − (2 s − π/ − π/ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ const( R ) x / , for x > R , therefore the conditions of Corollary 2.13 are satisfied and we have provedthe required convergence to zero of the intergal (2.12) for S ( x, y ). Thus the requiredconvergence is proved for the whole first term K Bessels ( x, y ) · K Bessels ( y, x ). The third part, Z D >R | ϕ ( R,T ) ( x ) − ϕ ( R,T ) ( y ) | ∂∂x K Bessels ( x, y ) Z yx K Bessels ( x, t ) dtdxdy. We will use the following notation: ∂∂x K Bessels ( x, y ) = − xy − / J s (2 x / ) J s − (2 y / ) − x / J s (2 y / ) J s − (2 x / )2( x − y ) + y − / J s (2 x / ) J s − ( y / ) + x / y − / J ′ s (2 x / ) J s − (2 y / )2( x − y ) −− / x − / J s (2 y / ) J s − (2 x / ) + J s (2 y / ) J ′ s − (2 x / )2( x − y ) ! ++ J s − (2 y / )2 y / J s − (2 x / )2 x / =: D ( x, y )( x − y ) + D ( x, y )( x − y ) + D ( x, y ) . And Z yx K Bessels ( x, t ) dt = Z y − x K Bessels ( x, t + x ) dt =2 Z ∞ (cid:18) xt + x (cid:19) / K Bessel , s − (4 x, t + x )) dt − Z ∞ y − x (cid:18) xt + x (cid:19) / K Bessel , s − (4 x, t + x )) dt ++ Z y / x / J s − (2 p ) dp Z ∞ x / J s − (2 p ) dp − Z y / x / J s − (2 p ) dp =:˜ I ( x ) + I ( x, y ) + I ( x, y ) + I ( x, y ) . We will first separate the main part of ˜ I ( x ), because the corresponding integrals arenot absolutely convergent. We have˜ I ( x ) = 2 Z ∞ (cid:18) xt + x (cid:19) / x / J s (2 x / ) J s − (2( t + x ) / )4 t −− ( t + x ) / J s (2( t + x ) / ) J s − (2 x / )4 t ! dt, and we use (2.24) to estimate it as follows: (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˜ I ( x ) − π Z ∞ x / cos(2 x / − sπ − π/ 4) cos(2( t + x ) / − (2 s − π/ − π/ t ( t + x ) / −− x / cos(2( t + x ) / − sπ − π/ 4) cos(2 x / − (2 s − π/ − π/ t ( t + x ) / ! dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Z const √ x dt + const x − / Z ∞ dtt ( t + x ) / ≤ const √ x . We also have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z ∞ x / cos(2 x / − sπ − π/ 4) cos(2( t + x ) / − (2 s − π/ − π/ t ( t + x ) / ×× x / ( t + x ) / − ! dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ x / Z dt ( t + x ) / ( x / + ( t + x ) / ) ++ x / (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z ∞ cos(2( t + x ) / − (2 s − π/ − π/ t + x ) / ( x / + ( t + x ) / ) dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ const( R ) √ x , for x > R, where we have used (2.13) once again to estimate the second term. Now we estimate themain term:12 π Z ∞ x / t ( t + x ) / (cid:18) cos(2 x / − sπ − π/ 4) cos(2( t + x ) / − (2 s − π/ − π/ −− cos(2( t + x ) / − sπ − π/ 4) cos(2 x / − (2 s − π/ − π/ (cid:19) dt == − π Z ∞ x / t ( t + x ) / sin(2( t + x ) / − x / ) dt = − π Z ∞ ( u + 1) / ( u + 1) − x / u ) du = − π Z ∞ sin(2 x / u ) u ( u + 1) / u + 2 du = − π Z ∞ sin( u ) u du + O (cid:18) √ x (cid:19) = − 14 + O (cid:18) √ x (cid:19) , where we have put u = ( t/x + 1) / − I ( x ) = − 14 + I ( x ) , Z yx K Bessels ( x, t ) dt = − 14 + I ( x ) + I ( x, y ) + I ( x, y ) + I ( x, y ) , where | I ( x ) | ≤ const( R ) / √ x for x ≥ R . N NUMBER RIGIDITY FOR PFAFFIAN POINT PROCESSES 29 Now we write ∂∂x K Bessels ( x, y ) Z yx K Bessels ( x, t ) dt = (cid:18) D ( x, y )( x − y ) + D ( x, y )( x − y ) + D ( x, y ) (cid:19) ×× (cid:18) − 14 + I ( x ) + I ( x, y ) + I ( x, y ) + I ( x, y ) (cid:19) == D ( x, y )( x − y ) × (cid:18) − 14 + I ( x ) + I ( x, y ) + I ( x, y ) (cid:19) + D ( x, y )( x − y ) × I ( x, y ) ! ++ D ( x, y )( x − y ) × (cid:18) I ( x ) + I ( x, y ) (cid:19) + D ( x, y ) × I ( x, y ) ! + D ( x, y )( x − y ) × I ( x, y )++ D ( x, y )( x − y ) (cid:18) − 14 + I ( x, y ) (cid:19) + D ( x, y ) × (cid:18) − 14 + I ( x ) + I ( x, y ) + I ( x, y ) (cid:19) , and we estimate the summands one by one.We use (2.24), (2.27), (2.28) and (2.13) to obtain the following simple estimates D ( x, y ) ≤ const( R ) λ / , D ( x, y ) ≤ const( R ) √ xλ / , D ( x, y ) ≤ const( R ) x / λ / ,I ( x, y ) ≤ const( R ) √ x , I ( x, y ) ≤ const( R ) x / . We additionally use (2.13) to obtain the following estimate:(2.37) | I ( x, y ) | = 2 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z ∞ y − x (cid:18) xt + x (cid:19) / x / J s ( x / ) J s − (( t + x ) / )2 t −− ( t + x ) / J s (( t + x ) / ) J s − ( x / )2 t ! dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤≤ const · x | J s ( x / ) | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z ∞ y − x t ( t + x ) / J s − (( t + x ) / )( t + x ) / dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ++ const · x / | J s − ( x / ) | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z ∞ y − x ( t + x ) / t J s (( t + x ) / )( t + x ) / dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤≤ const( R ) x / y − / + x / y / y − x ≤ const( R ) √ xλ / y − x . From the estimates above we obtain (cid:12)(cid:12)(cid:12)(cid:12) D ( x, y )( x − y ) × (cid:18) − 14 + I ( x ) + I ( x, y ) + I ( x, y ) (cid:19) + D ( x, y )( x − y ) × I ( x, y ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ const( R ) λ − / ( x − y ) ×× (cid:18) − 14 + const( R ) √ x + const( R ) x / (cid:19) + const( R ) √ xλ / × √ xλ / ( x − y ) ≤ const( R )( x − y ) , and we can apply Proposition 2.5(i). Then we have (cid:12)(cid:12)(cid:12)(cid:12) D ( x, y )( x − y ) × (cid:18) I ( x ) + I ( x, y ) (cid:19) + D ( x, y ) × I ( x, y ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ const( R ) √ xλ / ( y − x ) ×× (cid:18) const( R ) √ x + const( R ) √ x (cid:19) + const( R ) x / λ / × √ xλ / y − x ≤ const( R ) xλ / ( y − x ) , and we can apply Proposition 2.9(i).For D ( x, y ) / ( x − y ) × I ( x, y ) we obtain an estimate (cid:12)(cid:12)(cid:12)(cid:12) D ( x, y )( x − y ) × I ( x, y ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ const( R ) λ − / ( y − x ) × √ xλ / y − x ≤ const( R ) √ x ( y − x ) . The corresponding integral diverges in the neighborhood of the point λ = y/x = 1, so wesplit the domain into two parts: y − x ≤ x / and y − x > x / . For the first part we have (cid:12)(cid:12)(cid:12)(cid:12)Z y − x (cid:18) xt + x (cid:19) / K Bessel , s − (4 x, t + x )) dt (cid:12)(cid:12)(cid:12)(cid:12) ≤ x / max ≤ ξ ≤ x / K Bessel , s − (4 x, ξ + x )) == x / max ≤ ξ ≤ x / (cid:18) x / J s (2 x / ) J s − (2( ξ + x ) / ) − J s − (2 x / )4 ξ −− J s − (2 x / ) ( ξ + x ) / J s (2( ξ + x ) / ) − x / J s (2 x / )4 ξ (cid:19) ≤≤ x / | J s (2 x / ) | max ≤ ξ ≤ x / (cid:12)(cid:12)(cid:12)(cid:12) J ′ s − (2( ξ + x ) / ) (cid:12)(cid:12)(cid:12)(cid:12) + 14 | J s − (2 x / ) |×× (cid:18) x − / max ≤ ξ ≤ x / (cid:12)(cid:12)(cid:12)(cid:12) J s (2( ξ + x ) / ) (cid:12)(cid:12)(cid:12)(cid:12) + max ≤ ξ ≤ x / (cid:12)(cid:12)(cid:12)(cid:12) J ′ s (2( ξ + x ) / ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:19)! ≤ const( R ) , and we can apply Proposition 2.5(i) for this part. For the second part we have (cid:12)(cid:12)(cid:12)(cid:12) D ( x, y )( x − y ) × I ( x, y ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ const( R )( y − x ) , and we apply Proposition 2.5(i) once again.For D ( x, y ) / ( x − y )( − / I ( x, y )) we have2 D ( x, y ) = y − / J s (2 x / ) J s − ( y / ) − / x − / J s (2 y / ) J s − (2 x / ) ! ++ x / y − / J ′ s (2 x / ) J s − (2 y / ) + J s (2 y / ) J ′ s − (2 x / ) ! , and for the first term there is an easy estimate (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) y − / J s (2 x / ) J s − ( y / ) − / x − / J s (2 y / ) J s − (2 x / ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ const xλ / . N NUMBER RIGIDITY FOR PFAFFIAN POINT PROCESSES 31 For the second term we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z ba (cid:18) λ − / J ′ s (2 x / ) J s − (2 λ / x / ) + J s (2 λ / x / ) J ′ s − (2 x / ) (cid:19) ×× (cid:18) − − Z λ / x / x / J s − (2 p ) dp (cid:19) + 1 πλ / x / (cid:18) λ − / sin(2 x / − sπ − π/ ×× cos(2 λ / x / − (2 s − π/ − π/ 4) + cos(2 λ / x / − sπ − π/ ×× sin(2 x / − (2 s − π/ − π/ (cid:19)(cid:18) − − √ πx / sin(2 x / − (2 s − π/ − π/ √ π ( λx ) / sin(2 λ / x / − (2 s − π/ − π/ (cid:19) dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ const( R ) log( b ) λ / , and 1 πλ / x / (cid:18) λ − / sin(2 x / − sπ − π/ 4) cos(2 λ / x / − (2 s − π/ − π/ λ / x / − sπ − π/ 4) sin(2 x / − (2 s − π/ − π/ (cid:19) =1 πλ / x / (cid:18) − λ − / sin(2 x / − sπ − π/ 4) sin(2 λ / x / − sπ − π/ λ / x / − sπ − π/ 4) cos(2 x / − sπ − π/ (cid:19) =12 πλ / x / (cid:18) (1 + λ − / ) sin (cid:0) x / ( λ / + 1) − sπ (cid:1) ++ (1 − λ − / ) cos (cid:0) λ / − x / (cid:1)(cid:19) , thus we have (cid:12)(cid:12)(cid:12)(cid:12)Z ba D ( x, λx ) (cid:18) − / I ( x, λx ) (cid:19) dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ const( R ) log( b ) λ / +const( R ) λ / (cid:18) √ λ + 1 + 1 √ λ − √ λ + 1 + 12 √ λ − √ λ + 2 + 1 √ λ − (cid:19) for R < x < y, and we can use Proposition 2.9(i) and (ii).Regarding the product D ( x, y ) × (cid:18) − / I ( x ) + I ( x, y ) + I ( x, y ) (cid:19) , we can split thevariables for all the summands. For D ( x, y ) I ( x ) we have D ( x, y ) I ( x ) = J s − (2 y / )2 y / J s − (2 x / )2 x / I ( x ) , where (cid:12)(cid:12)(cid:12)(cid:12) J s − (2 x / )2 x / I ( x ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ const( R ) x / and(2.38) (cid:12)(cid:12)(cid:12)(cid:12) J s − (2 y / )2 y / − cos(2 y / − (2 s − π/ − π/ √ πy / (cid:12)(cid:12)(cid:12)(cid:12) ≤ const y / , and we can apply Corollary 2.13.For D ( x, y ) I ( x, y ) we write D ( x, y ) I ( x, y ) = J s − (2 y / )2 y / J s − (2 x / )2 x / Z ∞ x / J s − (2 p ) dp Z ∞ x / J s − (2 p ) dp −− J s − (2 y / )2 y / J s − (2 x / )2 x / Z ∞ y / J s − (2 p ) dp Z ∞ x / J s − (2 p ) dp where we can use the estimate (2.36) for J s − (2 x / ) / x / × R ∞ x / J s − (2 p ) dp . We alsohave (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) J s − (2 x / )2 x / (cid:18)Z ∞ x / J s − (2 p ) dp (cid:19) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ const( R ) x / , and we apply Corollary 2.13 once again.For D ( x, y ) I ( x, y ) we have D ( x, y ) I ( x, y ) = − J s − (2 y / )2 y / J s − (2 x / )2 x / Z ∞ x / J s − (2 p ) dp ++ 12 J s − (2 y / )2 y / J s − (2 x / )2 x / Z ∞ y / J s − (2 p ) dp. We use the estimates (2.38) and (cid:12)(cid:12)(cid:12)(cid:12) J s − (2 y / )2 y / Z ∞ y / J s − (2 p ) dp −− cos(2 y / − (2 s − π/ − π/ 4) sin(2 y / − (2 s − π/ − π/ πy (cid:12)(cid:12)(cid:12)(cid:12) == (cid:12)(cid:12)(cid:12)(cid:12) J s − (2 y / )2 y / Z ∞ y / J s − (2 p ) dp − cos(4 y / − sπ )4 πy (cid:12)(cid:12)(cid:12)(cid:12) ≤ const y / , and we apply Corollary 2.13 one more time.Finally for D ( x, y ) we use the estimate (2.38) and then Corollary 2.13 for the fourthtime.We have checked that the integral (2.12) tends to zero forΠ( x, y ) = ∂∂x K Bessels ( x, y ) Z yx K Bessels ( x, t ) dt, R < x < y. The fourth part, Z D For x < R , y > R we have | D ( x, y ) | ≤ const( R ) y − / , | D ( x, y ) | ≤ const( R ) x s − y − / , | D ( x, y ) | ≤ const( R ) x s − y − / . We also have (cid:12)(cid:12)(cid:12) ˜ I ( x ) (cid:12)(cid:12)(cid:12) = 2 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z ∞ (cid:18) xt + x (cid:19) / x / J s (2 x / ) J s − (2( t + x ) / )4 t −− ( t + x ) / J s (2( t + x ) / ) J s − (2 x / )4 t ! dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z x (cid:18) xt + x (cid:19) / K Bessel , s − (4 x, t + x )) dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ++ 2 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z ∞ x (cid:18) xt + x (cid:19) / K Bessel , s − (4 x, t + x )) dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , where (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z x (cid:18) xt + x (cid:19) / x / J s (2 x / ) J s − (2( t + x ) / )2 t −− ( t + x ) / J s (2( t + x ) / ) J s − (2 x / )2 t ! dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x / J s (2 x / ) Z x (cid:18) xt + x (cid:19) / J s − (2( t + x ) / ) − J s − (2 x / )2 t dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) J s (2 x / ) J s − (2 x / ) Z x (cid:18) xt + x (cid:19) / ( t + x ) / − x / t dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x / J s − (2 x / ) Z x J s (2( t + x ) / ) − J s (2 x / )2 t dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤≤ x | J s (2 x / ) | max ξ ∈ [0 ,x ] | J ′ s − (2( x + ξ ) / ) | + const · x / | J s (2 x / ) J s − (2 x / ) | ++ 12 x | J s − (2 x / ) | max ξ ∈ [0 ,x ] | J ′ s (2( x + ξ ) / ) | ≤ const( R ) , and (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z ∞ x (cid:18) xt + x (cid:19) / x / J s (2 x / ) J s − (2( t + x ) / )2 t −− ( t + x ) / J s (2( t + x ) / ) J s − (2 x / )2 t ! dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) J s (2 x / ) Z ∞ x t J s − (2( t + x ) / )( t + x ) / dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ++ x / (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) J s − (2 x / ) Z ∞ x ( t + x )2 t J s (2( t + x ) / ) t + x dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ const | J s (2 x / ) | + const · x / | J s − (2 x / ) | ≤ const( R ) x s ≤ const( R ) , where we have used (2.13) to estimate both terms. Finally we have (cid:12)(cid:12)(cid:12) ˜ I ( x ) (cid:12)(cid:12)(cid:12) ≤ const( R ) . We use the estimate (2.37) for x < R , y > R to obtain | I ( x, y ) | ≤ const · x | J s (2 x / ) | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z ∞ y − x t ( t + x ) / J s − (2( t + x ) / )( t + x ) / dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ++ const · x / | J s − (2 x / ) | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z ∞ y − x ( t + x ) / t J s (2( t + x ) / )( t + x ) / dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤≤ const( R ) x s +1 y − / + x s y / y − x ≤ const( R ) y / y − x . For I ( x, y ) + I ( x, y ) == Z ∞ y / J s − (2 p ) dp Z x / J s − (2 p ) dp − Z ∞ x / J s − (2 p ) dp Z x / J s − (2 p ) dp we have Z ∞ x / J s − (2 p ) dp Z x / J s − (2 p ) dp ≤ const( R )and Z ∞ y / J s − (2 p ) dp Z x / J s − (2 p ) dp ≤ const( R ) y / . Thus we can use Proposition 2.5(ii) for D ( x, y ) / ( x − y ) × (cid:18) ˜ I ( x ) + I ( x, y ) + I ( x, y ) (cid:19) and for D ( x, y ) / ( x − y ) × I ( x, y ). For D ( x, y ) / ( x − y ) × I ( x, y ) we obtain an estimate (cid:12)(cid:12)(cid:12)(cid:12) D ( x, y )( x − y ) × I ( x, y ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ const( R ) x s y − / ( y − x ) × x s y / y − x ≤ const( R )( y − x ) . N NUMBER RIGIDITY FOR PFAFFIAN POINT PROCESSES 35 We split the domain into two parts: x < R < y < R and x < R < R < y . For the firstpart we have max x Recall the definition of the matrix kernel K Bessel ,s ( x, y )in the introduction. We start with the following proposition, similar to Proposition 2.16. Proposition 2.19. Z ∞ Z ∞ det K Bessel ,s ( x, y ) dy − K ,s ( x, x ) ! dx = 0 . Proof. Since lim y →∞ K ,s ( y, x ) = 0 , we may use Corollary 2.3 again, and we will first simplify the expressions for the defectDef K Bessel ,s ( x ) and for the limits − K ,s ( y, x ) Z yx K ,s ( y, t ) dt (cid:12)(cid:12)(cid:12)(cid:12) ∞ y =0 . First, we see that K Bessel ,s +1 ( x, y ) = 14 Z J s +1 ( √ ux ) J s +1 ( √ uy ) du is an orthogonal projection onto the subspace of functions f ( x ) such that f ( x ) has itsHankel transform supported in [0 , J s +1 ( √ ux )lies in the image of K Bessel ,s +1 ( x, y ) for u ∈ [0 , 1] and is orthogonal to the image for u > x / Z ∞ x / J s +1 ( t ) dt = Z ∞ J s +1 ( x / t ) dt is orthogonal to Ran (cid:18) K Bessel ,s +1 ( x, y ) (cid:19) , that is Z ∞ y / Z ∞ y / J s +1 ( t ) dt · K Bessel ,s +1 ( x, y ) dy = 0 . Therefore we have Z ∞ K Bessel ,s ( x, y ) (cid:16) yx (cid:17) / K Bessel ,s +1 ( x, y ) dy = Z ∞ (cid:18)(cid:18) xy (cid:19) / K Bessel ,s +1 ( x, y )++ J s +1 ( y / )4 y / Z ∞ x / J s +1 ( t ) dt (cid:19) (cid:16) yx (cid:17) / K Bessel ,s +1 ( x, y ) dy == Z ∞ K Bessel ,s +1 ( x, y ) K Bessel ,s +1 ( x, y ) dy ++ 14 x / Z ∞ x / J s +1 ( t ) dt · Z ∞ J s +1 ( y / ) K Bessel ,s +1 ( x, y ) dy == K Bessel ,s +1 ( x, x ) + 14 x / Z ∞ x / J s +1 ( t ) dt · J s +1 ( x / ) = K Bessel ,s ( x, x )and Z ∞ (cid:18) xy (cid:19) / K Bessel ,s +1 ( x, y ) Z ∞ y / J s +1 ( t ) dtdy = 0 . We also have Z ∞ J s +1 ( y / )4 y / Z ∞ y / J s +1 ( t ) dtdy = − Z ∞ d (cid:18)Z ∞ y / J s +1 ( t ) dt (cid:19) = 14 . Thus Z ∞ K Bessel ,s ( x, y ) K Bessel ,s ( y, x ) dy = Z ∞ K Bessel ,s ( x, y ) (cid:16) yx (cid:17) / K Bessel ,s +1 ( x, y ) dy ++ J s +1 ( x / )4 x / Z ∞ (cid:18) xy (cid:19) / K Bessel ,s +1 ( x, y ) Z ∞ y / J s +1 ( t ) dtdy ++ J s +1 ( x / )4 x / Z ∞ x / J s +1 ( t ) dt Z ∞ J s +1 ( y / )4 y / Z ∞ y / J s +1 ( t ) dtdy = K Bessel ,s ( x, x ) + J s +1 ( x / )16 x / Z ∞ x / J s +1 ( t ) dt, and 2 Def K Bessel ,s ( x ) = J s +1 ( x / )8 x / Z ∞ x / J s +1 ( t ) dt. Now since K ,s (0 , x ) = J s +1 ( x / )4 x / and lim y →∞ K ,s ( y, x ) = 0 , we have − K ,s ( y, x ) Z yx K ,s ( y, t ) dt (cid:12)(cid:12)(cid:12)(cid:12) ∞ y =0 = J s +1 ( x / )4 x / (cid:18) lim y → Z xy (cid:16) yt (cid:17) / K Bessel ,s +1 ( y, t ) dt −− Z x J s +1 ( t / )4 t / Z ∞ J s +1 ( p ) dp (cid:19) dt = − J s +1 ( x / )8 x / Z x / J s +1 ( t ) dt. N NUMBER RIGIDITY FOR PFAFFIAN POINT PROCESSES 37 Write Z ∞ det K Bessel ,s ( x, y ) dy − K ,s ( x, x ) =2 Def K Bessel ,s ( x ) − K ,s ( y, x ) Z yx K ,s ( y, t ) dt (cid:12)(cid:12)(cid:12)(cid:12) ∞ y =0 == J s +1 ( x / )8 x / (cid:18)Z ∞ x / J s +1 ( t ) dt − Z x / J s +1 ( t ) dt (cid:19) . We directly see that(2.39) J s +1 ( x / )8 x / (cid:18)Z ∞ x / J s +1 ( t ) dt − Z x / J s +1 ( t ) dt (cid:19) = 0 , and therefore the relation (1.6) does not hold for the kernel K Bessel ,s . Nonetheless, we have Z ∞ J s +1 ( x / )8 x / (cid:18)Z ∞ x / J s +1 ( t ) dt − Z x / J s +1 ( t ) dt (cid:19)! dx == − Z ∞ d (cid:18)Z ∞ x / J s +1 ( t ) dt (cid:19) − Z ∞ d (cid:18)Z x / J s +1 ( t ) dt (cid:19) = 18 − 18 = 0 . (cid:3) Proof of the Theorem . Our plan is to expand the determinant,det K Bessel ,s ( x, y ) = K ,s ( x, y ) K ,s ( y, x ) + ∂∂x K ,s ( x, y ) Z yx K ,s ( x, t ) dt + sgn( x − y )2 ! and then to show that all the summands in the formula (2.11) for the variance, with f ( x ) = ϕ ( R,T ) ( x ), tend to zero term by term. The first part, Z R + | ϕ ( R,T ) ( x ) | (cid:18) K ,s ( x, x ) − Z R det K Bessel ,s ( x, y ) dy (cid:19) dx. We setΠ( x ) = K ,s ( x, x ) − Z ∞ det K Bessel ,s ( x, y ) dy == − J s +1 ( x / )8 x / (cid:18)Z ∞ x / J s +1 ( t ) dt − Z x / J s +1 ( t ) dt (cid:19) == J s +1 ( x / )8 x / (cid:18) − Z ∞ x / J s +1 ( t ) dt (cid:19) , we use Proposition 2.19 to write Z ∞ | ϕ ( R,T ) ( x ) | Π( x ) dx = Z ∞ (cid:18) | ϕ ( R,T ) ( x ) | − (cid:19) Π( x ) dx and then we use estimates (2.24) and (2.28) and Corollary 2.14 to see that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z ∞ (cid:18) | ϕ ( R,T ) ( x ) | − (cid:19) Π( x ) dx (cid:12)(cid:12)(cid:12)(cid:12) −−−→ T →∞ . The second part, Z D | ϕ ( R,T ) ( x ) − ϕ ( R,T ) ( y ) | K ,s ( x, y ) · K ,s ( y, x ) dxdy. We have K ,s ( x, y ) · K ,s ( y, x ) = (cid:18) xy (cid:19) / K Bessel ,s +1 ( x, y ) + J s +1 ( y / )4 y / Z ∞ x / J s +1 ( t ) dt ! ×× (cid:16) yx (cid:17) / K Bessel ,s +1 ( y, x ) + J s +1 ( x / )4 x / Z ∞ y / J s +1 ( t ) dt ! = (cid:0) K Bessel ,s +1 ( x, y ) (cid:1) + (cid:18) xy (cid:19) / K Bessel ,s +1 ( x, y ) J s +1 ( x / )4 x / Z ∞ y / J s +1 ( t ) dt ++ J s +2 ( x / ) J s +1 ( y / )8( x − y ) Z ∞ x / J s +1 ( t ) dt ! ++ y / x − / J s +2 ( y / ) J s +1 ( x / ) J s +1 ( y / )8( x − y ) Z ∞ x / J s +1 ( t ) dt ++ J s +1 ( x / )4 x / Z ∞ x / J s +1 ( t ) dt J s +1 ( y / )4 y / Z ∞ y / J s +1 ( t ) dt =:=: S ( x, y )( x − y ) + S ( x, y ) x − y + S ( x, y ) x − y + S ( x, y ) . The integral for the first term, S ( x, y ) / ( x − y ) = (cid:18) K Bessel ,s +1 ( x, y ) (cid:19) , was estimated in[2] by Proposition 2.5.We use (2.24) and (2.28) to obtain (cid:12)(cid:12)(cid:12)(cid:12) S ( x, y ) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:0) x / J s +2 ( x / ) J s +1 ( y / ) − y / J s +2 ( y / ) J s +1 ( x / ) (cid:1) × J s +1 ( x / )4 y / Z ∞ y / J s +1 ( t ) dt + J s +2 ( x / ) J s +1 ( y / )8 Z ∞ x / J s +1 ( t ) dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ const( R ) λ / x , for any R > x, y > R , and where we have set, as usual, λ = y/x . We see that theconditions of Proposition 2.92.14 are satisfied. Also we easily obtain (cid:12)(cid:12)(cid:12)(cid:12) S ( x, y ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ const(R , s) y − / for x < R, y > R, and we can apply Proposition 2.9. N NUMBER RIGIDITY FOR PFAFFIAN POINT PROCESSES 39 We also have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) S ( x, y ) − π √ x cos( y / − ( s + 2) π/ − π/ 4) cos( y / − ( s + 1) π/ − π/ J s +1 ( x / ) ×× Z ∞ x / J s +1 ( t ) dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ const (cid:12)(cid:12) J s +1 ( x / ) R ∞ x / J s +1 ( t ) dt (cid:12)(cid:12) √ xy , thus we can combine this estimate with (2.23), (2.24), (2.28) to use Proposition 2.9 inthis case.For the main term of S ( x, y ) we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z ba π √ x cos( y / − ( s + 2) π/ − π/ 4) cos( y / − ( s + 1) π/ − π/ J s +1 ( x / ) ×× Z ∞ x / J s +1 ( t ) dtdx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z ba π √ x cos(2 λ / x / − ( s + 2) π ) J s +1 ( x / ) ×× Z ∞ x / J s +1 ( t ) dtdx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ const log( b ) √ λ by Lemma 2.8, and we can use Proposition 2.9(i). We also haveconst( R )(log T ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) R Z T Z R log ( y − R + 1) cos(2 y / − ( s + 2) π ) J s +1 ( x / ) R ∞ x / J s +1 ( t ) dtx / ( y − x ) dydx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤≤ const( R )(log T ) R Z x s/ dx R ′ Z R log ( y − R + 1) y − R dy + √ R ′ log ( R ′ − R + 1) R ′ − R ! ≤ const( R )(log T ) , where the function √ y log ( y − R + 1) / ( y − R ) is decreasing for y ≥ R ′ . And (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) R Z ∞ Z T cos(2 y / − ( s + 2) π ) J s +1 ( x / ) R ∞ x / J s +1 ( t ) dtx / ( y − x ) dydx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ const( R ) √ TT − R R Z x s/ dx −−−→ T →∞ , thus we can use Corollary 2.11.For the last term S ( x, y ) we have S ( x, y ) = J s +1 ( x / )4 x / Z ∞ x / J s +1 ( t ) dt × J s +1 ( y / )4 y / Z ∞ y / J s +1 ( t ) dt, where the variables are split. We use the estimates (2.23), (2.24) to obtain (cid:12)(cid:12)(cid:12)(cid:12) J s +1 ( x / )4 x / Z ∞ x / J s +1 ( t ) dt (cid:12)(cid:12)(cid:12)(cid:12) ≤ const( R ) x s/ for x < R. Moreover, from the estimates (2.24), (2.28) we see that(2.40) (cid:12)(cid:12)(cid:12)(cid:12) J s +1 ( x / )4 x / Z ∞ x / J s +1 ( t ) dt −− πx sin( x / − ( s + 1) π/ − π/ 4) cos( x / − ( s + 1) π/ − π/ (cid:12)(cid:12)(cid:12)(cid:12) ≤ const( R ) x / , for x > R thus the conditions of Corollary 2.13 are satisfied and we have proved therequired convergence to zero of the intergal (2.12) for S ( x, y ), and therefore for all thefirst term K ,s ( x, y ) · K ,s ( y, x ). The third part, Z D >R | ϕ ( R,T ) ( x ) − ϕ ( R,T ) ( y ) | ∂∂x K ,s ( x, y ) Z yx K ,s ( x, t ) dt + sgn( x − y )2 ! dxdy. We will use the following notation: ∂∂x K ,s ( x, y ) = − xy − / J s +2 ( x / ) J s +1 ( y / ) − x / J s +2 ( y / ) J s +1 ( x / )2( x − y ) + y − / J s +2 ( x / ) J s − ( y / ) + x / y − / J ′ s +2 ( x / ) J s +1 ( y / )2( x − y ) −− / x − / J s +2 ( y / ) J s +1 ( x / ) + J s +2 ( y / ) J ′ s +1 ( x / )2( x − y ) ! ++ J s +1 ( y / )4 y / J s +1 ( x / )2 x / =: D ( x, y )( x − y ) + D ( x, y )( x − y ) + D ( x, y ) . And Z yx K ,s ( x, t ) dt + 12 sgn( x − y ) = Z y − x K ,s ( x, t + x ) dt − 12 = Z ∞ (cid:18) xt + x (cid:19) / K Bessel ,s +1 ( x, t + x ) dt − Z ∞ y − x (cid:18) xt + x (cid:19) / K Bessel ,s +1 ( x, t + x ) dt ++ 12 Z y / x / J s +1 ( p ) dp Z ∞ x / J s +1 ( p ) dp − 12 =: ˜ I ( x ) + I ( x, y ) + I ( x, y ) − , N NUMBER RIGIDITY FOR PFAFFIAN POINT PROCESSES 41 where we have used that sgn( x − y ) = − y > x . We will first separate the main partof ˜ I ( x ), because the corresponding integrals are not absolutely convergent. We have˜ I ( x ) = Z ∞ (cid:18) xt + x (cid:19) / x / J s +2 ( x / ) J s +1 (( t + x ) / )2 t −− ( t + x ) / J s +2 (( t + x ) / ) J s +1 ( x / )2 t ! dt, and we use (2.24) to estimate it as follows: (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˜ I ( x ) − π Z ∞ x / cos( x / − ( s + 2) π/ − π/ 4) cos(( t + x ) / − ( s + 1) π/ − π/ t ( t + x ) / −− x / cos(( t + x ) / − ( s + 2) π/ − π/ 4) cos( x / − ( s + 1) π/ − π/ t ( t + x ) / ! dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Z const x / dt + const x − / Z ∞ dtt ( t + x ) / ≤ const x / . We also have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z ∞ x / cos( x / − ( s + 2) π/ − π/ 4) cos(( t + x ) / − ( s + 1) π/ − π/ t ( t + x ) / x / ( t + x ) / − ! dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ x / Z dt t + x ) / ( x / + ( t + x ) / ) + x / (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z ∞ cos(( t + x ) / − ( s + 1) π/ − π/ t + x ) / ( x / + ( t + x ) / ) dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤≤ const( R ) √ x , for x > R, where we have used (2.13) once again to estimate the second term. We proceed now withthe estimate of the main term2 π Z ∞ x / t ( t + x ) / (cid:18) cos( x / − ( s + 2) π/ − π/ 4) cos(( t + x ) / − ( s + 1) π/ − π/ −− cos(( t + x ) / − ( s + 2) π/ − π/ 4) cos( x / − ( s + 1) π/ − π/ (cid:19) dt == − π Z ∞ x / t ( t + x ) / sin(( t + x ) / − x / ) dt = − π Z ∞ ( u + 1) / ( u + 1) − x / u ) du = − π Z ∞ sin( x / u ) u ( u + 1) / u + 2 du = − π Z ∞ sin( u ) u du + O (cid:18) √ x (cid:19) = − 12 + O (cid:18) √ x (cid:19) , where we have put u = ( t/x + 1) / − I ( x ) = − 12 + I ( x ) , Z yx K ,s ( x, t ) dt − 12 = − I ( x ) + I ( x, y ) + I ( x, y ) , where | I ( x ) | ≤ const( R ) / √ x for x ≥ R . Now we write ∂∂x K ,s ( x, y ) Z yx K ,s ( x, t ) dt − ! = (cid:18) D ( x, y )( x − y ) + D ( x, y )( x − y ) + D ( x, y ) (cid:19) ×× (cid:18) − I ( x ) + I ( x, y ) + I ( x, y ) (cid:19) = D ( x, y )( x − y ) × (cid:18) − I ( x ) + I ( x, y ) (cid:19) + D ( x, y )( x − y ) × I ( x, y ) ! + D ( x, y )( x − y ) × (cid:18) I ( x ) + I ( x, y ) (cid:19) + D ( x, y ) × I ( x, y ) ! ++ D ( x, y )( x − y ) × I ( x, y ) − D ( x, y )( x − y ) + D ( x, y ) × (cid:18) − I ( x ) + I ( x, y ) (cid:19) , and we estimate the summands one by one.We use (2.24), (2.27), (2.28) and (2.13) to obtain the following simple estimates D ( x, y ) ≤ const( R ) λ / , D ( x, y ) ≤ const( R ) √ xλ / , D ( x, y ) ≤ const( R ) x / λ / , I ( x, y ) ≤ const( R ) √ x . We additionally use (2.13) to obtain the following estimate:(2.41) | I ( x, y ) | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z ∞ y − x (cid:18) xt + x (cid:19) / x / J s +2 ( x / ) J s +1 (( t + x ) / )2 t −− ( t + x ) / J s +2 (( t + x ) / ) J s +1 ( x / )2 t ! dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤≤ const · x | J s +2 ( x / ) | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z ∞ y − x t ( t + x ) / J s +1 (( t + x ) / )( t + x ) / dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ++ const · x / | J s +1 ( x / ) | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z ∞ y − x ( t + x ) / t J s +2 (( t + x ) / )( t + x ) / dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤≤ const( R ) x / y − / + x / y / y − x ≤ const( R ) √ xλ / y − x . From the estimates above we obtain (cid:12)(cid:12)(cid:12)(cid:12) D ( x, y )( x − y ) × (cid:18) − I ( x ) + I ( x, y ) (cid:19) + D ( x, y )( x − y ) × I ( x, y ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ const( R ) λ − / ( x − y ) ×× (cid:18) R ) √ x + const( R ) √ x (cid:19) + const( R ) √ xλ / × √ xλ / ( x − y ) ≤ const( R )( x − y ) , and we can apply Proposition 2.5(i).Then we have (cid:12)(cid:12)(cid:12)(cid:12) D ( x, y )( x − y ) × (cid:18) I ( x ) + I ( x, y ) (cid:19) + D ( x, y ) × I ( x, y ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ const( R ) √ xλ / ( y − x ) ×× (cid:18) const( R ) √ x + const( R ) √ x (cid:19) + const( R ) x / λ / × √ xλ / y − x ≤ const( R ) xλ / ( y − x ) , N NUMBER RIGIDITY FOR PFAFFIAN POINT PROCESSES 43 and we can apply Proposition 2.9(i).For D ( x, y ) / ( x − y ) × I ( x, y ) we obtain an estimate (cid:12)(cid:12)(cid:12)(cid:12) D ( x, y )( x − y ) × I ( x, y ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ const( R ) λ − / ( y − x ) × √ xλ / y − x ≤ const( R ) √ x ( y − x ) . The corresponding integral diverges in the neighborhood of the point λ = y/x = 1, so wesplit the domain into two parts: y − x ≤ x / and y − x > x / . For the first part we have (cid:12)(cid:12)(cid:12)(cid:12)Z y − x (cid:18) xt + x (cid:19) / K Bessel ,s +1 ( x, t + x ) dt (cid:12)(cid:12)(cid:12)(cid:12) ≤ x / max ≤ ξ ≤ x / K Bessel ,s +1 ( x, ξ + x ) == x / max ≤ ξ ≤ x / (cid:18) x / J s ( x / ) J s − (( ξ + x ) / ) − J s − ( x / ) ξ −− J s − ( x / ) ( ξ + x ) / J s (( ξ + x ) / ) − x / J s ( x / ) ξ (cid:19) ≤≤ x / | J s ( x / ) | max ≤ ξ ≤ x / (cid:12)(cid:12)(cid:12)(cid:12) J ′ s − (( ξ + x ) / ) (cid:12)(cid:12)(cid:12)(cid:12) + 12 | J s − ( x / ) |×× (cid:18) x − / max ≤ ξ ≤ x / (cid:12)(cid:12)(cid:12)(cid:12) J s (( ξ + x ) / ) (cid:12)(cid:12)(cid:12)(cid:12) + max ≤ ξ ≤ x / (cid:12)(cid:12)(cid:12)(cid:12) J ′ s (( ξ + x ) / ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:19)! ≤ const( R ) , and we can apply Proposition 2.5(i) for this part. For the second part we have (cid:12)(cid:12)(cid:12)(cid:12) D ( x, y )( x − y ) × I ( x, y ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ const( R )( y − x ) , and we apply Proposition 2.5(i) once again.For D ( x, y ) / ( x − y ) we have2 D ( x, y ) = y − / J s +2 ( x / ) J s +1 ( y / ) − / x − / J s +2 ( y / ) J s +1 ( x / ) ! ++ x / y − / J ′ s +2 ( x / ) J s +1 ( y / ) + J s +2 ( y / ) J ′ s +1 ( x / ) ! , and for the first term there is an easy estimate (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) y − / J s +2 ( x / ) J s +1 ( y / ) − / x − / J s +2 ( y / ) J s +1 ( x / ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ const xλ / . For the second term we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z ba λ − / J ′ s +2 ( x / ) J s +1 ( λ / x / ) + J s +2 ( λ / x / ) J ′ s +1 ( x / )++ 2 πλ / x / (cid:18) λ − / sin( x / − ( s + 2) π/ − π/ 4) cos( λ / x / − ( s + 1) π/ − π/ λ / x / − ( s + 2) π/ − π/ 4) sin( x / − ( s + 1) π/ − π/ (cid:19) dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ const( R ) log( b ) λ / , and 2 πλ / x / (cid:18) λ − / sin( x / − ( s + 2) π/ − π/ 4) cos( λ / x / − ( s + 1) π/ − π/ λ / x / − ( s + 2) π/ − π/ 4) sin( x / − ( s + 1) π/ − π/ (cid:19) =2 πλ / x / (cid:18) − λ − / sin( x / − ( s + 2) π/ − π/ 4) sin( λ / x / − ( s + 2) π/ − π/ λ / x / − ( s + 2) π/ − π/ 4) cos( x / − ( s + 2) π/ − π/ (cid:19) =2 πλ / x / (cid:18) (1 + λ − / ) sin (cid:0) x / ( λ / + 1) − ( s + 2) π (cid:1) ++ (1 − λ − / ) cos (cid:0) ( λ / − x / (cid:1)(cid:19) , thus we have (cid:12)(cid:12)(cid:12)(cid:12)Z ba D ( x, λx ) dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ const( R ) log( b ) λ / for R < x < y, and we can use Proposition 2.9(i).Regarding the product D ( x, y ) × (cid:18) − I ( x ) + I ( x, y ) (cid:19) , we can split the variablesfor all the summands. For D ( x, y ) I ( x ) we have D ( x, y ) I ( x ) = J s +1 ( y / )4 y / J s +1 ( x / )2 x / I ( x ) , where (cid:12)(cid:12)(cid:12)(cid:12) J s +1 ( x / )2 x / I ( x ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ const( R ) x / and(2.42) (cid:12)(cid:12)(cid:12)(cid:12) J s +1 ( y / )4 y / − cos( y / − ( s + 1) π/ − π/ √ πy / (cid:12)(cid:12)(cid:12)(cid:12) ≤ const y / , and we can apply Corollary 2.13.For D ( x, y ) I ( x, y ) we write D ( x, y ) I ( x ) = J s +1 ( y / )4 y / J s +1 ( x / )4 x / Z ∞ x / J s +1 ( p ) dp Z ∞ x / J s +1 ( p ) dp −− J s +1 ( y / )4 y / J s +1 ( x / )4 x / Z ∞ y / J s +1 ( p ) dp Z ∞ x / J s +1 ( p ) dp where we can use the estimate (2.40) for J s +1 ( x / ) / x / × R ∞ x / J s +1 ( p ) dp . We also have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) J s +1 ( x / )4 x / (cid:18)Z ∞ x / J s +1 ( p ) dp (cid:19) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ const( R ) x / , and we apply Corollary 2.13 once again. N NUMBER RIGIDITY FOR PFAFFIAN POINT PROCESSES 45 Finally for D ( x, y ) we use the estimate (2.42) and then Corollary 2.13 for the thirdtime.We have checked that the integral (2.12) tends to zero forΠ( x, y ) = ∂∂x K ,s ( x, y ) Z yx K ,s ( x, t ) dt − ! , R < x < y. The fourth part, Z D We use the estimate (2.41) for x < R , y > R to obtain | I ( x, y ) | ≤ const · x | J s +2 ( x / ) | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z ∞ y − x t ( t + x ) / J s +1 (( t + x ) / )( t + x ) / dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ++ const · x / | J s +1 ( x / ) | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z ∞ y − x ( t + x ) / t J s +2 (( t + x ) / )( t + x ) / dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤≤ const( R ) x s/ y − / + x s/ y / y − x ≤ const( R ) y / y − x . For I ( x, y ) = 12 Z ∞ x / J s +1 ( p ) dp Z ∞ x / J s +1 ( p ) dp − Z ∞ y / J s +1 ( p ) dp Z ∞ x / J s +1 ( p ) dp we have Z ∞ x / J s +1 ( p ) dp Z ∞ x / J s +1 ( p ) dp ≤ const( R )and Z ∞ y / J s +1 ( p ) dp Z ∞ x / J s +1 ( p ) dp ≤ const( R ) y / . Thus we can use Proposition 2.5(ii) for D ( x, y ) / ( x − y ) × (cid:18) ˜ I ( x ) + I ( x, y ) (cid:19) and for D ( x, y ) / ( x − y ) × I ( x, y ). For D ( x, y ) / ( x − y ) × I ( x, y ) we have obtained an estimate (cid:12)(cid:12)(cid:12)(cid:12) D ( x, y )( x − y ) × I ( x, y ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ const( R ) y − / ( y − x ) × y / y − x ≤ const( R )( y − x ) . We split the domain into two parts: x < R < y < R and x < R < R < y . For the firstpart we have max x The spectral measure of linear statistics. In this section, we deal with thespectral measure of continuous time stationary processes. In what follows, a simple con-figuration ξ on R will be identified with its support X = supp( ξ ) ⊂ R , which is a locallyfinite countable subset of R .Recall that the spectral measure µ M for a complex-valued continuous time L -boundedstationary (in the weak sense) stochastic process M = ( M t ) t ∈ R is defined as the uniquemeasure µ M on R such that c µ M ( t ) = Z R e − i πλt dµ M ( λ ) = E h ( M − E ( M ))( M t − E ( M )) i . Consider a stationary point process P on R . Assume that the first and second correlationfunctions of P exist. Let X denote a random configuration on R with distribution P andlet g ∈ L ( R ). We consider an additive functional (linear statistics): S g ( X ) = X x ∈ X g ( x ) . The spectral measure for the centralized linear statistics (cid:16) S g ( X + t ) − E S g ( X ) (cid:17) t ∈ R (3.43)is given as follows. For avoiding the analysis of convergence of integrals, in what follows,we assume for simplicity that g is bounded and compactly supported. We have E h(cid:16) S g ( X ) − E S g ( X ) (cid:17)(cid:16) S g ( X + t ) − E S g ( X ) (cid:17)i = E h X x,y ∈ X g ( x ) g ( y + t ) i −| E ( S g ( X )) | = Z R g ( x ) g ( y + t ) ρ (2) P ( x, y ) dxdy + Z R g ( x ) g ( x + t ) ρ (1) P ( x, x ) dx − (cid:12)(cid:12)(cid:12) Z R g ( x ) ρ (1) P ( x, x ) dx (cid:12)(cid:12)(cid:12) = Z R g ( x ) g ( y + t ) ρ (2 ,T ) P ( x, y ) dxdy + Z R g ( x ) g ( x + t ) ρ (1) P ( x ) dx, where ρ (2 ,T ) P ( x, y ) = ρ (2) P ( x, y ) − ρ (1) P ( x ) ρ (1) P ( y ) . (3.44)The stationarity of P implies that there exists a function F : R → R and a constant ρ > ρ (2 ,T ) P ( x, y ) = F ( x − y ) and ρ (1) P ( x ) = ρ. (3.45)Denote by g t ( x ) := g ( x + t ), then b g t ( λ ) = e i πλt b g ( λ ), where we use the definition of theFourier transform: b g ( λ ) = Z R g ( x ) e − i πλx dx. By Fubini theorem and the Plancherel identity, Z R g ( x ) g ( y + t ) ρ (2 ,T ) P ( x, y ) dxdy = Z R g ( x ) g t ( y ) F ( x − y ) dxdy == D g, g t ∗ F E L ( R ) = Db g, e i πλt b g · b F E L ( R ) = Z R | b g ( λ ) | e − i πλt b F ( λ ) dλ. N NUMBER RIGIDITY FOR PFAFFIAN POINT PROCESSES 49 On the other hand, Z R g ( x ) g ( x + t ) ρ (1) P ( x ) dx = ρ h g, g t i L ( R ) = ρ h b g, e i πλt b g i L ( R ) = ρ Z R | b g ( λ ) | e − i πλt dλ. Thus E h(cid:16) S g ( X ) − E S g ( X ) (cid:17)(cid:16) S g ( X + t ) − E S g ( X ) (cid:17)i = Z R | b g ( λ ) | e − i πλt ( b F ( λ ) + ρ ) dλ. Thus we have Proposition 3.1. The spectral measure for the centralized linear statistics (3.43) is givenby | b g ( λ ) | ( b F ( λ ) + ρ ) dλ, where F and ρ are given by (3.45) . Proof of Proposition 1.4. The proof of Proposition 1.4 below is similar to that of[4, Lemma 3.3]. Proof of Proposition . It suffices to prove that for any fixed R > n ∈ N , we can construct a real-valued Schwartz function ϕ n such that | ϕ n ( x ) | ≤ , sup x ∈ [ − R,R ] | ϕ n ( x ) − | ≤ /n and Var P ( S ϕ n ) ≤ /n. By Proposition 3.1 and the assumption (1.7), we haveVar P ( S ϕ ) = Z R | b ϕ ( λ ) | ( b F ( λ ) − b F (0)) dλ ≤ C Z R | b ϕ ( λ ) | | λ | dλ. Fix n ∈ N . Let k ≥ n be large enough such that for any | t | ≤ Rk − , we have | e i πt − | ≤ n − . We claim that there exists a non-negative even function ψ n ∈ C ∞ c ( R ) supported in a( k )-neighbourhood of 0, such that Z R ψ n ( λ ) dλ = 1 and Z R | λ | ψ n ( λ ) dλ ≤ Cn . (3.46)Indeed, since the function Cn | λ | χ | λ |≤ /k is not integrable, there exists a Schwartz function ψ n such that R R ψ n = 1 and ψ n ( λ ) ≤ Cn | λ | χ | λ |≤ /k , for any λ ∈ R . This inequality implies supp( ψ n ) ⊂ [ − /k, /k ] and Z R | λ | ψ n ( λ ) dλ ≤ (cid:16) sup λ | λ | ψ n ( λ ) (cid:17) · Z R ψ n ( λ ) dλ ≤ Cn . Now set ϕ n ( x ) = ˇ ψ n ( x ) = Z R ψ n ( λ ) e i πxλ dλ. Then ϕ n ∈ S ( R ), ϕ n (0) = 1 and | ϕ n ( x ) | ≤ 1. Since ψ n is even and real-valued, ϕ n isreal-valued. By (3.46), we haveVar P ( S ϕ n ) ≤ C Z R | λ || b ϕ n ( λ ) | dλ = C Z R | λ || ψ n ( λ ) | dλ ≤ n − . Moreover, by our choice of k , if | λ | ≤ k − and | x | ≤ R , then | e i πxλ − | ≤ n − . Hence forany | x | ≤ R , | ϕ n ( x ) − | = | ϕ n ( x ) − ϕ n (0) | ≤ Z R | e i πxλ − || ψ n ( λ ) | dλ = Z | λ |≤ k − | e i πxλ − || ψ n ( λ ) | dλ ≤ n − . This completes the proof of the proposition. (cid:3) Proof of Propositions 1.5 and 1.6.Lemma 3.2. We have the following identity Z R ρ (2 ,T )sine , ( x, y ) dy = − ρ (1)sine , ( x ) = − . (3.47) Proof. By the reproducing property of the sine kernel, we have Z R S ( x − y ) dy = S ( x − x ) = 1 . Therefore, Z R ρ (2 ,T )sine , ( x, y ) dy = − Z R det K sine , ( x, y ) dy = − Z R S ( x − y ) dy + Z R S ′ ( x − y ) IS ( x − y ) dy − Z R S ′ ( x − y ) ε ( x − y ) dy = − IS ( y − x ) S ( y − x ) (cid:12)(cid:12)(cid:12) ∞ y = −∞ − Z R S ( y − x ) dy −− Z ∞ x S ′ ( y − x ) dy + 12 Z x −∞ S ′ ( y − x ) dy = − . (cid:3) Lemma 3.3. The limit lim R,M →∞ Z M − R ρ (2 ,T )sine , ( x, y ) dy exists and we have lim R,M →∞ Z M − R ρ (2 ,T )sine , ( x, y ) dy = − ρ (1)sine , ( x ) = − . N NUMBER RIGIDITY FOR PFAFFIAN POINT PROCESSES 51 Proof. Indeed,lim R,M →∞ Z M − R ρ (2 ,T )sine , ( x, y ) dy = − 14 lim R,M →∞ (cid:18)Z M − R S ( x − y ) dy − Z M − R IS ( x − y ) S ′ ( x − y ) dy (cid:19) = − 14 lim R,M →∞ (cid:18) Z M − R S ( y − x ) dy − IS ( y − x ) S ( y − x ) (cid:12)(cid:12)(cid:12) My = − R (cid:19) = − (cid:18) Z ∞−∞ S ( y − x ) dy − IS ( y − x ) S ( y − x ) (cid:12)(cid:12)(cid:12) ∞ y = −∞ (cid:19) = − . (cid:3) Denote Z R ρ (2 ,T )sine , ( x, y ) dy := lim R,M →∞ Z M − R ρ (2 ,T )sine , ( x, y ) dy. We have Z R ρ (2 ,T )sine , ( x, y ) dy = − ρ (1)sine , ( x ) . (3.48)In the orthogonal and symplectic cases, the equalities (3.47) and (3.48) can be in-terpreted as b F (0) = − ρ . Therefore, in these two cases, the spectral measure for thecentralized linear statistics (3.43) is given by | b g ( λ ) | ( b F ( λ ) − b F (0)) dλ. (3.49) 4. Appendix. Another proof of rigidity for pfaffian sine-processes can be given by considering station-ary processes of occupation numbers of consecutive intervals and applying the Kolmogorovcriterion, in the spirit of [3] for the determinantal case.Let P be a stationary point process on R . For any bounded Borel subset B ⊂ R , wedenote by B the function that associates any configuration ξ ∈ Conf( R ) to ξ ( B ). Forany λ > n ∈ Z , set I ( λ ) n := [ nλ − λ/ , nλ + λ/ . The sequence ( X ( λ ) n ) n ∈ Z := ( I ( λ ) n ) n ∈ Z (4.50)defines a stationary stochastic process on the probability space (Conf( R ) , P ). Assumethat P admits up to second order correlation measures. Then in particular, X ( λ ) n aresquare integrable for any λ > 0. The number rigidity of P directly follows once thereexists a sequence of positive real numbers ( λ k ) k ∈ N such that λ k → + ∞ and for any k ∈ N ,we have X ( λ k )0 − E ( X ( λ k )0 ) ∈ span n X ( λ k ) n − E ( X ( λ k ) n ) : n ∈ Z \ { } o L . (4.51) Proposition 4.1 ([3, Theorem 3.1]) . Let Z = ( Z n ) n ∈ Z be a stationary stochastic process.If sup N ≥ N X | n |≥ N | Cov( Z , Z n ) | < ∞ , (4.52) and X n ∈ Z Cov( Z , Z n ) = 0 . (4.53) Then Z − E ( Z ) ∈ span n Z n − E ( Z n ) : n ∈ Z \ { } o L . Remark 4.2. The absolute summability P n ∈ Z | Cov( Z , Z n ) | < ∞ is clear from (4.52) . Consider now a Pfaffian point process P K induced by a matrix kernel K such that (1.3)is satisfied. We have ρ (1) P K ( x ) = Pf (cid:20) K ( x, x ) − K ( x, x ) 0 (cid:21) = K ( x, x ) , (4.54)and also ρ (2) P K ( x, y ) = Pf K ( x, x ) − K ( x, y ) K ( x, y ) − K ( x, x ) 0 − K ( x, y ) K ( x, y ) − K ( y, x ) K ( y, x ) 0 K ( y, y ) − K ( y, x ) K ( y, x ) − K ( y, y ) 0 = K ( x, x ) K ( y, y ) − K ( x, y ) K ( x, y ) + K ( x, y ) K ( x, y )= ρ (1) P K ( x ) ρ (1) P K ( y ) − det K ( x, y ) . (4.55)In what follows, we denote the truncated second correlation function ρ (2 ,T ) P K ( x, y ) := ρ (2) P K ( x, y ) − ρ (1) P K ( x ) ρ (1) P K ( y ) = − det K ( x, y ) . (4.56) Lemma 4.3. Fix λ > . Then for ( X ( λ ) n ) n ∈ Z defined by (4.50) we have Cov( X ( λ )0 , X ( λ ) n ) = Z I ( λ )0 ρ (1) P K ( x ) dx + Z I ( λ )0 × I ( λ )0 ρ (2 ,T ) P K ( x, y ) dxdy, n = 0; Z I ( λ )0 × I ( λ ) n ρ (2 ,T ) P K ( x, y ) dxdy, n = 0 . . Proof. Fix λ > 0. For brevity, in this proof, we denote X ( λ ) n , I ( λ ) n by X n , I n respectively.Let X denote a random configuration on R with distribution P K . If n = 0, then E ( X X n ) = E (cid:16) X x ∈ X I I ( x ) X y ∈ X I I n ( y ) (cid:17) = E (cid:16) X x,y ∈ X ,x = y I I ( x ) I I n ( y ) (cid:17) = Z I × I n ρ (2) P K ( x, y ) dxdy = Z I × I n h ρ (2 ,T ) P K ( x, y ) + ρ (1) P K ( x ) ρ (1) P K ( y ) i dxdy = Z I × I n ρ (2 ,T ) P K ( x, y ) dxdy + E ( X ) E ( X n ) . N NUMBER RIGIDITY FOR PFAFFIAN POINT PROCESSES 53 It follows thatCov( X , X n ) = E ( X X n ) − E ( X ) E ( X n ) = Z I × I n ρ (2 ,T ) P K ( x, y ) dxdy. If n = 0, then E ( X ) = E (cid:16) X x,y ∈ X I I ( x ) I I ( y ) (cid:17) = E (cid:16) X x ∈ X I I ( x ) (cid:17) + E (cid:16) X x,y ∈ X ,x = y I I ( x ) I I ( y ) (cid:17) . By similar computation as above, we obtain E ( X ) = Z I ρ (1) P K ( x ) dx + Z I × I ρ (2 ,T ) P K ( x, y ) dxdy + E ( X ) E ( X )and thus Cov( X , X ) = Z I ρ (1) P K ( x ) dx + Z I × I ρ (2 ,T ) P K ( x, y ) dxdy. (cid:3) The orthogonal sine process. Proof of Proposition . It suffices to show that for any λ > 0, the hypothesis of Propo-sition 4.1 is satisfied for the stochastic process ( X ( λ ) n ) n ∈ Z defined in (4.50). For brevity, inthis proof, we will omit the superscript λ in the notation X ( λ ) n , I ( λ ) n .Claim A1: We have sup N ∈ N (cid:16) N X | n |≥ N | Cov( X , X n ) | (cid:17) < ∞ . (4.57)We will use the estimate of ρ (2 ,T )sine , ( x, y ) in Forrester [6, formula (7.135)]: ρ (2 ,T )sine , ( x, 0) = O (cid:16) x (cid:17) as | x | → ∞ .In other words, there exists C > 0, such that | ρ (2 ,T )sine , ( x, | ≤ Cx . By Lemma 4.3 and note that ρ (2 ,T )sine , ( x, y ) = ρ (2 ,T )sine , ( x − y, X | n |≥ N | Cov( X , X n ) | ≤ X | n |≥ N Z I × I n | ρ (2 ,T )sine , ( x, y ) | dxdy = Z I dx Z | y |≥ Nλ − λ/ | ρ (2 ,T )sine , ( x − y, | dy ≤ Z I sup x ∈ I (cid:16) Z | y |≥ Nλ − λ/ | ρ (2 ,T )sine , ( x − y, | dy (cid:17) dx ≤ Z I (cid:16) Z | z |≥ Nλ − λ | ρ (2 ,T )sine , ( z, | dz (cid:17) dx ≤ λ Z | z |≥ Nλ − λ Cz dz = 2 CN − . We thus obtain the inequality (4.57). Claim B1: We have X n ∈ Z Cov( X , X n ) = 0 . (4.58)Indeed, by (4.57), we already know that the above series converges absolutely. And byLemma 4.3, we have X n ∈ Z Cov( X , X n ) = Z I ρ (1)sine , ( x ) dx + Z I × R ρ (2 ,T )sine , ( x, y ) dxdy. Then by using the equality (3.47), we obtain the desired equality (4.58).The proof of Proposition 1.5 is complete. (cid:3) The symplectic sine process. In what follows, we will use the following estimateof ρ (2 ,T )sine , ( x, y ) in Forrester [6, formula (7.94)]: ρ (2 ,T )sine , ( x, 0) = cos( πx )8 x + O (cid:16) x (cid:17) as | x | → ∞ .That is, there exists C > 0, such that (cid:12)(cid:12)(cid:12) ρ (2 ,T )sine , ( x, − cos( πx )8 x (cid:12)(cid:12)(cid:12) ≤ Cx . (4.59)Note that Z R | ρ (2 ,T )sine , ( x, y ) | dy = Z R | ρ (2 ,T )sine , ( t, | dt = ∞ . Proof of Proposition . It suffices to show that for any positive even integer 2 k ∈ N ,the hypothesis of Proposition 4.1 is satisfied for the stochastic process ( X (2 k ) n ) n ∈ Z definedin (4.50). Let us now fix the positive even integer 2 k ∈ N . For brevity, in this proof, wewill omit the superscript k in the notation X (2 k ) n , I (2 k ) n .Claim A2: We have sup N ∈ N (cid:16) N X | n |≥ N | Cov( X , X n ) | (cid:17) < ∞ . (4.60)By (4.59) and Lemma 4.3 and note that ρ (2 ,T )sine , ( x, y ) = ρ (2 ,T )sine , ( x − y, X | n |≥ N | Cov( X , X n ) | = X | n |≥ N (cid:12)(cid:12)(cid:12) Z I × I n ρ (2 ,T )sine , ( x, y ) dxdy (cid:12)(cid:12)(cid:12) ≤ X | n |≥ N (cid:12)(cid:12)(cid:12) Z I × I n cos( π ( x − y ))8( x − y ) dxdy | {z } denoted σ n (cid:12)(cid:12)(cid:12) + C X | n |≥ N Z I × I n x − y ) dxdy. For the term σ n , we have σ n = Z I dx Z I n cos( π ( x − y ))8( x − y ) dy = Z I dx Z I n d sin( π ( y − x ))8 π ( x − y ) == Z I dx (cid:20) sin( π ( y − x ))8 π ( x − y ) (cid:12)(cid:12)(cid:12) nk + ky =2 nk − k − Z I n sin( π ( y − x ))8 π ( x − y ) dy (cid:21) . N NUMBER RIGIDITY FOR PFAFFIAN POINT PROCESSES 55 Note that since the function y sin( π ( y − x )) is 2 π -periodic, we have (cid:12)(cid:12)(cid:12) sin( π ( y − x ))8 π ( x − y ) (cid:12)(cid:12)(cid:12) nk + ky =2 nk − k (cid:12)(cid:12)(cid:12) = | sin( π (2 nk + k − x ) | π Z I n y − x ) dy ≤ Z I n π ( y − x ) dy. Therefore, we have | σ n | ≤ Z I × I n π ( x − y ) dxdy + Z I × I n | sin( π ( y − x )) | π ( x − y ) dxdy ≤ Z I × I n π ( x − y ) dxdy. Consequently, X | n |≥ N | Cov( X , X n ) | ≤ ( C + 14 π ) X | n |≥ N Z I × I n x − y ) dxdy = ( C + 14 π ) Z I dx Z | y |≥ Nk − k/ x − y ) dy ≤ ( C + 14 π ) Z I (cid:16) Z | z |≥ Nk − k z dz (cid:17) dx = C + π N − . Claim B2: We have X n ∈ Z Cov( X , X n ) = 0 . (4.61)Indeed, by (4.60), we already know that the above series converges absolutely. And byLemma 4.3, we have X n ∈ Z Cov( X , X n ) = Z I ρ (1)sine , ( x ) dx + X n ∈ Z Z I × I n ρ (2 ,T )sine , ( x, y ) dxdy = Z I ρ (1)sine , ( x ) dx + Z I dx Z R ρ (2 ,T )sine , ( x, y ) dy. Then by using the equality (3.48), we obtain the desired equality (4.61).The proof of Proposition 1.6 is complete. (cid:3) Comments. For the symplectic sine process, let us now consider the stationarystochastic process ( X (1) n ) n ∈ Z . We show that X n ∈ Z | Cov( X (1)0 , X (1) n ) | = ∞ . (4.62)If follows that the assumptions of Proposition 4.1 doesn’t hold in this case, see Remark 4.2.Indeed, by Lemma 4.3 and the inequality (4.59), for n = 0, we have(4.63) | Cov( X (1)0 , X (1) n ) | = (cid:12)(cid:12)(cid:12) Z I (1)0 × I (1) n ρ (2 ,T )sine , ( x, y ) dxdy (cid:12)(cid:12)(cid:12) ≥≥ (cid:12)(cid:12)(cid:12) Z I (1)0 × I (1) n cos( π ( x − y ))8( x − y ) dxdy | {z } denoted σ (1) n (cid:12)(cid:12)(cid:12) − C Z I (1)0 × I (1) n x − y ) dxdy. Note that by integration by parts, we obtain σ (1) n = Z I (1)0 dx (cid:20) sin( π ( y − x ))8 π ( x − y ) (cid:12)(cid:12)(cid:12) n +1 / y = n − / − Z I (1) n sin( π ( y − x ))8 π ( x − y ) dy (cid:21) . Using the identitysin h π ( n + 1 / − x ) i = − sin h π ( n − / − x ) i = ( − n cos( πx ) , we obtain thatsin( π ( y − x ))8 π ( x − y ) (cid:12)(cid:12)(cid:12) n +1 / y = n − / = ( − n cos( πx )8 π h x − n − / x − n + 1 / i . For any positive integer n > 0, we have (cid:12)(cid:12)(cid:12) Z I (1)0 sin( π ( y − x ))8 π ( x − y ) (cid:12)(cid:12)(cid:12) n +1 / y = n − / dx (cid:12)(cid:12)(cid:12) = Z / − / cos( πx )8 π h n + 1 / − x + 1 n − / − x i dx ≥≥ Z / cos( πx )8 π · n − / − x dx ≥ Z / π · n − / − x dx ≥ π ( n − / . Therefore, | σ (1) n | ≥ (cid:12)(cid:12)(cid:12) Z I (1)0 sin( π ( y − x ))8 π ( x − y ) (cid:12)(cid:12)(cid:12) n +1 / y = n − / dx (cid:12)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:12) Z I (1)0 dx Z I (1) n sin( π ( y − x ))8 π ( x − y ) dy (cid:12)(cid:12)(cid:12) ≥≥ π ( n − / − Z I (1)0 dx Z I (1) n π ( x − y ) dy. Combining the above inequality with (4.63), we get | Cov( X (1)0 , X (1) n ) | ≥ π ( n − / − ( C + 18 π ) Z I (1)0 dx Z I (1) n x − y ) dy By the argument in the proof of Proposition 1.6, we know that X n ∈ Z Z I (1)0 dx Z I (1) n x − y ) dy < ∞ . Therefore, we get the claimed divergence X n ∈ Z | Cov( X (1)0 , X (1) n ) | ≥ X n ≥ π ( n − / − ( C + 18 π ) X n ≥ Z I (1)0 dx Z I (1) n x − y ) dy = ∞ . Acknowledgements. We are deeply grateful to Alexei Klimenko for useful discussions.The research of A. Bufetov on this project has received funding from the European Re-search Council (ERC) under the European Union’s Horizon 2020 research and innovationprogramme under grant agreement No 647133 (ICHAOS). Yanqi Qiu’s research is sup-ported by the National Natural Science Foundation of China, grants NSFC Y7116335K1,NSFC 11801547 and NSFC 11688101. The research of P. Nikitin is supported by theRFBR grant 17-01-00433. N NUMBER RIGIDITY FOR PFAFFIAN POINT PROCESSES 57 References [1] M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, National Bureau of Standards,Department of Commerce of the United States of America, Tenth Printing, 1972.[2] Alexander I. 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Rigidity and tolerance in point processes: Gaussian zeros andGinibre eigenvalues. Duke Math. J. 166 (2017), no.10, 1789–1858.[9] C. A. Tracy and H. Widom. Level spacing distributions and the Bessel kernel. Comm. Math. Phys. Alexander I. BUFETOV: Aix-Marseille Universit´e, Centrale Marseille, CNRS, Insti-tut de Math´ematiques de Marseille, UMR7373, 39 Rue F. Joliot Curie 13453, Marseille,France; Steklov Mathematical Institute of RAS, Moscow, Russia E-mail address : [email protected], [email protected] Pavel P. Nikitin: St. Petersburg Department of V.A.Steklov Institute of Mathemat-ics of the Russian Academy of Sciences, 27 Fontanka, 191023, St. Petersburg, Russia;St. Petersburg State University, St. Petersburg, Russia E-mail address : [email protected] Yanqi QIU: Institute of Mathematics, Academy of Mathematics and Systems Science,Chinese Academy of Sciences, Beijing 100190, China E-mail address ::