Number variance of random zeros on complex manifolds, II: smooth statistics
aa r X i v : . [ m a t h . C V ] N ov NUMBER VARIANCE OF RANDOM ZEROS ON COMPLEXMANIFOLDS, II: SMOOTH STATISTICS
BERNARD SHIFFMAN AND STEVE ZELDITCH
To Joseph J. Kohn on the occasion of his 75th birthday
Abstract.
We consider the zero sets Z N of systems of m random polynomials of degree N in m complex variables, and we give asymptotic formulas for the random variables given bysumming a smooth test function over Z N . Our asymptotic formulas show that the variancesfor these smooth statistics have the growth N m − . We also prove analogues for the integralsof smooth test forms over the subvarieties defined by k < m random polynomials. Suchlinear statistics of random zero sets are smooth analogues of the random variables given bycounting the number of zeros in an open set, which we proved elsewhere to have variances oforder N m − / . We use the variance asymptotics and off-diagonal estimates of Szeg˝o kernelsto extend an asymptotic normality result of Sodin-Tsirelson to the case of smooth linearstatistics for zero sets of codimension one in any dimension m . Introduction
This article is concerned with zero sets of systems of Gaussian random polynomials (ormore generally, of sections of a positive holomorphic line bundle over a compact K¨ahlermanifold M m ) as the degree N → ∞ . One of the most fundamental statistical quantities isthe number N UN ( p N , . . . , p Nm ) of zeros in a bounded open set U ⊂ C m of a system { p N , . . . , p Nm } of m independent Gaussian random polynomials. The expected value of this random variablewas shown in [SZ1] to be the integral of the K¨ahler volume form over U (times a universalconstant). In a recent article [SZ4], we gave an asymptotic formula for the variance of thisrandom variable. We also give analogous results for the volume of the simultaneous zeroset of k < m polynomials or sections. In this article we apply the methods of [SZ4] to theanalogous ‘smooth linear statistics’, i.e. the sum (or integral) of a smooth test functionover the zeros of a system of random polynomials. Such smooth linear statistics arise assmooth approximations for discontinuous random variables such as N UN and also arise ina number of other problems (see the discussion in § N than in the non-smooth case. Further, we prove that in thecodimension one case, the smooth linear statistics are asymptotically normal, extending aresult of Sodin-Tsirelson [ST].To state our results precisely, we need some notation and background. We let ( L, h ) → M be a positively curved Hermitian holomorphic line bundle over a compact complex manifold M , and we give H ( M, L N ) the Hermitian Gaussian measure induced by h and the K¨ahler Research of the first author partially supported by NSF grants DMS-0100474 and DMS-0600982; researchof the second author partially supported by NSF grants DMS-0302518 and DMS-0603850. form ω = i Θ h (see Definition 2.1). For m independent random sections s Nj ∈ H ( M, L N ),1 ≤ j ≤ m , the number of simultaneous zeros of the sections in a smooth domain U ⊂ M isgiven by N UN ( s N , . . . , s Nm ) := { z ∈ U : s N ( z ) = · · · = s Nm ( z ) = 0 } . In [SZ1], we proved that the expected value E (cid:0) N UN (cid:1) of the random variable N UN has theasymptotics E (cid:0) N UN (cid:1) = N m π m Z U ω m + O ( N m − ) , (1)and in [SZ4], the variance of the random variable is shown to have the asymptotics,Var (cid:0) N UN (cid:1) = N m − / h ν mm Vol m − ( ∂U ) + O ( N − + ε ) i , (2)where ν mm is a universal positive constant ([SZ4, Theorem 1.1]). Analogous results are provedfor the volumes of zero sets of k ≤ m − s Nj ∈ H ( M, L N ),1 ≤ j ≤ k : in this case,Var (cid:0) Vol m − k [ Z s N ,...,s Nk ∩ U ] (cid:1) = N k − m − / h ν mk Vol m − ( ∂U ) + O ( N − + ε ) i , (3)where ν mk is a universal constant ([SZ4, Theorem 1.4]); in particular, ν m = π m − / ζ ( m + ).More generally, the domain could be piecewise smooth without cusps.In this article, we are interested in the smooth analogue of N UN where we integrate asmooth test function rather than the characteristic function of a smooth domain over thezero set. Given a test function ϕ ∈ D ( M ), we consider the random variable( Z s N ,...,s Nm , ϕ ) = X s N ( z )= ··· = s Nm ( z )=0 ϕ ( z ) . When the system is not full, we define( Z s N ,...,s Nk , ϕ ) = Z s N ( z )= ··· = s Nk ( z )=0 ϕ ( z ) , ϕ ∈ D m − k,m − k ( M ) . The expected value of ( Z s N ,...,s Nm , ϕ ) is given by (see (14)) E ( Z s N ,...,s Nk , ϕ ) ≈ N k π − k Z M ω k ∧ ϕ. (4)The main result of this article is an asymptotic formula for the variance: Theorem
Let ( L, h ) be a positive Hermitian holomorphic line bundle over a compactK¨ahler manifold ( M, ω ) , where ω = i Θ h , and let ≤ k ≤ m . We give H ( M, L N ) theHermitian Gaussian measure induced by h, ω (see Definition 2.1).Let ϕ be a real ( m − k, m − k ) -form on M with C coefficients. Then for independentrandom sections s N , . . . , s Nk ∈ H ( M, L N ) , we have Var (cid:0) Z s N ,...,s Nk , ϕ (cid:1) = N k − m − (cid:20)Z M B mk (cid:0) ∂ ¯ ∂ϕ, ∂ ¯ ∂ϕ (cid:1) Ω M + O ( N − + ε ) (cid:21) , UMBER VARIANCE OF RANDOM ZEROS ON COMPLEX MANIFOLDS II 3 where Ω M is the volume form on M , and B mk is a universal Hermitian form on the bundle T ∗ m − k +1 ,m − k +1 ( M ) . When k = 1 , we have B m ( f Ω M , f Ω M ) = π m − ζ ( m +2)4 | f | , and hence Var (cid:0) Z s N , ϕ (cid:1) = N − m h π m − ζ ( m +2)4 k ∂ ¯ ∂ϕ k L + O ( N − + ε ) i . In particular, for the complex curve case m = 1, we note that | ∂ ¯ ∂ϕ | = | ∆ ϕ | , and thusVar (cid:0) Z s N , ϕ (cid:1) = N − (cid:20) ζ (3)16 π k ∆ ϕ k + O ( N − + ε ) (cid:21) . (5)The leading term in (5) was obtained by Sodin and Tsirelson [ST] for the case of randompolynomials s N ∈ H ( CP , O ( N )) as well as for random holomorphic functions on C and onthe disk. (The constant ζ (3)16 π was given in a private communication from M. Sodin.)Here we say that B mk is universal if there exists a Hermitian inner product B mk on T ∗ m − k +1 ,m − k +10 ( C m ), independent of M and L , such that for all w ∈ M and all unitarytransformations τ : T ∗ ( C m ) → T ∗ w ( M ), we have B mk ( w ) = τ ∗ B mk . The global inner prod-uct (cid:0) ϕ, ψ (cid:1) = R M B mk (cid:0) ∂ ¯ ∂ϕ, ∂ ¯ ∂ψ (cid:1) Ω M is certainly positive semi-definite on D m − k,m − k ( M ),since the variance is nonnegative. We believe that, in fact, B mk is positive definite on T ∗ m − k +1 ,m − k +10 ( C m ). This follows for k = 1 from the above formula for B m ; one should beable to verify positivity for k > B mk .Thus the variance of the ‘smooth statistic’ ( Z s N ,...,s Nk , ϕ ) is of lower order than the varianceof the number and volume statistics given by (2)–(3), as expected. In view of (4), it is alsoself-averaging in the sense that its fluctuations are of smaller order than its typical values.An application of our methods is an extension of the Sodin-Tsirelson [ST] asymptotic nor-mality result for smooth statistics to general one-dimensional ensembles and to codimensionone zero sets in higher dimensions: Theorem
Let ( L, h ) → ( M, ω ) be as in Theorem 1.1 and give H ( M, L N ) the HermitianGaussian measure induced by h, ω . Let ϕ be a real ( m − , m − -form on M with C coef-ficients, such that ∂ ¯ ∂ϕ . Then for random sections s N in H ( M, L N ) , the distributionsof the random variables ( Z s N , ϕ ) − E ( Z s N , ϕ ) p Var( Z s N , ϕ ) converge weakly to the standard Gaussian distribution N (0 , as N → ∞ . Sodin and Tsirelson [ST] obtained the asymptotics of Theorem 1.2 for random functions on C , CP , and the disk. The proof of Theorem 1.2 is a relatively straightforward application ofthe fundamental Szeg˝o kernel asymptotics underlying Theorem 1.1 to the argument in [ST].(One easily sees that the random variable ( Z s N , ϕ ) is constant for all N if ∂ ¯ ∂ϕ ≡ Z s N , ϕ ) from (4) and Theorem1.1, respectively, we have: Corollary
With the same notation and hypotheses as in Theorem 1.2, the distribu-tions of the random variables N m/ ( Z s N − Nπ ω, ϕ ) converge weakly to N (0 , √ κ m k ∂ ¯ ∂ϕ k ) as N → ∞ , where κ m = π m − ζ ( m +2)4 . BERNARD SHIFFMAN AND STEVE ZELDITCH
Here, N (0 , σ ) denotes the (real) Gaussian distribution of mean zero and variance σ .We now summarize the key ideas in the proofs in [SZ4] and in this paper. The variance inTheorem 1.1, as well as the number and volume variances in [SZ4], can be expressed in termsof the variance currents Var ( Z s N ,...,s Nk ) of the random currents Z s N ,...,s Nk . In joint work withP. Bleher in 2000 [BSZ1], we introduced a bipotential Q N for the ‘pair correlation function’ K N of the volume density of zeros of random sections in H ( M, L N ); this bipotential satisfies∆ z ∆ w Q N ( z, w ) = K N ( z, w ) . The bipotential Q N is a universal function of the normalized Szeg˝o kernel (see (16) and (22)).Sodin and Tsirelson [ST] obtained a variance formula as well as asymptotic normality forzeros of certain model one-dimensional random holomorphic functions by implicitly usingthis bipotential.In [SZ4], we showed that Q N is actually a ‘pluri-bipotential’ for the codimension-onevariance current; i.e., ( i∂ ¯ ∂ ) z ( i∂ ¯ ∂ ) w Q N ( z, w ) = Var (cid:0) Z s N (cid:1) . (6)We further found a formula (Theorem 3.1) for the higher codimension variance current Var ( Z s N ,...,s Nk ) in terms of Q N and its derivatives of order ≤
4. We then applied the off-diagonal asymptotics of the Szeg˝o kernel Π N ( z, w ) in [SZ2] to obtain asymptotics of thebipotential Q N ( z, w ) and then of the number variance (2) as well as the volume variance (3).In this paper, we begin by reviewing basic facts about the Szeg˝o kernel and summarizingthe asymptotics from [SZ4] of the bipotential Q N as N → ∞ as well as d ( z, w ) →
0. Toillustrate our ideas, we apply these asymptotics to (6) to derive the codimension one formula(i.e., the case k = 1) of Theorem 1.1. We then prove in § § § § Background
In this section we summarize results from [SZ4] used in this paper.We let (
L, h ) be a Hermitian holomorphic line bundle over a compact K¨ahler manifold M .We consider a local holomorphic frame e L over a trivializing chart U . If s = f e L is a sectionof L over U , its Hermitian norm is given by k s ( z ) k h = a ( z ) − | f ( z ) | where a ( z ) = k e L ( z ) k − h . (7)The curvature form of ( L, h ) is given locally byΘ h = ∂ ¯ ∂ log a , and the Chern form c ( L, h ) is given by c ( L, h ) = √− π Θ h = √− π ∂ ¯ ∂ log a . (8)The current of integration Z s over the zeros of a section s ∈ H ( M, L ) is then given by the
Poincar´e-Lelong formula , Z s = √− π ∂ ¯ ∂ log | f | = √− π ∂ ¯ ∂ log k s k h + c ( L, h ) , (9) UMBER VARIANCE OF RANDOM ZEROS ON COMPLEX MANIFOLDS II 5 where the second equality is a consequence of (7)–(8).We now assume that the Hermitian metric h has strictly positive curvature and we give M the K¨ahler form ω = i h = πc ( L, h ) . (10)Next we describe the natural Gaussian probability measures on the spaces H ( M, L N ) ofholomorphic sections of tensor powers L N = L ⊗ N of the line bundle L : Definition
Let ( L, h ) → ( M, ω ) be as above, and let h N denote the Hermitian metricon L N induced by h . We give H ( M, L N ) the inner product induced by the K¨ahler form ω and the Hermitian metric h N : h s , ¯ s i = Z M h N ( s , s ) 1 m ! ω m , s , s ∈ H ( M, L N ) . (11) The
Hermitian Gaussian measure on H ( M, L N ) is the complex Gaussian probability mea-sure γ N induced by the inner product (11) : dγ N ( s ) = 1 π m e −| c | dc , s = d N X j =1 c j S Nj , where { S N , . . . , S Nd N } is an orthonormal basis for H ( M, L N ) . It is of course independent ofthe choice of orthonormal basis. The Gaussian ensembles ( H ( M, L N ) , γ N ) were also studied in [SZ1, SZ2, BSZ1, BSZ2];for the case of polynomials in one variable, they become the SU(2) ensembles in [BBL, Ha,NV, Zh]; for polynomials in m complex variables, they are the SU( m + 1) ensembles (see,e.g., [SZ1, Zr]).We consider the diagonal Szeg˝o kernelsΠ N ( z, z ) := d N X j =1 k S Nj ( z ) k h N ( z ) , where the S Nj are as in the above definition. It follows from the leading terms of the asymp-totic expansion of the diagonal Szeg˝o kernel of [Ca, Ti, Ze] thatΠ N ( z, z ) = 1 π m N m (1 + O ( N − )) . (12)The expected value of the zero current of a random holomorphic section in H ( M, L N ) isgiven by the basic formula E Z s N = i π ∂ ¯ ∂ log Π N ( z, z ) + Nπ ω , (13)and the expected values of simultaneous zero currents are given by E (cid:0) Z s N ,...,s Nk (cid:1) = (cid:2) E (cid:0) Z s N ) (cid:3) ∧ k = (cid:18) i π ∂ ¯ ∂ log Π N ( z, z ) + Nπ ω (cid:19) k = N k π k ω k + O ( N k − ) , (14)for 1 ≤ k ≤ m (see [SZ1, SZ4]). The final equality of (14) is a consequence of the asymptoticformula (12). BERNARD SHIFFMAN AND STEVE ZELDITCH
For the variance asymptotics, we need the properties of the off-diagonal Szeg˝o kernel: | Π N ( z, w ) | := (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) d N X j =1 S Nj ( z ) ⊗ S Nj ( w ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) h N ( z ) ⊗ h N ( w ) . (15)In particular, our variance formulas are expressed in terms of the normalized Szeg˝o kernel P N ( z, w ) := | Π N ( z, w ) | Π N ( z, z ) Π N ( w, w ) . (16)In [SZ4], we used the off-diagonal asymptotics for Π N from [SZ2] to provide the off-diagonalestimates given below for the normalized Szeg˝o kernel P N . As in [BSZ1, SZ1, SZ2, SZ4]), weobtained these asymptotics by identifying the line bundle Szeg˝o kernel | Π N ( z, w ) | of (15) withthe absolute value of a scalar Szeg˝o kernel Π N ( x, y ) on the unit circle bundle X ⊂ L − → M associated to the Hermitian metric h .Our estimates are of two types: (1) ‘near-diagonal’ asymptotics (Proposition 2.3) for P N ( z, w ) where the distance d ( z, w ) between z and w satisfies an upper bound d ( z, w ) ≤ b (cid:0) log NN (cid:1) / ( b ∈ R + ); (2) ‘far-off-diagonal’ asymptotics (Proposition 2.2) where d ( z, w ) ≥ b (cid:0) log NN (cid:1) / : Proposition [SZ4, Prop. 2.6]
Let ( L, h ) → ( M, ω ) be as in Theorem 1.1, and let P N ( z, w ) be the normalized Szeg˝o kernel for H ( M, L N ) given by (16) . For b > √ j + 2 k , j, k ≥ , we have ∇ j P N ( z, w ) = O ( N − k ) uniformly for d ( z, w ) ≥ b r log NN .
Here, ∇ j stands for the j -th covariant derivative. The normalized Szeg˝o kernel P N alsosatisfies Gaussian decay estimates valid very close to the diagonal. To give this estimate, weconsider a local normal coordinate chart ρ : U, z → C m , z ∈ U ⊂ M ,and we write, by abuse of notation, P N ( z + u, z + v ) := P N ( ρ − ( u ) , ρ − ( v )) . Proposition [SZ4, Prop. 2.7–2.8]
Let P N ( z, w ) be as in Proposition 2.2, and let z ∈ M . For ε, b > , there are constants C j = C j ( M, ε, b ) , j ≥ , independent of the point z ,such that P N (cid:16) z + u √ N , z + v √ N (cid:17) = e − | u − v | [1 + R N ( u, v )] , where | R N ( u, v ) | ≤ C | u − v | N − / ε , |∇ R N ( u ) | ≤ C | u − v | N − / ε , |∇ j R N ( u, v ) | ≤ C j N − / ε j ≥ , for | u | + | v | < b √ log N . UMBER VARIANCE OF RANDOM ZEROS ON COMPLEX MANIFOLDS II 7
The pluri-bipotential for the variance.
For random codimension k zeros, we havethe variance current of Z s N ,...,s Nk : Var (cid:0) Z s N ,...,s Nk (cid:1) = E (cid:0) Z s N ,...,s Nk ⊠ Z s N ,...,s Nk (cid:1) − E (cid:0) Z s N ,...,s Nk (cid:1) ⊠ E (cid:0) Z s N ,...,s Nk (cid:1) ∈ D ′ k, k ( M × M ) . (17)The variance for the ‘smooth zero statistics’ is given by:Var (cid:0) Z s N ,...,s Nk , ϕ (cid:1) = (cid:16) Var (cid:0) Z s N ,...,s Nk (cid:1) , ϕ ⊠ ϕ (cid:17) . (18)Here we write R ⊠ S = π ∗ R ∧ π ∗ S ∈ D ′ p + q ( M × M ) , for R ∈ D ′ p ( M ) , S ∈ D ′ q ( M ) , where π , π : M × M → M are the projections to the first and second factors, respectively.For a current T on M × M , we shall write ∂T = ∂ T + ∂ T , ∂ = X dz j ∂∂z j , ∂ = X dw j ∂∂w j , where z , . . . , z m are local coordinates on the first factor, and w , . . . , w m are local coordinateson the second factor of M × M . In particular, ∂ ( R ⊠ S ) = ( ∂R ) ⊠ S and ∂ ( R ⊠ S ) = R ⊠ ( ∂S ).We similarly write ¯ ∂T = ¯ ∂ T + ¯ ∂ T .
In [SZ4], we constructed a pluri-bipotential for the variance current in codimension one,i.e. a function Q N ∈ L ( M × M ) such that Var (cid:0) Z s N (cid:1) = − ∂ ¯ ∂ ∂ ¯ ∂ Q N = ( i∂ ¯ ∂ ) z ( i∂ ¯ ∂ ) w Q N ( z, w ) . (19)To describe our pluri-bipotential Q N ( z, w ), we define the function e G ( t ) := − π Z t log(1 − s ) s ds = 14 π ∞ X n =1 t n n , ≤ t ≤ . (20)Alternatively, e G ( e − λ ) = − π Z ∞ λ log(1 − e − s ) ds , λ ≥ . (21) Theorem [SZ4, Theorem 3.1]
Let ( L, h ) → ( M, ω ) be as in Theorem 1.1. Let Q N : M × M → [0 , + ∞ ) be the function given by Q N ( z, w ) = e G ( P N ( z, w )) = − π Z P N ( z,w ) log(1 − s ) s ds , (22) where P N ( z, w ) is the normalized Szeg˝o kernel given by (16) . Then Var (cid:0) Z s N (cid:1) = − ∂ ¯ ∂ ∂ ¯ ∂ Q N . Theorem 2.4 says thatVar( Z s N , ϕ ) = (cid:0) − ∂ ¯ ∂ ∂ ¯ ∂ Q N , ϕ ⊠ ϕ (cid:1) = Z M × M Q N ( z, w ) i∂ ¯ ∂ϕ ( z ) ∧ i∂ ¯ ∂ϕ ( w ) , (23)for all real ( m − , m − ϕ on M with C coefficients. BERNARD SHIFFMAN AND STEVE ZELDITCH
Since P N ∈ C ∞ ( M × M ) and P N ( z, w ) < z = w , for sufficiently large N (so that theKodaira map for L N is an embedding), it follows from (20) that Q N is C ∞ off the diagonal,for N ≫ Lemma [SZ4, Lemma 3.4]
For b > √ j + q + 1 , j ≥ , we have |∇ j Q N ( z, w ) | = O (cid:18) N q (cid:19) , for d ( z, w ) > b √ log N √ N .
Proposition 2.3 yields the near-diagonal asymptotics:
Lemma [SZ4, Lemma 3.5]
For b ∈ R + , we have Q N (cid:16) z , z + v √ N (cid:17) = e G ( e − | v | ) + O ( N − / ε ) , for | v | ≤ b p log N .
Recalling (21), we write, F ( λ ) := e G ( e − λ ) = − π Z ∞ λ log(1 − e − s ) ds ( λ ≥ , (24)so that Q N = F ◦ ( − log P N ). By Proposition 2.3, − log P N (cid:18) z , z + v √ N (cid:19) = | v | + O ( | v | N − / ε ) for | v | < b p log N . (25)It follows from Lemma 2.6 and (25) that Q N ∈ C ( M × M ) and the first partial derivativesof Q N vanish along the diagonal in M × M , for N ≫
0. (We note that Q N is C ∞ off thediagonal, but is not C at all points on the diagonal in M × M , as the computations in [SZ4]show.) We furthermore have the near-diagonal asymptotics: Lemma [SZ4, Lemma 3.7]
There exist a constant C m ∈ R + (depending only on thedimension m ) and N = N ( M ) ∈ Z + such that for N ≥ N , we have: i) The coefficients of the current ¯ ∂ ¯ ∂ Q N are locally bounded functions (given by point-wise differentiation of Q N ), and we have the pointwise estimate | ¯ ∂ ¯ ∂ Q N ( z, w ) | ≤ C m N for < | w − z | < b r log NN . ii) If m ≥ , the coefficients of the current ∂ ¯ ∂ ∂ ¯ ∂ Q N are locally L m − functions, andwe have the estimate | ∂ ¯ ∂ ∂ ¯ ∂ Q N ( z, w ) | ≤ C m N | w − z | for < | w − z | < b r log NN .
Lemma [SZ4, Lemma 3.9]
For N sufficiently large, − ∂ ¯ ∂ ∂ ¯ ∂ Q N ( z , z + v √ N ) = N Var z ∞ ( v )+ O (cid:0) | v | − N / ε (cid:1) for < | v | < b √ log N , (26)
UMBER VARIANCE OF RANDOM ZEROS ON COMPLEX MANIFOLDS II 9 where
Var z ∞ ∈ T ∗ , z ( M ) ⊗ D ′ , ( C m ) is given by Var z ∞ ( v ) := − F (4) ( | v | ) (¯ v · dz )( v · d ¯ z )(¯ v · dv )( v · d ¯ v ) − F (3) ( | v | ) (cid:2) ( dz · d ¯ z )(¯ v · dv )( v · d ¯ v ) + ( v · d ¯ z )(¯ v · dv )( dz · d ¯ v )+(¯ v · dz )( d ¯ z · dv )( v · d ¯ v ) + (¯ v · dz )( v · d ¯ z )( dv · d ¯ v ) (cid:3) − F ′′ ( | v | ) (cid:2) ( d ¯ z · dv )( dz · d ¯ v ) + ( dz · d ¯ z )( dv · d ¯ v ) (cid:3) . (27)Differentiating (24), we note that F ′′ ( λ ) = 1 π e λ − , F (3) ( λ ) = − π csch λ , F (4) ( λ ) = 1 π coth λ csch λ . (28)Thus, F ( j ) ( λ ) = ( − j ( j − π λ − j +1 + O (1) ( λ > , for j ≥
2, and hence
Var z ∞ ( v ) = (cid:26) O ( | v | − ) for | v | > O ( | v | e −| v | ) for | v | > . (29)3. The sharp variance estimate: Proof of Theorem 1.1
The codimension one case.
To illustrate the basic ideas of the argument, we beginwith the proof for the case k = 1. By Theorem 2.4, we haveVar (cid:0) Z s N , ϕ (cid:1) = Z M I N ( z ) i∂ ¯ ∂ϕ ( z ) , (30)where I N ( z ) = Z { z }× M Q N ( z, w ) i∂ ¯ ∂ϕ ( w ) . (31)We let Ω M = 1 m ! ω m denote the volume form of M , and we write i∂ ¯ ∂ϕ = ψ Ω M , ψ ∈ C R ( M ) , (32)so that I N ( z ) = Z { z }× M Q N ( z, w ) ψ ( w )Ω M ( w ) . (33)To evaluate I N ( z ) at a fixed point z ∈ M , we choose a normal coordinate chart centered at z as in §
2, and we make the change of variables w = z + v √ N . By Lemma 2.5 and (32)–(33),we can approximate I N ( z ) by integrating (33) over a small ball about z : I N ( z ) = Z | v |≤ b √ log N Q N (cid:18) z , z + v √ N (cid:19) ψ (cid:18) z + v √ N (cid:19) Ω M (cid:18) z + v √ N (cid:19) + O (cid:18) N m (cid:19) , (34)where b = √ m + 2. Since ω = i ∂ ¯ ∂ log a = i ∂ ¯ ∂ [ | z | + O ( | z | )] in normal coordinates, we have ω (cid:18) z + v √ N (cid:19) = i X (cid:20) δ jk + O (cid:18) | v |√ N (cid:19)(cid:21) N dv j ∧ d ¯ v k = i N ∂ ¯ ∂ | v | + O (cid:18) | v | N / (cid:19) , (35)for | v | ≤ b √ log N . HenceΩ M (cid:18) z + v √ N (cid:19) = 1 m ! (cid:20) i N ∂ ¯ ∂ | v | + O (cid:18) | v | N / (cid:19)(cid:21) m = 1 N m " Ω E ( v ) + O r log NN ! , (36)for | v | ≤ b √ log N , whereΩ E ( v ) = 1 m ! (cid:18) i ∂ ¯ ∂ | v | (cid:19) m = m Y j =1 i dv j ∧ d ¯ v j denotes the Euclidean volume form. Since ϕ ∈ C and hence ψ ( z + v √ N ) = ψ ( z )+ O ( | v | / √ N ),we then have by Lemma 2.6 and (34)–(36), I N ( z ) = 1 N m (cid:20)Z | v |≤ b √ log N n e G ( e − | v | ) + O ( N − / ε ) o (cid:8) ψ ( z ) + O ( N − / ε ) (cid:9) × (cid:8) Ω E ( v ) + O ( N − / ε ) (cid:9) (cid:21) + O (cid:0) N − m − (cid:1) = ψ ( z ) N m (cid:20)Z | v |≤ b √ log N e G ( e − | v | )Ω E ( v ) + O (cid:0) N − / ε (cid:1)(cid:21) . (37)Since e G ( e − λ ) = O ( e − λ ) and hence Z | v |≥ b √ log N e G ( e − | v | ) Ω E ( v ) = O ( N − m − ) , (38)we can replace the integral over the ( b √ log N )-ball with one over all of C m , and therefore I N ( z ) = ψ ( z ) N m (cid:20)Z C m e G ( e − | v | )Ω E ( v ) + O ( N − / ε ) (cid:21) . (39)Recalling (20), we have Z C m e G ( e − | v | ) Ω E ( v ) = 14 π ∞ X k =1 Z C m e − k | v | k Ω E ( v )= 14 π ∞ X k =1 π m k m +2 = π m − ζ ( m + 2) . (40)Therefore, by (30) and (39)–(40),Var (cid:0) Z s N , ϕ (cid:1) = 1 N m Z M (cid:20) π m − ζ ( m + 2) + O ( N − / ε ′ ) (cid:21) ψ ( z ) Ω M ( z ) . (41)The variance formula of Theorem 1.1 for the case k = 1 follows from (32) and (41). UMBER VARIANCE OF RANDOM ZEROS ON COMPLEX MANIFOLDS II 11
An explicit formula for the variance.
In this section, we give an integral formulafor the variance of simultaneous zero currents in higher codimension (Corollary 3.3), whichwe shall use in the next section to derive the asymptotics of Theorem 1.1. This integralformula is a modification of the following formula for the variance:
Theorem [SZ4, Theorem 3.13]
Let ≤ k ≤ m . Then for N sufficiently large, Var (cid:0) Z s N ,...,s Nk (cid:1) = ∂ ∂ " k X j =1 ( − j (cid:18) kj (cid:19) ¯ ∂ ¯ ∂ Q N ∧ (cid:0) ∂ ¯ ∂ ∂ ¯ ∂ Q N (cid:1) j − ∧ (cid:0) E Z s N ⊠ E Z s N (cid:1) k − j . where the current inside the brackets is an L current on M × M given by pointwise multi-plication, Q N is given by (22) , and E Z s N is given by (13) . Furthermore, Var (cid:0) Z s N ,...,s Nk (cid:1) isan L current on M × M if k ≤ m − . By an L current on the compact manifold M × M , we mean a current whose localcoefficients are L functions. Corollary
Let ≤ k ≤ m . Then for N sufficiently large, Var (cid:0) Z s N ,...,s Nk (cid:1) = ∂ ¯ ∂ ∂ ¯ ∂ " k X j =1 (cid:18) kj (cid:19) Q N (cid:0) − ∂ ¯ ∂ ∂ ¯ ∂ Q N (cid:1) j − ∧ (cid:0) E Z s N ⊠ E Z s N (cid:1) k − j . where the current inside the brackets is an L current on M × M given by pointwise multi-plication.Proof. Let T = k X j =1 (cid:18) kj (cid:19) Q N (cid:0) − ∂ ¯ ∂ ∂ ¯ ∂ Q N (cid:1) j − ∧ (cid:0) E Z s N ⊠ E Z s N (cid:1) k − j denote the expression inside the brackets in Corollary 3.2, regarded as a (4 k − M × M r ∆, where ∆ = { ( z, z ) : z ∈ M } denotes the diagonal. By Lemma 2.7(ii), T = O (cid:0) d ( z, w ) − k +2 (cid:1) and hence T defines an L current on M × M . It suffices to show that¯ ∂ ¯ ∂ T is an L current and thus ¯ ∂ ¯ ∂ can be moved outside the brackets in Theorem 3.1.Let U ε = { ( z, w ) ∈ M × M : d ( z, w ) < ε } denote the ε -neighborhood of the diagonal ∆.For a test form ϕ with m − k + 1 dz j ’s, d ¯ z j ’s, and dw j ’s , and m − k d ¯ w j ’s, we have( ¯ ∂ T, ϕ ) = − lim ε → Z M × M r U ε T ∧ ¯ ∂ ϕ = lim ε → Z M × M r U ε ¯ ∂ T ∧ ϕ + lim ε → Z ∂U ε T ∧ ϕ . Since T ∧ ϕ = o ( ε − m +1 ) on ∂U ε , the boundary integral goes to 0. By Lemma 2.7(ii) andthe fact that Q N ∈ C ( M × M ), the pointwise-defined form ¯ ∂ T is also O ( d ( z, w ) − k +2 ) andthus ¯ ∂ T is an L current on M × M given by pointwise differentiation.Repeating the same argument with T replaced by ¯ ∂ T and using part (i) of Lemma 2.7 aswell as part (ii), we then conclude that ¯ ∂ ¯ ∂ T is an L current. (cid:3) Corollary 3.2 can also be shown directly, using the argument in the proof of Theorem 3.1in [SZ4].
Corollary
The variance in Theorem 1.1 is given by:
Var (cid:0) [ Z s N ,...,s Nk ] , ϕ (cid:1) = k X j =1 (cid:18) kj (cid:19) Z M × M Q N (cid:0) − ∂ ¯ ∂ ∂ ¯ ∂ Q N (cid:1) j − ∧ (cid:0) E Z s N ⊠ E Z s N (cid:1) k − j ∧ ( ∂ ¯ ∂ϕ ⊠ ∂ ¯ ∂ϕ ) , where the integrands are in L ( M × M ) . Higher codimensions.
Recalling (14), we write the formula of Corollary 3.3 as follows:Var (cid:0) [ Z s N ,...,s Nk ] , ϕ (cid:1) = k X j =1 (cid:18) kj (cid:19) V Nj ( ϕ ) , (42) V Nj ( ϕ ) = (cid:18) Nπ (cid:19) k − j Z M × M Q N ( z, w ) (cid:2) − ∂ ¯ ∂ ∂ ¯ ∂ Q N ( z, w ) (cid:3) j − ∧ (cid:20) ω ( z ) k − j + O (cid:18) N (cid:19)(cid:21) ∧ (cid:20) ω ( w ) k − j + O (cid:18) N (cid:19)(cid:21) ∧ i∂ ¯ ∂ϕ ( z ) ∧ i∂ ¯ ∂ϕ ( w )= (cid:18) Nπ (cid:19) k − j Z M I Nj ∧ (cid:20) ω k − j ∧ i∂ ¯ ∂ϕ + O (cid:18) N (cid:19)(cid:21) , (43)where I Nj ( z ) = Z { z }× M Q N ( z, w ) (cid:2) − ∂ ¯ ∂ ∂ ¯ ∂ Q N ( z, w ) (cid:3) j − ∧ (cid:20) ω ( w ) k − j ∧ i∂ ¯ ∂ϕ ( w ) + O (cid:18) N (cid:19)(cid:21) ∈ T ∗ j − ,j − z ( M ) . (44)The integrand in (44) is regarded as an ( m, m )-form (in the w variable) with values in T ∗ j − ,j − z ( M ).Fix a point z ∈ M , and let 1 ≤ j ≤ k ≤ m . To evaluate I Nj ( z ), we write ω k − j ∧ i∂ ¯ ∂ϕ = X ψ JK ( w ) dw J ∧ d ¯ w K = 1 N m − j +1 X ψ JK (cid:18) z + v √ N (cid:19) dv J ∧ d ¯ v K , | J | = | K | = m − j + 1 . By Lemma 2.5, we can replace integration over M in (44) with integration over the smallball of radius b p log N/N , with b = √ m + 4, to obtain: I Nj ( z ) = N − m + j − Z | v |≤ b √ log N Q N (cid:18) z , z + v √ N (cid:19) (cid:20) − ∂ ¯ ∂ ∂ ¯ ∂ Q N (cid:18) z , z + v √ N (cid:19)(cid:21) j − ∧ X (cid:20) ψ JK (cid:18) z + v √ N (cid:19) + O (cid:18) N (cid:19)(cid:21) dv J ∧ d ¯ v K + O (cid:0) N − m − (cid:1) . By Lemma 2.7, the above integrand is L , and hence by Lemma 2.8, I Nj ( z ) = N j − − m (cid:20)Z | v |≤ b √ log N F ( | v | ) n Var z ∞ ( v ) o j − X ψ JK ( z ) dv J ∧ d ¯ v K + O ( N − + ε ) (cid:21) = N j − − m (cid:20)X ψ JK ( z ) Z v ∈ C m F ( | v | ) n Var z ∞ ( v ) o j − dv J ∧ d ¯ v K + O ( N − + ε ) (cid:21) . (45) UMBER VARIANCE OF RANDOM ZEROS ON COMPLEX MANIFOLDS II 13
Here, we replaced the integral over the ( b √ log N )-ball with one over all of C m , since by (29)we have F ( | v | ) { Var z ∞ ( v ) } j − = O ( e −| v | ) for | v | >
1, and hence Z | v | >b √ log N F ( | v | ) n Var z ∞ ( v ) o j − dv J ∧ d ¯ v K = O ( N − m ) . It follows from (43) and (45) that V Nj ( ϕ ) = N k − m − (cid:20)Z M X B jJKAB ψ JK ¯ ψ AB Ω M + O ( N − + ε ) (cid:21) , (46)where B j = { B jJKAB } is a universal Hermitian form on T ∗ m − j +1 ,m − j +1 ( M ). Theorem 1.1then follows from (42) and (46) with B mk ( α, α ) = k X j =1 B j ( ω k − j ∧ α, ω k − j ∧ α ) . (cid:3) Asymptotic normality: Proof of Theorem 1.2
The proof of Theorem 1.2 is an application of Propositions 2.2–2.3 to a general result ofSodin-Tsirelson [ST] on asymptotic normality of nonlinear functionals of Gaussian processes.Following [ST], we define a normalized complex Gaussian process to be a complex-valuedrandom function w ( t ) on a measure space ( T, µ ) of the form w ( t ) = X c j g j ( t ) , where the c j are i.i.d. complex Gaussian random variables (of mean 0, variance 1), and the g j are (fixed) complex-valued measurable functions such that X | g j ( t ) | = 1 for all t ∈ T. We let w , w , w , . . . be a sequence of normalized complex Gaussian processes on a finitemeasure space ( T, µ ). Let f ( r ) ∈ L ( R + , e − r / rdr ) and let ψ : T → R be bounded measur-able. We write Z ψN ( w N ) = Z T f ( | w N ( t ) | ) ψ ( t ) dµ ( t ) . Theorem [ST, Theorem 2.2]
Let ρ N ( s, t ) be the covariance functions for the Gaussianprocesses w N ( t ) . Suppose that i) lim inf N →∞ R T R T | ρ N ( s, t ) | α ψ ( s ) ψ ( t ) dµ ( s ) dµ ( t )sup s ∈ T R T | ρ N ( s, t ) | dµ ( t ) > , for α = 1 if f is monotonically increasing, or for all α ∈ Z + otherwise; ii) lim N →∞ sup s ∈ T Z T | ρ N ( s, t ) | dµ ( t ) = 0 . Then the distributions of the random variables Z ψN − E Z ψN q Var( Z ψN ) converge weakly to N (0 , as N → ∞ . We apply this result with f ( r ) = log r and ( T, µ ) = ( M, Ω M ). To define our normalizedGaussian processes w N on M , we choose a measurable section σ L : M → L of L with k σ L ( z ) k h = 1 for all z ∈ M , and we let S Nj = F Nj σ ⊗ NL , j = 1 , . . . , d N , be an orthonormal basis for H ( M, L N ) with respect to its Hermitian Gaussian measure, foreach N ∈ Z + . We then let g Nj ( z ) := F Nj ( z ) p Π N ( z, z ) , j = 1 , . . . , d N . Since | F Nj | = k S Nj k h N , it follows that w N = P c j g Nj defines a normalized complex Gauss-ian process, for each N ∈ Z + (where the c j are i.i.d. standard complex Gaussian randomvariables). In fact, | w N ( z ) | = k s N ( z ) k h N p Π N ( z, z ) , where s N = p Π N ( z, z ) w N σ ⊗ NL = X c j S Nj is a random holomorphic section in H ( M, L N ). The covariance functions ρ N ( z, w ) for theseGaussian processes satisfy | ρ N ( z, w ) | = P N ( z, w ) . We now let ϕ be a fixed C real ( m − , m − M and we write iπ ∂ ¯ ∂ϕ = ψ Ω M . Then ψ ∈ C , and Z ψN ( w N ) = Z M (cid:16) log k s N ( z ) k h N − log p Π N ( z, z ) (cid:17) iπ ∂ ¯ ∂ϕ ( z ) = (cid:0) Z s N , ϕ (cid:1) + k N , where the k N are constants (depending on L → M and ϕ , but independent of the randomsections s N ). Hence Z ψN ( w N ) has the same variance as the smooth linear statistic (cid:0) Z s N , ϕ (cid:1) ,and it suffices by Theorem 4.1 to check that the covariance function satisfies conditions(i)–(ii) of the theorem. We start with (ii): by Proposition 2.2,lim N →∞ sup z ∈ M Z d ( z,w ) >b √ log NN P N ( z, w ) Ω M ( w ) = 0 . On the other hand, since P N ( z, w ) ≤
1, it is obvious that the same limit holds for d ( z, w ) ≤ b q log NN , verifying (ii).To check (i), we again break up the integral into the near diagonal d ( z, w ) ≤ b q log NN andthe off-diagonal d ( z, w ) > b q log NN . As before, the integrals over the off-diagonal set tend tozero rapidly and can be ignored in both the numerator and denominator. UMBER VARIANCE OF RANDOM ZEROS ON COMPLEX MANIFOLDS II 15
On the near diagonal, we replace P N by its asymptotics in Proposition 2.3. The asymptoticformula for P N has a universal leading term independent of z and has uniform remainder,so condition (i) (with α = 1) becomeslim inf N →∞ R M Ω M ( z ) R | u | . Since ψ ∈ C , the ratio clearly tends to 2 − m R M ψ ( z ) Ω M >
0, which verifies (i) and completesthe proof of Theorem 1.2. (cid:3) Open problems on smooth and counting statistics
In this section, we present a number of open problems on smooth and discontinuous linearstatistics of zeros.(1) Asymptotic normality of the smooth linear statistics ( Z s N ,...,s Nk , ϕ ) has only beenproved in codimension one, i.e. when k = 1. But these random variables are likelyto be asymptotically normal for all dimensions m and codimensions k . It would beinteresting to prove (or disprove) this statement, in particular for k = m .(2) To our knowledge, no results are known to date regarding the asymptotic normalityof the counting statistics N UN ( s N , . . . , s Nm ), even when m = 1. This is analogous to,but presumably harder than, the smooth linear statistic when k = m .(3) In [Zh], Qi Zhong obtained surprising results on the expected value of the ‘energy’random variable E G ( s N ) = X i = j G ( a i , a j ) , Z s N = { a , . . . , a N } , (47)summing the values of the Green’s function G over pairs of distinct zeros of a randompolynomial or section s N ∈ H ( M, L N ) of the N -th power of a positive line bundle( L, h ) over a compact Riemann surface M . (The Green’s function is normalized toequal + ∞ on the diagonal.) Zhong proved that when G is the Green’s function forthe Riemannian metric induced by the curvature of h , the expected energy has theasymptotics E E G ( p N ) ∼ − π N log N . It is known (N. Elkies) that − π N log N isalso the asymptotic minimum for the energy sum (47). The energy is a partiallysmooth linear statistic ( G, Z s N ⊗ Z s N − ∆ Z sN ) (48)on M × M , where ∆ Z sN represents the diagonal terms of Z s N ⊗ Z s N . The statistic(48) is not smooth since G has a logarithmic singularity and since we subtracted thediagonal current. It is a random variable of one section s N in dimension one, but G is a function on M × M , so the variance of E G involves the rather complicated four-point correlation function of Z s rather than the pair correlation, for which Theorem2.4 gives a useful formula. It would be interesting to investigate the variance of theenergy E G . It seems that it should tend to zero with N since the Gaussian measureis concentrated on ‘polynomials’ whose zeros are asymptotic minimizers.(4) The expected distribution of zeros can have quite disparate asymptotics when theensembles are given Gaussian measures induced from inner products on the space of polynomials (or sections) which use non-smooth volume forms or non-positivelycurved line bundle metrics. For instance, in the case where the measure is supportedon an analytic plane domain Ω ⊂ C or on its boundary, it was shown in [SZ3] that thethe expected distributions of random zeros of random polynomials of degree N tendto the equilibrium measure of Ω. This result was generalized to higher dimensionsand more general metrics and measures in a sequence of papers [Bl1, BS, Be] inwhich it is shown that the expected distribution of zeros tends to an equilibriummeasure adapted to the measure and metric. In [Sh], an upper bound was given forthe variances of the smooth linear statistics when the inner products are defined byarbitrary measures (and also for more general sequences of ensembles of increasingdegrees). This bound is sufficient to prove that sequences of random zeros in theseensembles almost surely converge to their equilibrium measure, although the boundis not always sharp. The Szeg˝o kernels for the inner products in [SZ3, Bl1, BS, Be]are quite different from Szeg˝o kernels for positive line bundles in this article, and sothe asymptotics of the variances might be quite different. It would be interesting todetermine them.(5) Results on expected values for the analogous SO(2) and SO( m + 1) ensembles ofrandom real polynomials in one or several variables were given by [BD, EK, Ro, SS],and a (global) variance result for real zeros was given in [Ws]. Maslova [Ma] provedthe asymptotic normality of the number of real zeros for the Kac ensemble [Kac] ofrandom real polynomials on R (as well as for some non-Gaussian ensembles). But asfar as we are aware, asymptotic normality for numbers of real zeros in the SO( m + 1)ensemble has not been investigated. References [Be] R. Berman, Bergman kernels and weighted equilibrium measures of C n , preprint(arXiv:math/0702357).[BD] P. Bleher and X. Di, Correlations between zeros of a random polynomial, J. Statist. Phys.
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