Correlations Between Zeros and Supersymmetry

Transcription

1 Commun Math Phys 224, (200) Communications in Mathematical Physics Springer-Verlag 200 Correlations Between Zeros and Supersymmetry Pavel Bleher, Bernard Shiffman 2, Steve Zelditch 2 Department of Mathematical Sciences, IUPUI, Indianapolis, IN 46202, USA bleher@mathiupuiedu 2 Department of Mathematics, Johns Hopkins University, Baltimore, MD 228, USA shiffman@mathjhuedu; zelditch@mathjhuedu Received: 3 November 2000 / Accepted: 23 February 200 To Joel Lebowitz on his 70th birthday Abstract: In our previous work [BSZ2], we proved that the correlation functions for simultaneous zeros of random generalized polynomials have universal scaling limits and we gave explicit formulas for pair correlations in codimensions and 2 The purpose of this paper is to compute these universal limits in all dimensions and codimensions First, we use a supersymmetry method to express the n-point correlations as Berezin integrals Then we use the Wick method to give a closed formula for the limit pair correlation function for the point case in all dimensions Introduction This paper is a continuation of our articles [BSZ, BSZ2, BSZ3] on the correlations between zeros of random holomorphic polynomials in m complex variables and their generalization to holomorphic sections of positive line bundles L M over general Kähler manifolds of dimension m and their symplectic counterparts These correlations are defined by the probability density Knk N (z,,z n ) of finding joint zeros of k independent sections at the points z,,z n M (see Sect 2) To obtain universal uantities, we rescale the correlation functions in normal coordinates by a factor of N Our main result from [BSZ2, BSZ3] is that the (normalized) correlation functions have a universal scaling limit, ) K nkm (z,,z n ) lim N KN k (z 0) n Knk (z N 0 + z,,z 0 + zn, () N N which is independent of the manifold M, the line bundle L and the point z 0 ; K nkm depends only on the dimension m of the manifold and the codimension k of the zero set The Research partially supported by NSF grants #DMS (first author), #DMS (second author), #DMS (third author)

2 256 P Bleher, B Shiffman, S Zelditch problem then arises of calculating these universal functions explicitly and analyzing their small distance and large distance behavior In [BSZ, BSZ2], we gave explicit formulas for the pair correlation functions K 2km (z,z 2 ) in codimensions k, 2, respectively The purpose of this paper is to complete these results by giving explicit formulas for K nkm in all dimensions and codimensions Our first formula expresses the correlation as a supersymmetric (Berezin) integral involving the matrices (z), A (z) used in our prior formulas, as well as a matrix of fermionic variables described below Theorem The limit n-point correlation functions are given by K nkm (z,,z n [(m k)!] n ) (m!) n [det A (z)] k det[i + (z) ] dη Here, is the nkm nkm matrix ( ) ) pj p j (δ p p δ η p j η p j ( p, p n, j,j k,, m), (2) where the η j, η j are anti-commuting (fermionic) variables, and dη j,p dηp j d ηp j The integral in Theorem is a Berezin integral, which is evaluated by simply taking the coefficient of the top degree form of the integrand det[i + (z) ] (see Sect 3) Hence the formula in Theorem is a purely algebraic expression in the coefficients of (z) and A (z), which are given in terms of the Szegö kernel of the Heisenberg group and its derivatives (see Sect 2) We remark that supersymmetric methods have also been applied to limit correlations in random matrix theory by Zirnbauer [Zi] In the case n 2, K 2km (z,z 2 ), depends only on the distance between the points z,z 2, since it is universal and hence invariant under rigid motions Hence it may be written as: K 2km (z,z 2 ) κ km ( z z 2 ) (3) We refer to [BSZ2] for details In [BSZ] we gave an explicit formula for κ m (using the Poincaré Lelong formula ), and in [BSZ2] we evaluated κ 2m (The pair correlation function κ (r) was first determined by Hannay [Ha] in the case of zeros of SU(2) polynomials in one complex variable) In Sect 3, we use Theorem to give the following new Berezin integral formula for κ km : Corollary 2 The pair correlation functions are given by (m k)!2 κ km (r) dη, m! 2 ( e r2 ) k m where det [I + P( + 2 ) + T 2 ], P r2 e r2 e, T r2 e r2 r4 e r2 e, r2 ] det [I ( e r2 ) 2

3 Correlations Between Zeros and Supersymmetry 257 Here,, 2 are the k k matrices ( ) p η p j η p j j,j k, p, 2 We then expand the formula as a (finite) series (32), which we use to compute explicit formulas for κ km The most vivid case is when k m, where the simultaneous zeros of k-tuples of sections almost surely form a set of discrete points Our second result is an explicit formula for the point pair correlation functions κ mm in all dimensions: Theorem 3 The point pair correlation functions are given by κ mm (r) [ ] m( v m+ )( v) + r 2 (2m + 2)(v m+ v) + r 4 v m+ + v m + ({m + }v + )(v m v)/(v ) m( v) m+2, for m For small values of r, we have v e r2, (4) κ mm (r) m + r 4 2m + O(r 8 2m ), as r 0 (5) 4 We prove Theorem 3 in Sect 4 without making use of supersymmetry Our proof uses instead the Wick formula expansion of the Gaussian integral representation of the correlation It is interesting to observe the dimensional dependence of the short distance behavior of κ mm (r) When m,κ mm (r) 0asr 0 and one has zero repulsion When m 2, κ mm (r) 3/4 asr 0 and one has a kind of neutrality With m 3, κ mm (r) as r 0 and there is some kind of attraction between zeros More r Fig The limit pair correlation function κ 33

4 258 P Bleher, B Shiffman, S Zelditch precisely, in dimensions greater than 2, one is more likely to find a zero at a small distance r from another zero than at a small distance r from a given point; ie, zeros tend to clump together in high dimensions Indeed, in all dimensions, the probability of finding another zero in a ball of small scaled radius r about another zero is r 4 We give in Fig a graph of κ 33 ; graphs of κ and κ 22 can be found in [BSZ2] Remark Theorem 3 says that the expected number of zeros in the punctured ball of scaled radius r about a given zero is r 0 κ mm(t)t 2m dt r 4 Also, one can show that for balls of small scaled radii r, the expected number of zeros approximates the probability of finding a zero 2 Background We begin by recalling the scaling limit zero correlation formula of [BSZ2] Consider a random polynomial s of degree N in m variables More generally, s can be a random section of the N th power L N of a positive line bundle L on an m-dimensional compact complex manifold M (or a symplectic 2m-manifold; see [SZ3, BSZ3]) We give M the Kähler metric induced by the curvature form ω of the line bundle L The probability measure on the space of sections is the complex Gaussian measure induced by the Hermitian inner product s, s 2 h N (s, s 2 )dv M, M where h N is the metric on L N and dv M is the volume measure induced by ω (For further discussion of the topics of this section, see [BSZ2]) In particular, if L is the hyperplane section bundle over CP m, then random sections of L N are polynomials of degree N in m variables of the form P(z,,z m ) C J (N J )!j! j m! zj zj m m (J (j,,j m )), J N where the C J are iid Gaussian random variables with mean 0; they are called SU(m+ )-polynomials We consider k-tuples s (s,,s k ) of iid random polynomials (or sections) s j ( k m) The zero correlation density Knk N (z,,z n ) is defined as the expected joint volume density of zeros of sections of L N at the points z,,z n In the case k m, where the zero sets are discrete points, Knk N (z,,z n ) can be interpreted as the probability density of finding simultaneous zeros at these points For instance, the zero density function Kk N (z) c kn k as N, where c k is independent of the point z (see [SZ]) In [BSZ2, BSZ3], we gave generalized forms of the Kac-Rice formula [Kac, Ri], which we used to express Knk N (z,,z n ) in terms of the joint probability distribution (JPD) of the random variables s(z ),,s(z n ), s(z ),, s(z n ) We then showed that the scaling limit correlation function K nkm given by () can be expressed in terms of the scaling limit of the JPD The central result of [BSZ2] is that the limit JPD is universal and can be expressed in terms of the Szegö kernel H for the Heisenberg group: H (z, θ; w, ϕ) π m ei(θ ϕ+iz w) 2 z w 2 π m ei(θ ϕ)+z w 2 ( z 2 + w 2) (6)

5 Correlations Between Zeros and Supersymmetry 259 To be precise, the limit JPD is a complex Gaussian measure with covariance matrix given by: ( ) (z) m! A (z) B (z) π m B (z) C, (7) (z) where π m A (z) p p H (zp, 0; z p, 0), π m B (z) p p H z p (zp, 0; z p p, 0) (z z p ) H (zp, 0; z p, 0), 2 π m C (z) p p z p H z p (zp, 0; z p, 0) ) (δ + ( z p z p )(z p z p ) H (zp, 0; z p, 0) (8) (Here A, B, C are n n, n mn, mn mn matrices, respectively) In the seuel, we shall use the matrix (z) : C (z) B (z) A (z) B (z) (9) We note that A (z) and (z) are positive definite whenever z,,z n are distinct points In [BSZ2], we gave the following key formula for the limit correlation functions: K nkm (z,,z n ) [(m k)!] n (m!) n [det A (z)] k C kmn n det j,j p k m ξj ξ p p j dγ (z) (ξ), where γ (z) is the Gaussian measure with (nkm nkm) covariance matrix ( ) ( ) (z) : (z) pj p j δ j j (z) p p () (z) pj p j ) For the pair correlation case (n 2), E (0) be- (Ie, ξj ξ p p comes: κ km (r) [ j γ (z) m! (m k)! (0) ] 2 det A(r) k m det ξj ξ j C 2km j,j k det j,j k m ξj ξ 2 j 2 dγ (r) (ξ), where A(r) A (z,z 2 ), (r) (z,z 2 ), z z 2 r The computations in this paper are all based on formula (0) (2)

6 260 P Bleher, B Shiffman, S Zelditch 3 Supersymmetric Approach to n-point Correlations We now prove Theorem using our formula (0) for the limit n-point correlation function, which we restate as follows: where K nkm (z,,z n ) G nkm (z) n C kmn j,j p k [(m k)!] n (m!) n [det A (z)] k G nkm, (3) det m ξj ξ p p j dγ (z) (ξ) (4) Our approach is to represent the determinant in (4) as a Berezin integral and then to exchange the order of integration We introduce anti-commuting (or fermionic ) variables η p j, ηp j ( j k, p n), which can be regarded as generators of the Grassmann algebra C 2l 2l t t0 C 2l, l nk The Berezin integral on C 2l is the linear functional I : C 2l C given by ( ) I t C 2l 0 for t<2l, I j,p ηp j ηp j Elements f C 2l are considered as functions of anti-commuting variables, and we write I(f ) fdη f j,p dηp j d ηp j ( ) (See for example [Ef, Chapter 2], [ID, Sect 2]) If H H pj p j is an l l Hermitian matrix, we have the supersymmetric formula for the determinant: det H e Hη, η dη, Hη, η η p j H pj p j η p j (5) j,p,j,p We now use (5) to compute G nkm : let ξ p ξ p m ξ p ξ p k ξ p km (where {ξ p j } are ordinary bosonic variables) We also write ξ ξ ξ n : C mn C kn Then ξ n m ξ 0 det ξj ξ p p j,j p k j det(ξξ ) det (6) 0 ξ n ξ n

7 Correlations Between Zeros and Supersymmetry 26 Applying (5) with H ξξ,wehave G nkm π nkm det(ξξ )e ξ, ξ dξ det C nkm π nkm e ξ, ξ ξξ η, η dηdξ, (7) det C nkm ξξ η, η ξj ξ p p j ηp j η p j ξ, ξ, (8) p,,j,j where is given by (2) Note that the entries of commute, since they are of degree 2 Furthermore, adopting the supersymmetric definition of the conjugate [Ef], (η p j ) η p j, ( ηp j ) η p j, we see that the matrix is superhermitian; ie,, where t Thus by (7) (8), we have G nkm π nkm e ( + )ξ, ξ dηdξ (9) det C nkm We recall that π nkm e Pξ, ξ dξ det P, (20) C nkm for a positive definite, Hermitian (nkm nkm) matrix P Furthermore, (20) holds when P is the superhermitian matrix + ; we give a short proof of this fact below Reversing the order of integration in (9) and applying (20) with P +, we have G nkm det det( + ) dη (2) det(i + ) dη We now verify by formal substitution that (20) holds when P + : Suppose that, t are even fermionic functions; ie, j even C 2l Let S be the homomorphism from the algebra C{z,,z t, z,, z t } of convergent power series onto even C 2l given by substitution: Let S [f(z,,z t, z,, z t )]f(,, t,,, t ) f(z, Z) e ( +Z+Z )ξ, ξ dξ e (Z+Z )ξ, ξ e ξ, ξ dξ det( + Z + Z ), for Z (z αβ ) gl(nkm, C), where the last euality is by (20) We easily see that f is a convergent power series in {z αβ, z αβ } and that the integrand e (Z+Z )ξ, ξ can be written as an absolutely convergent power series in {z αβ, z αβ } with values in L (e ξ, ξ dξ) We now let αβ 2 αβ 2 βα ; the conclusion follows by applying S to (22) (22)

8 262 P Bleher, B Shiffman, S Zelditch 3 Pair correlation In this section, we prove Corollary 2 To illustrate the computation, we consider first the case k m of zero correlations in dimension one: We have ( ) ( ) 2 η η 0 2 2, 2 0 η 2 η 2, ( ) + I + η η 2 η2 η 2 2 η η η2 η 2 We easily compute det(i + ) + η η η2 η 2 + (det )η η η 2 η 2, det(i + ) η η 2 2 η2 η 2 + (2 2 2 det )η η η 2 η 2, det(i + ) d η dη d η 2 dη det Hence by Theorem, we have κ (r) (23) det A To obtain the explicit formula for κ (r) [Ha,BSZ], we set z (r, 0,,0), z 2 0 and substitute in (23) the resulting values of p p and det A (see (24) (26) below) To obtain formulas for κ km (r) for higher k, m, we again set z (r, 0,,0), z 2 0 Using (6), (8) and (9), we see that pj p j 0 for (j, ) (j, ) and where j j 2j j j j 2j j j 2j 2j 2j j 2j 2j 2j ( ) P Q, Q P ( ) R S for 2, S R ( P e r2 r 2 e r2 e 2 r2 e r2 r 2), Q, e r2 e r2 R, S e 2 r2 (24) (25) We also have and thus ( e 2 r2 A(r) e 2 r2 ), det A e r2 (26)

9 Correlations Between Zeros and Supersymmetry 263 We can write the 2km 2km matrices, in block form: ,, ( ) ( ) ( PIk QI k, RIk SI k, 0 QI k PI k SI k RI k 0 2 where I k is the k k identity matrix and η p ηp η p k ηp p, p, 2 ), η p ηp k η p k ηp k Hence det(i + ) m, (27) where det(i + ), det(i + ) (28) We note that and thus ( ) I + I + P Q 2, Q I + P 2 ] det(i + P ) det [I + P 2 Q 2 (I + P ) 2 ] det [I + P( + 2 ) + (P 2 Q 2 ) 2 (29) Similarly, ] det [I + R( + 2 ) + (R 2 S 2 ) 2 (30) We recall that by Theorem, [(m k)!] 2 κ km (r) (m!) 2 ( e r2 ) k dη det(i + ) (3) Combining (25) (3), we obtain Corollary 2 To obtain explicit formulas for the pair correlation in a fixed codimension k (for all dimensions m), we write ( ) and our formula becomes: [(m k)!] 2 2k ( ) m + t 2 κ km (r) (m!) 2 ( e r2 ) k t ( ) t dη (32) t0

10 264 P Bleher, B Shiffman, S Zelditch Using Maple TM, we evaluate (32) to obtain the following pair correlation formulas: [ κ m (r) P (m ) PR+ Q 2 + (m ) 2 R 2 + (m ) S 2], κ 2m (r) κ 3m (r) m 2 det A [ m 2 (m ) det A 2 4(m )P 2 R 2 + 2P 2 S (m )(m 2) PR 3 + 4(m 2)P RS (m ) Q 2 R 2 + 4Q 2 S 2 + (m )(m 2) 2 R (m 2) 2 R 2 S (m 2) S 4], [ m 2 (m )(m 2) det A 3 9 (m )(m 2) P 2 R (m 2) P 2 R 2 S 2 + 6P 2 S (m 3)(m )(m 2) PR (m 3)(m 2) PR 3 S (m 3) PRS (m )(m 2) Q 2 R (m 2) Q 2 R 2 S 2 + 8Q 2 S 4 + (m )(m 2)(m 3) 2 R (m 2)(m 3) 2 R 4 S (m 3) 2 R 2 S (m 3) S 6] Recalling (25) (26), we then obtain power series expansions of the pair correlation function in codimensions, 2, 3: κ m (r) m m r 2 + m (m + 2)(m + ) + 2 m 2 m 2 r 2 (m + 6)(m + 5) m 2 r 0 (m + 8)(m + 7) m 2 r 4, (m + 4)(m + 3) 720 m 2 r 6 κ 2m (r) m 2 m r 4 + m 2 m r 2 + 5m2 7m + 2 (m 2)(m + 2)(m + ) + 2(m )m 2(m )m 2 r 2 (m + 3)(m + 2) (m 2)(m + 4)(m + 3) + 240(m )m r4 720(m )m 2 r 6 +, κ 3m (r) m 3 m r 6 + 3(m 3) 2 m r 4 + m2 4 m + 6 (m 2) m + (m + 2)(m + ) ( 9 m 2 79 m + 20 ) 240 m 2 (m )(m 2) (m 3)(m + 3)(m + 2) + 60 m (m )(m 2) r4 r 2 + (m 3) ( 3 m 2 m + 8 ) 8 m (m )(m 2) (The power series for κ m and κ 2m were given in [BSZ] and [BSZ2] respectively) r 2 4 The Point Case We now prove Theorem 3: For the case k m, where the zero set is discrete, (2) becomes: G m (r) κ mm (r) (m!) 2 det A(r) m, (33)

11 Correlations Between Zeros and Supersymmetry 265 where G m (r) G 2mm (r) C 2m2 det j, m ( ) ξj det j, m ( ) ξj 2 2 dγ (r) (ξ) (34) We let (r) dγ (r) denote the expected value with respect to the Gaussian probability measure γ (r) Thus G m (r) det(ξ ) det(ξ 2 ) det( ξ ) det( ξ 2 ) (r) )( )( )( ) ξα ξ µ ξ ν 2, (35) α,β,µ,ν( ) α+β+µ+ν ( where the sum is over all 4-tuples α, β, µ, ν S m ( permutations of {,,m}), and ( ) σ stands for the sign of the permutation σ We shall compute the terms of (35) using the Wick formula rather than directly from the Berezin integral formula The computation simplifies considerably since the matrix (r) is sparse In fact, we shall see that the sign ( ) α+β+µ+ν is positive whenever the corresponding moment is nonzero Let us now evaluate the moments of order 4m in (35): ( )( )( )( ) M αβµν : ξα ξ µ ξ ν 2 (36) ξ 2 β Recall that the Wick formula expresses M αβµν as a sum of products of second moments with respect to the Gaussian measure γ (r) Since this Gaussian is complex, these second moments come from pairings of ξ s with ξ s We write ξ 2 β pj p j δ j j p p (r) (r) ξ p j ξ p j (r) (37) Hence the term M αβµν euals the permanent of the submatrix of ( pj ) p j formed from the rows corresponding to the variables ξα,,ξ α m m,ξ2 β,,ξ2 β m m and columns corresponding to ξ µ(),, ξ µ(m)m, ξ ν() 2,, ξ ν(m)m 2 : α µ α µ m m α 2ν α 2ν m m α mm µ α mm µ m m α mm 2ν α mm 2ν m m 2β µ 2β µ m m 2β 2ν 2β (38) 2ν m m 2β mm µ 2β mm µ m m 2β mm 2ν 2β mm 2ν m m (Recall that permanent(v ij ) σ i V iσ i, where the sum is over all permutations σ ) To compute κ mm (r), we can set z (r, 0,,0), z 2 0, as before Recalling (24) and the fact that pj p j 0 for (j, ) (j, ), we observe that (38) is made up of 4 diagonal matrices For example, if m 3, then (38) becomes

12 266 P Bleher, B Shiffman, S Zelditch δ α µ P 0 0 δ α ν Q δ α 2 µ 2 R 0 0 δ α 2 ν 2 S δ α 3 µ 3 R 0 0 δ α 3 ν 3 S δ β µ Q 0 0 δ β ν P δ β 2 µ 2 S 0 0 δ β 2 ν 2 R δ β 3 µ 3 S 0 0 δ β 3 ν 3 R We conclude that M αβµν is a product of 2 2 permanents: M αβµν ( δ α µ δ β ν P 2 + δ α ν δ β In particular, M αβµν vanishes unless µ Q 2) m 2 ( δ α µ δ β ν R 2 + δ α ν δ β µ S 2) (39) {µ,ν }{α,β } for m (40) We now claim that (40) implies that ( ) α+β+µ+ν : First of all, by multiplying the four permutations by α on the left, we can assume without loss of generality that α i i for all i Now write β as a product of disjoint cycles Then one sees that µ is a product of some of these cycles and ν is a product of the other cycles, and the positivity of the product of signs easily follows Hence E (35) becomes G m (r) M αβµν (4) α,β,µ,ν We now use (35) to evaluate G m for arbitrary dimension m Lemma 4 G m (m )!m! [ P 2 f m (R 2,S 2 ) + Q 2 f m (S 2,R 2 ) ], where f m (x, y) y m + 2xy m 2 + +(m )x m 2 y + mx m mxm+ + y m+ (m + ) x m y (x y) 2 Proof We use induction on m The identity holds trivially for m, since f and G P 2 + Q 2 Let m 2 and suppose the identity has been verified for,,m Since M αβµν is unchanged if we multiply all the permutations on the left by α,we have G m m!g 0 m, where G0 m β,µ,ν M eβµν (e identity) For i m, let C i S m denote the collection of i-cycles of the form (a 2 a i ) For each σ C i, let σ denote those permutations τ S m that fix the elements,a,,a i We claim that M eβµν (m i)!(p 2 R 2i 2 + Q 2 S 2i 2 )g m i (R 2,S 2 ), (42) β σ µ,ν where g l (x, y) x l + x l y + +xy l + y l

13 Correlations Between Zeros and Supersymmetry 267 To verify (42), we can assume without loss of generality that σ ( i) Recall that we need only consider permutations µ that are products of some of the cycles in β (and ν is determined by the pair (β, µ), since ν is the product of the other cycles of β when M eβµν 0) For the P 2 -terms of the sum, ν contains the cycle ( i)so that µ, ν σ β for,,i(fortheq 2 -terms, µ contains ( i)) Hence by (39), we have M eβµν β σ µ,ν P 2 R 2i 2 m β σ µ,ν i+ ( δ µ δ β ν R 2 + δ ν δ β µ S 2) + terms with Q 2, (43) where is over those µ, ν with µ, ν β, for i To compute the double sum in the right side of (43), we notice by (39) and (4) that it euals (m i)! G m i with P,Q replaced by R,S respectively Hence by our inductive assumption, we have m β σ µ,ν i+ ( δ µ δ β ν R 2 + δ ν δ β µ S 2) (m i )![R 2 f m i (R 2,S 2 ) + S 2 f m i (S 2,R 2 )] (m i)!g m i (R 2,S 2 ) The computation of the Q 2 terms is similar, and hence (42) holds We now have by (42), m G m m! M eβµν i σ C i β σ µ,ν m! m (#C i )(m i)!(p 2 R 2i 2 + Q 2 S 2i 2 )g m i (R 2,S 2 ) i Noting that (#C i )(m i)!(m )! and summing over i, we obtain the desired formula We now complete the proof of Theorem 3 By (33) and Lemma 4, we have κ mm (r) P 2 f m (R 2,S 2 ) + Q 2 f m (S 2,R 2 ) m(det A(r)) m (44) Substituting (25) (26) into (44), we obtain (4) Finally, to verify (5), we note by (25) that P 2 r2 +,Q 2 r2 +,R, S +, and hence κ mm 2f m(, )(r 4 /4) + mr 2m f m(, ) + 2m r4 2m + m + r 4 2m + 4 The following proposition yields the remainder estimate of (5)

14 268 P Bleher, B Shiffman, S Zelditch Proposition 42 If m is odd, resp even, then κ mm (r) is an odd, resp even, function of r 2 Proof Let P, Q be the functions given by P(r) P (u), Q(r) Q(u), where u r 2 From (44), we have κ mm P (u) 2 f m (,e u ) + Q(u) 2 f m (e u, ) m( e u ) m We observe from (25) that P( u) e u/2 Q(u) and thus κ mm ( u) eu Q(u) 2 f m (,e u ) + e u P (u) 2 f m (e u, ) m( e u ) m ( ) m κ mm (u), since f m is homogeneous of order m The expansions of (4) are easily obtained using Maple TM : κ (r) 2 r2 36 r r r r r22, κ 22 (r) r4 288 r r r r20, κ 33 (r) r r2 260 r r r r8, κ 44 (r) 5 4 r r r r r6, κ 55 (r) 3 2 r r r r r r4, κ 66 (r) 7 4 r r r r r2 References [BSZ] Bleher, P, Shiffman, B and Zelditch, S: Poincaré Lelong approach to universality and scaling of correlations between zeros Commun Math Phys 208, (2000) [BSZ2] Bleher, P, Shiffman, B and Zelditch, S: Universality and scaling of correlations between zeros on complex manifolds Invent Math 42, (2000) [BSZ3] Bleher, P, Shiffman, B and Zelditch, S: Universality and scaling of zeros on symplectic manifolds Random Matrices and Their Applications, P Bleher and A Its (eds) MSRI Publications 40 Cambridge: Cambridge Univ Press, 200, pp 3 69 [Ef] Efetov, K: Supersymmetry in disorder and chaos Cambridge: Cambridge Univ Press, 996 [Ha] Hannay, JH: Chaotic analytic zero points: Exact statistics for those of a random spin state J Phys A: Math Gen 29, 0 05 (996) [ID] Itzykson, C and Drouffe, JM: Statistical field theory, Vol Cambridge: Cambridge Univ Press, 989

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