UNIVERSALITY AND SCALING OF CORRELATIONS BETWEEN ZEROS ON COMPLEX MANIFOLDS

PAVEL BLEHER, BERNARD SHIFFMAN, AND STEVE ZELDITCH

Abstract. We study the limit as N → ∞ of the correlations between simultaneous zeros of random sections of the powers L^{N} of a positive holomorphic line bundle L over a compact complex manifold M, when distances are rescaled so that the average density of zeros is independent of N. We show that the limit correlation is independent of the line bundle and depends only on the dimension of M and the codimension of the zero sets. We also provide some explicit formulas for pair correlations. In particular, we provide an alternate derivation of Hannay’s limit pair correlation function for SU(2) polynomials, and we show that this correlation function holds for all compact Riemann surfaces.

Introduction

This paper is concerned with the local statistics of the simultaneous zeros of k random holomorphic sections s_{1}, …, s_{k} ∈ H^{0}(M, L^{N}) of the N^{th} power L^{N} of a positive Hermitian holomorphic line bundle (L, h) over a compact Kähler manifold M (where k ≤ m = dim M). The terms ‘random’ and ‘statistics’ are with respect to a natural Gaussian probability measure dν_{N} on H^{0}(M, L^{N}) which we define below. In the special case where M = ℂℙ^{m} and L is the hyperplane section bundle O(1), sections of L^{N} correspond to holomorphic polynomials of degree N, and (H^{0}(ℂℙ^{m}, O(N)), dν_{N}) is known as the ensemble of SU(m+1) polynomials in the physics literature. To obtain local statistics, we expand a ball U around a given point z^{0} by a factor \sqrt{N} so that the average density of simultaneous scaled zeros is independent of N. We then ask whether the simultaneous scaled zeros behave as if thrown independently in \sqrt{N}U or how they are correlated. Correlations between (unscaled) zeros are measured by the so-called n-point zero correlation function K_{nk}^{N}(z^{1}, …, z^{n}), and those between scaled zeros are measured by the scaled correlation function K_{nk}^{N}(\frac{z^{1}}{\sqrt{N}} , …, \frac{z^{n}}{\sqrt{N}}). Our main result is that the large N limits of the scaled n-point correlation functions K_{nk}^{N}(\frac{z^{1}}{\sqrt{N}} , …, \frac{z^{n}}{\sqrt{N}}) exist and are universal, i.e. are independent of M, L and h as well as the point z^{0}. Moreover, the scaling limit correlation functions can be calculated explicitly. We find that the limit correlations are short range, i.e. that simultaneous scaled zeros behave quite independently for large distances. On the other hand, nearby zeros exhibit some degree of repulsion.

To state our problems and results more precisely, we begin with provisional definitions of the correlation functions K_{nk}^{N}(z^{1}, …, z^{n}) and of the scaling limit. (See §§1–2 for the complete definitions and notation.) In order to provide a standard yardstick for our universality results, we give M the Kähler metric ω given by the (positive) curvature form of h. The metrics h and ω then induce a Hilbert space inner product on the space H^{0}(M, L^{N}) of holomorphic sections of L^{N}, for each N ≥ 1. In the spirit of [SZ] we use this L^{2}-norm to define a Gaussian probability measure dν_{N} on H^{0}(M, L^{N}). When we speak of a random section, we mean a section drawn at random from this ensemble. More generally, we can draw k sections (s_{1}, …, s_{k}) independently and at random from this ensemble. Let Z_{(s_{1},...,s_{k})} denote their simultaneous zero set and let |Z_{(s_{1},...,s_{k})}| denote the “delta measure” with support on Z_{(s_{1},...,s_{k})} and with density given by the natural Riemannian volume (2m − 2k)-form defined by the metric ω. To define the n-point zero correlation measure K_{nk}^{N}(z^{1}, …, z^{n}) we form the product measure

Research supported in part by NSF grants DMS-9623214 (first author), DMS-9800479 (second author), DMS-9703775 (third author). Research at MSRI supported in part by by NSF grant DMS-9701755.

|Z_{(s_{1},...,s_{k})}|^{n} = ( \underbrace{|Z_{(s_{1},...,s_{k})}| × ⋯ × |Z_{(s_{1},...,s_{k})}|}_{n} ) on M^{n} := \underbrace{M × ⋯ × M}_{n} .

To avoid trivial self-correlations, we puncture out the generalized diagonal in M^{n} to get the punctured product space

M_{n} = {(z^{1}, …, z^{n}) ∈ M^{n} : z^{p} ≠ z^{q} for p ≠ q} .

We then restrict |Z_{(s_{1},...,s_{k})}|^{n} to M_{n} and define K_{nk}^{N}(z^{1}, …, z^{n}) to be the expected value E(|Z_{(s_{1},...,s_{k})}|^{n}) of this measure with respect to ν_{N}. When k = m, the simultaneous zeros almost surely form a discrete set of points and so this case is perhaps the most vivid. Roughly speaking, K_{nk}^{N}(z^{1}, …, z^{n}) gives the probability density of finding simultaneous zeros at (z^{1}, …, z^{n}).

The first correlation function K_{1kN} just gives the expected distribution of simultaneous zeros of k sections. In a previous paper [SZ] by two of the authors, it was shown (among other things) that the expected distribution of zeros is asymptotically uniform; i.e.

K_{1k}^{N}(z^{0}) = c_{mk}N^{k} + O(N^{k−1}) ,

for any positive line bundle (see [SZ, Prop. 4.4]). The question then arises of determining the higher correlation functions. As was first observed by [BBL] and [Han] for SU(2) polynomials and by [BD] for real polynomials in one variable, the zeros of a random polynomial are nontrivially correlated, i.e. the zeros are not thrown down like independent points. We will prove the same for all SU(m + 1) polynomials and hence, by universality of the scaling limit, for any M, L, h.

To introduce the scaling limit, let us return to the case k = m where the simultaneous zeros form a discrete set of points. Since an m-tuple of sections of L^{N} will have N^{m} times as many zeros as m-tuples of sections of L, it is natural to expand U by a factor of \sqrt{N} to get a density of zeros that is independent of N. That is, we choose coordinates {z_{q}} for which z^{0} = 0 and ω(z^{0}) = \frac{i}{2} ∑_{q} dz_{q} ∧ dz̄_{q} and then rescale z ↦ \frac{z}{\sqrt{N}}. Were the zeros thrown independently and at random on U, the conditional probability density of finding a simultaneous zero at a point w given a zero at z would be a constant independent of (z, w).

Non-trivial correlations (for any codimension k ∈ {1, …, m}) are measured by the difference between 1 and the (normalized) n-point scaling limit zero correlation function

K̃_{nkm}^{∞}(z^{1}, …, z^{n}) = lim_{N→∞} (c_{mk}N^{k})^{−n} K_{nk}^{N} (\frac{z^{1}}{\sqrt{N}} , …, \frac{z^{n}}{\sqrt{N}} ) , (z^{1}, …, z^{n}) ∈ U_{n} .

Our main result (Theorem 3.4) is universality of the scaling limit correlation functions:

The n-point scaling limit zero correlation function K̃_{nkm}^{∞}(z^{1}, …, z^{n}) is given by a universal rational function, homogeneous of degree 0, in the values of the function e^{iℑ(z·w̄)− \frac{1}{2} |z−w|^{2}} and its first and second derivatives at the points (z, w) = (z^{p}, z^{p′}), 1 ≤ p, p′ ≤ n. Alternately it is a rational function in z_{q}^{p}, z̄_{q}^{p}, e^{z^{p}·z̄^{p′}}

The function e^{iℑ(z·w̄)− \frac{1}{2} |z−w|^{2}} which appears in the universal scaling limit is (up to a constant factor) the Szegö kernel Π_{1}^{H}(z, w) of level one for the reduced Heisenberg group H_{red}^{n} (cf. §1). Its appearance here owes to the fact that the correlation functions can be expressed in terms of the Szegö kernels Π_{N}(x, y) of L^{N}. I.e., let X denote the circle bundle over M consisting of unit vectors in L^{∗}; then Π_{N}(x, y) is the kernel of the orthogonal projection Π_{N} : L^{2}(X) → H_{N}^{2}(X) ≈ H^{0}(M, L^{N}). Indeed we have (Theorem 2.4):

The n-point correlation K̃_{nk}^{N}(z^{1}, …, z^{n}) is given by the above universal rational function, applied this time to the values of the Szegö kernel Π_{N} and its first and second derivatives at the points (z^{p}, z^{p′}).

In view of this relation between the correlation functions and the Szegö kernel, it suffices for the proof of the universality theorem 3.4 to determine the scaling limit of the Szegö kernel Π_{N} and to show its universality. Indeed we shall show (Theorem 3.1) that:

Let (z_{1}, …, z_{m}, θ) denote local coordinates in a neighborhood Ũ ≈ U ×S^{1} of a point (z^{0}, λ) ∈ X (where (z_{1}, …, z_{m}) are the above local coordinates about z_{0} ∈ M). We then have

N^{−m}Π_{N} (\frac{z}{\sqrt{N}} , \frac{θ}{N}; \frac{z′}{\sqrt{N}} , \frac{θ′}{N} ) = Π_{1}^{H}(z, θ; z′, θ′) + O(N^{−1/2}) .

The fact that the correlation functions can be expressed in terms of the Szegö kernel may be explained in (at least) two ways. The first is that the correlation functions may be expressed in terms of the joint probability density D_{nk}^{N}(x, ξ; z)dxdξ of the (vector-valued) random variable

(x, ξ) = [x^{p}, ξ^{p}]_{1≤p≤n} , x^{p} = (s_{1}(z^{p}), …, s_{k}(z^{p})), ξ^{p} = (∇s_{1}(z^{p}), …, ∇s_{k}(z^{p}))

given by the values of the k sections and of their covariant derivatives at the n points {z^{p}}. Our method of computing the correlation functions is based on the following probabilistic formula (Theorem 2.1):

For N sufficiently large so that the density D_{nk}^{N}(x, ξ; z) is given by a continuous function, we have

K_{nk}^{N}(z) = ∫ dξ D_{nk}^{N}(0, ξ; z) ∏_{p=1}^{n} det (ξ_{j}^{p}ξ_{j′}^{p∗})_{1≤j,j′≤k} , z = (z^{1}, …, z^{n}) ∈ M_{n} ,

where ξ = (ξ^{1}, …, ξ^{n}) and ξ_{j}^{p∗} : L_{z^{p}}^{N} → T_{M,z^{p}} denotes the adjoint to ξ_{j}^{p} : T_{M,z^{p}} → L_{z^{p}}^{N}.

This formula, which is valid in a more general setting, is based on the approach of Kac [Ka] and Rice [Ri] (see also [EK]) for zeros of functions on ℝ^{1}, and of [Hal] for zeros of (real) Gaussian vector fields. Since our probability measure dν_{N} (on the space of sections) is Gaussian, it follows that D_{nk}^{N} is also a Gaussian density. It will be proved in §2.3 that the covariance matrix of this Gaussian may be expressed entirely in terms of Π_{N} and its covariant derivatives. This type of formula for the correlation function of zeros was previously used in [BD], [Han] and the works cited above. We believe that this formula will have interesting applications in geometry.

A second link between correlation functions and Szegö kernels is given by the Poincaré-Lelong formula. In fact, this was our original approach to computing the correlation functions in the codimension 1 case. For the sake of brevity, we will not discuss this approach here; instead we refer the reader to our companion article [BSZ].

From the universality of our answers, it follows that the scaling limit pair correlation functions depend only on the distance between points:

K̃_{2km}^{∞}(z^{1}, z^{2}) = κ_{km}(r) , r = |z^{1} − z^{2}| ,

where κ_{km} depends only on the dimension m of M and the codimension k of the zero set. In §4, we give explicit formulas for the limit pair correlation functions κ_{km} in some special cases. Our calculation uses the Heisenberg model, which (although noncompact) is the most natural one since the scaled Szegö kernels are all equal to Π_{1}, and there is no need in this case to take a limit. We also discuss the hyperplane section bundle O(1) → ℂℙ^{m}, which is the most studied, since the sections of its powers are the SU(m + 1) polynomials—homogeneous polynomials in m + 1 variables—and the case m = 1 (the SU(2) polynomials) appears frequently in the physics literature (e.g., [BBL, FH, Han, KMW, PT]). We give expressions for the zero correlations K_{nk}^{N} for the SU(m + 1) polynomials and by letting N → ∞, we obtain an alternate derivation of our universal formula for the scaling limit correlation.

We show (Theorem 4.1) that κ_{km}(r) = 1 + O(r^{4}e^{−r^{2}}) as r → +∞, and hence these correlations are short range in that they differ from the case of independent random points by an exponentially decaying term. We observe that when dim M = 1, there is a strong repulsion between nearby zeros in the sense that κ_{11}(r) → 0 as r → 0, as was noted by Hannay [Han] and Bogomolny-Bohigas-Leboeuf [BBL] for the case of SU(2) polynomials. These asymptotics are illustrated by the following graph (see also [Han]):

Figure 1. The 1-dimensional limit pair correlation function κ_{11}

For dim M = 2, the simultaneous scaled zeros of a random pair (s_{1}, s_{2}) of sections still exhibit a mild repulsion (lim_{r→0} κ_{22}(r) = \frac{3}{4}), as illustrated in Figure 2 below.

Figure 2. The limit pair correlation function κ_{22}

The function κ_{mm}(r) can be interpreted as the normalized conditional probability of finding a zero near a point z^{1} given that there is a zero at a second point a scaled distance r from z^{1} (in the case of discrete zeros in m dimensions). The above graphs show that for dimensions 1 and 2, there is a unique scaled distance where this probability is maximized. It would be interesting to explore the dependence of the correlations on the dimension. To ask one concrete question, do the simultaneous scaled zeros in the point case become more and more independent in the sense that κ_{mm}(r) → 1 as the dimension m → ∞?

When k < m, the zero sets are subvarieties of positive dimension m − k; in this case the expected volume of the zero set in a small spherical shell of radius r and thickness ε about a point in the zero set must be ∼ εr^{2m−2k−1}. Hence we have κ_{km}(r) ∼ r^{−2k}, for small r. The graph of the limit correlation function for the case m = 2, k = 1 is given in Figure 3 below.

Figure 3. The limit pair correlation function κ_{12}

To end this introduction, we would like to link our methods and results at least heuristically to a long tradition of (largely heuristic) results on universality and scaling in statistical mechanics (cf. [FFS]). One may view the rescaling transformation on U as generating a renormalization group. The intuitive picture in statistical mechanics is that the renormalization group should carry a given system (read “L → M”) to the fixed point of the renormalization group, i.e. to the scale invariant situation. We observe that the local rescaling of U is nothing other than the Heisenberg dilations δ\sqrt{N} on H_{red}^{m}. Since these dilations are automorphisms of the (unreduced) Heisenberg group, the Szegö kernel of H^{m} is invariant under these dilations; i.e., it is the fixed point of the renormalization group. As predicted by this intuitive picture, we find that in the scaling limit all the invariants of the line bundle, in particular its zero-point correlation functions, are drawn to their values for the fixed point system (read “Heisenberg model”).

1. Notation

We begin with some notation and basic properties of sections of holomorphic line bundles, their zero sets, Szegö kernels, and Gaussian measures. We also provide two examples that will serve as model cases for studying correlations of zeros of sections of line bundles in the high power limit.

1.1. Sections of holomorphic line bundles. In this section, we introduce the basic complex analytic objects: holomorphic sections and the currents of integration over their zero sets. We also introduce Gaussian probability measures on spaces of holomorphic sections. For background in complex geometry, we refer to [GH].

Let M be a compact complex manifold and let L → M be a holomorphic line bundle with a smooth Hermitian metric h; its curvature 2-form Θ_{h} is given locally by

Θ_{h} = −∂ ¯∂log ∥e_{L}∥_{h}^{2} , (1)

where e_{L} denotes a local holomorphic frame (= nonvanishing section) of L over an open set U ⊂ M, and ∥e_{L}∥_{h} = h(e_{L}, e_{L})^{1/2} denotes the h-norm of e_{L}. We say that (L, h) is positive if the (real) 2-form ω = \frac{\sqrt{−1}}{2}Θ_{h} is positive, i.e., if ω is a Kähler form. We henceforth assume that (L, h) is positive, and we give M the Hermitian metric corresponding to the Kähler form ω and the induced Riemannian volume form

dV_{M} = \frac{1}{m!}ω^{m} . (2)

Since \frac{1}{π}ω is a de Rham representative of the Chern class c_{1}(L) ∈ H^{2}(M, ℝ), the volume of M equals \frac{π^{m}}{m!} c_{1}(L)^{m}.

The space H^{0}(M, L^{N}) of global holomorphic sections of L^{N} = L ⊗ ⋯ ⊗ L is a finite dimensional complex vector space. (Its dimension, given by the Riemann-Roch formula for large N, grows like N^{m}. By the Kodaira embedding theorem, the global sections of L^{N} give an embedding into a projective space for N ≫ 0, and hence M is a projective algebraic manifold.) The metric h induces Hermitian metrics h^{N} on L^{N} given by ∥s^{⊗N}∥_{h^{N}} = ∥s∥_{h}^{N}. We give H^{0}(M, L^{N}) the Hermitian inner product

⟨s_{1}, s_{2}⟩ = ∫_{M} h^{N}(s_{1}, s_{2})dV_{M} (s_{1}, s_{2} ∈ H^{0}(M, L^{N}) ) , (3)

and we write |s| = ⟨s, s⟩^{1/2}.

We now explain our concept of a “random section.” We are interested in expected values and correlations of zero sets of k-tuples of holomorphic sections of powers L^{N}. Since the zeros do not depend on constant factors, we could suppose our sections lie in the unit sphere in H^{0}(M, L^{N}) with respect to the Hermitian inner product (3), and we pick random sections with respect to the spherical measure. Equivalently, we could suppose that s is a random element of the projectivization ℙH^{0}(M, L^{N}). Another equivalent approach is to use Gaussian measures on the entire space H^{0}(M, L^{N}). We shall use the third approach, since Gaussian measures seem the best for calculations. Precisely, we give H^{0}(M, L^{N}) the complex Gaussian probability measure

dν_{N}(s) = \frac{1}{π^{m}} e^{−|c|^{2}}dc , s = ∑_{j=1}^{d_{N}} c_{j}S_{j}^{N} , (4)

where {S_{j}^{N}} is an orthonormal basis for H^{0}(M, L^{N}) and dc is 2d_{N}-dimensional Lebesgue measure. This Gaussian is characterized by the property that the 2d_{N} real variables ℜc_{j}, ℑc_{j} (j = 1, …, d_{N}) are independent random variables with mean 0 and variance \frac{1}{2}; i.e.,

E c_{j} = 0, E c_{j}c_{k} = 0, E c_{j}c̄_{k} = δ_{jk} .

Here and throughout this paper, E denotes expectation.

In general, a complex Gaussian measure (with mean 0) on a finite dimensional complex vector space V is a measure ν of the form (4), where the c_{j} are the coordinates with respect to some basis. Explicitly, the complex Gaussian measures on ℂ^{m} are the probability measures of the form

\frac{e^{−⟨Δ^{−1}z,z⟩}}{π^{m} det Δ} dz (5)

where Δ = (Δ_{k}^{j}) is a positive definite Hermitian matrix and

⟨ζ, z⟩ = ζ · z̄ = ∑_{q=1}^{m} ζ_{q}z̄_{q}

denotes the standard Hermitian inner product in ℂ^{m}. For the Gaussian measure (5), we have

E (z_{j}z_{k}) = 0, E (z_{j}z̄_{k}) = Δ_{k}^{j} . (6)

If ν is a complex Gaussian on V and τ : V → Ṽ is a surjective linear transformation, then τ_{∗}ν is a complex Gaussian on Ṽ. In particular, if Ṽ = ℂ^{m}, then, τ_{∗}ν is of the form (5), where the covariance matrix Δ is given by (6) with z_{j} = z_{j} ∘ τ : V → ℂ.

We shall consider the space S = H^{0}(M, L^{N})^{k} (1 ≤ k ≤ m) with the probability measure dμ = dν × ⋯ × dν, which is also Gaussian. Picking a random element of S means picking k sections of H^{0}(M, L^{N}) independently and at random. For s = (s_{1}, …, s_{k}) ∈ S, we let

Z_{s} = {z ∈ M : s_{1}(z) = ⋯ = s_{k}(z) = 0}

denote the zero set of s. Note that if N is sufficiently large so that L^{N} is base point free, then for μ-a.a. s ∈ S, we have codim Z_{s} = k. (Indeed, the set of s where codim Z_{s} < k is a proper algebraic subvariety of H^{0}(M, L^{N})^{k}. In fact, by Bertini’s theorem, the Z_{s} are smooth submanifolds of complex dimension m − k for almost all s, provided N is large enough so that the global sections of L^{N} give a projective embedding of M, but we do not need this fact here.) For these s, we let |Z_{s}| denote Riemannian (2m − 2k)-volume along the regular points of Z_{s}, regarded as a measure on M:

(7) (|Z_{s}|, φ) = ∫_{Z_{s}^{reg}} φdVol_{2m−2k} = \frac{1}{(m − k)!} ∫_{Z_{s}^{reg}} φω^{m−k} .

It was shown by Lelong [Le] (see also [GH]) that the integral in (7) converges. (In fact, |Z_{s}| can be regarded as the total variation measure of the closed current of integration over Z_{s}.) We regard |Z_{s}| as a measure-valued random variable on the probability space (S, dμ); i.e., for each test function φ ∈ C^{0}(M), (|Z_{s}|, φ) is a complex-valued random variable.

1.2. Szegö kernels. As in [Ze, SZ] we now lift the analysis of holomorphic sections over M to a certain S^{1} bundle X → M. This is a useful approach to the asymptotics of powers of line bundles and goes back at least to [BG].

We let L^{∗} denote the dual line bundle to L, and we consider the circle bundle X = {λ ∈ L^{∗} : ∥λ∥_{h^{∗}} = 1}, where h^{∗} is the norm on L^{∗} dual to h. Let π : X → M denote the bundle map; if v ∈ L_{z}, then ∥v∥_{h} = |(λ, v)|, λ ∈ X_{z} = π^{−1}(z). Note that X is the boundary of the disc bundle D = {λ ∈ L^{∗} : ρ(λ) > 0}, where ρ(λ) = 1 − ∥λ∥_{h^{∗}}^{2}. The disc bundle D is strictly pseudoconvex in L^{∗}, since Θ_{h} is positive, and hence X inherits the structure of a strictly pseudoconvex CR manifold. Associated to X is the contact form α = −i∂ρ|_{X} = i¯∂ρ|_{X}. We also give X the volume form

dV_{X} = \frac{1}{m!}α ∧ (dα)^{m} = α ∧ π^{∗}dV_{M} . (8)

The setting for our analysis of the Szegö kernel is the Hardy space H^{2}(X) ⊂ L^{2}(X) of square-integrable CR functions on X, i.e., functions that are annihilated by the Cauchy-Riemann operator ¯∂_{b} (see [St, pp. 592–594]) and are L^{2} with respect to the inner product

⟨F_{1}, F_{2}⟩ = \frac{1}{2π} ∫_{X} F_{1}\overline{F_{2}}dV_{X} , F_{1}, F_{2} ∈ L^{2}(X) . (9)

Equivalently, H^{2}(X) is the space of boundary values of holomorphic functions on D that are in L^{2}(X). We let r_{θ}x = e^{iθ}x (x ∈ X) denote the S^{1} action on X and denote its infinitesimal generator by \frac{∂}{∂θ}. The S^{1} action on X commutes with ¯∂_{b}; hence H^{2}(X) = ⨁_{N=0}^{∞} H_{N}^{2}(X) where H_{N}^{2}(X) = {F ∈ H^{2}(X) : F (r_{θ}x) = e^{iNθ}F (x)}. A section s of L determines an equivariant function ŝ on L^{∗} by the rule ŝ(λ) = (λ, s(z)) (λ ∈ L_{z}^{∗}, z ∈ M). It is clear that if τ ∈ ℂ then ŝ(z, τλ) = τŝ. We henceforth restrict ŝ to X and then the equivariance property takes the form ŝ(r_{θ}x) = e^{iθ}ŝ(x). Similarly, a section s_{N} of L^{N} determines an equivariant function ŝ_{N} on X: put

ŝ_{N}(λ) = (λ^{⊗N}, s_{N}(z)) , λ ∈ X_{z} ,

where λ^{⊗N} = λ⊗⋯⊗λ; then ŝ_{N}(r_{θ}x) = e^{iNθ}ŝ_{N}(x). The map s ↦ ŝ is a unitary equivalence between H^{0}(M, L^{N}) and H_{N}^{2}(X). (This follows from (8)–(9) and the fact that α = dθ along the fibers of π : X → M.)

We let Π_{N} : L^{2}(X) → H_{N}^{2}(X) denote the orthogonal projection. The Szegö kernel Π_{N}(x, y) is defined by

Π_{N}F (x) = ∫_{X} Π_{N}(x, y)F (y)dV_{X}(y) , F ∈ L^{2}(X) . (10)

It can be given as

Π_{N}(x, y) = ∑_{j=1}^{d_{N}} Ŝ_{j}^{N}(x)\overline{Ŝ_{j}^{N}(y)} , (11)

where S_{1}^{N}, …, S_{d_{N}}^{N} form an orthonormal basis of H^{0}(M, L^{N}). Pick a local holomorphic frame e_{L} for L over an open subset U ⊂ M, let e_{L}^{∗} denote the dual frame, and write h(z) = h(e_{L}(z), e_{L}(z)) = ∥e_{L}∥_{h}^{2}. The map (z, e^{iθ}) ↦ e^{iθ}h(z)^{1/2}e_{L}^{∗}(z) gives an isomorphism U × S^{1} ≈ π^{−1}(U) ⊂ X, and we use the coordinates (z, θ) to identify points of π^{−1}(U). For s ∈ H^{0}(M, L^{N}), we have

ŝ(z, θ) = ⟨s(z), e^{iNθ}h(z)^{N/2}e_{L}^{∗}(z)⟩ = e^{iNθ}h(z)^{N/2}f(z), s = fe_{L}^{⊗N} . (12)

Although the Szegö kernel is defined on X, its absolute value is well-defined on M as follows: writing S_{j}^{N} = f_{j}^{N}e_{L}^{⊗N}, we have

Π_{N}(z, θ; w, φ) = e^{iN(θ−φ)}Π_{N}(z, 0; w, 0) = e^{iN(θ−φ)}h(z)^{N/2}h(w)^{N/2} ∑_{j=1}^{d_{N}} f_{j}^{N}(z)\overline{f_{j}^{N}(w)} , (13)

for z, w ∈ U. (Here we may take U to be the disjoint union of connected neighborhoods of z and w, if z is not close to w.) Thus we can write

|Π_{N}(z, w)| = |Π_{N}(z, 0; w, 0)| ,

which is independent of the choice of local frame e_{L}. On the diagonal we have

Π_{N}(z, z) = Π_{N}(z, θ; z, θ) = ∑_{j=1}^{d_{N}} ∥S_{j}^{N}(z)∥_{h^{N}} .

The Hermitian connection ∇ on L induces the decomposition T_{X} = T_{X}^{H}⊕T_{X}^{V} into horizontal and vertical components, and we let t^{H} denote the horizontal lift (to X) of a vector field t in M. We consider the horizontal operators on X:

d_{z_{q}}^{H} =^{def} d_{(∂/∂z_{q})^{H}} , dH ; z̄_{q} =^{def} d_{(∂/∂z̄_{q})^{H}} ,

where z_{1}, …, z_{m}, θ denote local coordinates on X. We note that

d_{z_{q}}^{H} ŝ = (∇_{z_{q}}^{N}s) , s ∈ H^{0}(M, L^{N}) , (14)

where ∇^{N} is the induced connection on L^{N}. We then have

d_{z_{q}}^{H}Π_{N}(z, θ; w, φ) = ∑_{j=1}^{d_{N}} (∇_{z_{q}}^{N}S_{j}^{N}) (x)\overline{Ŝ_{j}^{N}(y)}

= e^{iN(θ−φ)}h(z)^{N/2}h(w)^{N/2} ∑_{j=1}^{d_{N}} f_{j;q}^{N}(z)\overline{f_{j}^{N}(w)} ,

(15)  d_{z_{p}}^{H}dH ; w̄_{q}Π_{N}(z, θ; w, φ) = ∑_{j=1}^{d_{N}} (∇_{z_{p}}^{N}S_{j}^{N}) (x)\overline{(∇_{w_{q}}^{N}S_{j}^{N}) (y)}

= e^{iN(θ−φ)}h(z)^{N/2}h(w)^{N/2} ∑_{j=1}^{d_{N}} f_{j;p}^{N}(z)\overline{f_{j;q}^{N}(w)} ,

∇_{z_{q}}^{N} = ∇_{∂/∂z_{q}}^{N} , f_{j;q}^{N} = \frac{∂f}{∂z_{q}} + Nf_{j}^{N}h^{−1} \frac{∂h}{∂z_{q}} .

We can also use (12) and (14) to describe the horizontal lift in local coordinates:

d_{z_{q}}^{H} = \frac{∂}{∂z_{q}} − \frac{i}{2} \frac{∂ log h}{∂z_{q}} \frac{∂}{∂θ} . (16)

1.3. Model examples. In two special cases we can work out the Szegö kernels and their derivatives explicitly, namely for the hyperplane section bundle over ℂℙ^{m} and for the Heisenberg bundle over ℂ^{m}, i.e. the trivial line bundle with curvature equal to the standard symplectic form on ℂ^{m}. These cases will be important after we have proven universality, since scaling limits of correlation functions for all line bundles coincide with those of the model cases.

In fact, the two models are locally equivalent in the CR sense. In the case of ℂℙ^{m}, the circle bundle X is the 2m + 1 sphere S^{2m+1}, which is the boundary of the unit ball B^{2m+2} ⊂ ℂ^{m+1}. In the case of ℂ^{m}, the circle bundle is the reduced Heisenberg group H_{red}^{m}, which is a discrete quotient of the simply connected Heisenberg group ℂ^{m} × ℝ. As is well-known, the latter is equivalent (in the CR and contact sense) to the boundary of B^{2m+2} ([St]).

1.3.1. SU(m + 1)-polynomials. For our first example, we let M = ℂℙ^{m} and take L to be the hyperplane section bundle O(1). Sections s ∈ H^{0}(ℂℙ^{m}, O(1)) are linear functions on ℂ^{m+1}; the zero divisors Z_{s} are projective hyperplanes. The line bundle O(1) carries a natural metric h_{FS} given by

∥s∥_{h_{FS}}([w]) = \frac{|(s, w)|}{|w|} , w = (w_{0}, …, w_{m}) ∈ ℂ^{m+1} , (17)

for s ∈ ℂ^{m+1∗} ≡ H^{0}(ℂℙ^{m}, O(1)), where |w|^{2} = ∑_{j=0}^{m} |w_{j}|^{2} and [w] ∈ ℂℙ^{m} is the complex line through w. The Kähler form on ℂℙ^{m} is the Fubini-Study form

(18) ω_{FS} = \frac{\sqrt{−1}}{2}Θ_{h_{FS}} = \frac{\sqrt{−1}}{2}∂ ¯∂log |w|^{2} .

The dual bundle L^{∗} = O(−1) is the affine space ℂ^{m+1} with the origin blown up, and X = S^{2m+1} ⊂ ℂ^{m+1}. The N-th tensor power of O(1) is denoted O(N). Elements s_{N} ∈ H^{0}(ℂℙ^{m}, O(N)) are homogeneous polynomials on ℂ^{m+1} of degree N, and ŝ_{N} = s_{N}|_{S^{2m−1}}. The monomials

s_{J}^{N} = [\frac{(N + m)!}{π^{m}j_{0}! ⋯ j_{m}!}]^{\frac{1}{2}} z^{J} , z^{J} = z_{0}^{j_{0}} ⋯ z_{m}^{j_{m}}, J = (j_{0}, …, j_{m}), |J| = N (19)

form an orthonormal basis for H^{0}(ℂℙ^{m}, O(N)). (See [SZ, §4.2]; the extra factor (\frac{m!}{π^{m}} )^{1/2} in (19) comes from the fact that here ℂℙ^{m} has the usual volume \frac{π^{m}}{m!} , whereas in [SZ], the volume of ℂℙ^{m} is normalized to be 1.) Hence the Szegö kernel for O(N) is given by

Π_{N}(x, y) = ∑_{J} \frac{(N + m)!}{π^{m}j_{0}! ⋯ j_{m}!}x^{J}ȳ^{J} = \frac{(N + m)!}{π^{m}N!}⟨x, y⟩^{N} . (20)

Note that

Π(x, y) = ∑_{N=1}^{∞} Π_{N}(x, y) = \frac{m!}{π^{m}} (1 − ⟨x, y⟩)^{−(m+1)} = 2π × [classical Szegö kernel on S^{2m+1}] .

(The factor 2π is due to our normalization (9).)

1.3.2. The Heisenberg model. Our second example is the linear model ℂ^{m} × ℂ → ℂ^{m} for positive line bundles L → M over Kähler manifolds and their associated Szegö kernels. It is most illuminating to consider the associated principal S^{1} bundle ℂ^{m} × S^{1} → ℂ^{m}, which may be identified with the boundary of the disc bundle D ⊂ L^{∗} in the dual line bundle. This S^{1} bundle is the reduced Heisenberg group H_{red}^{m} (cf. [Fo], p. 23).

Let us recall its definition and properties. We start with the usual (simply connected) Heisenberg group H^{m} (cf. [Fo] [St]; note that different authors differ by factors of 2 and π in various definitions). It is the group ℂ^{m} × ℝ with group law

(ζ, t) · (η, s) = (ζ + η, t + s + ℑ(ζ · η̄)).

The identity element is (0, 0) and (ζ, t)^{−1} = (−ζ, −t). Abstractly, the Lie algebra of H_{m} is spanned by elements Z_{1}, …, Z_{m}, Z̄_{1}, …, Z̄_{m}, T satisfying the canonical commutation relations [Z_{j}, Z̄_{k}] = −iδ_{jk}T (all other brackets zero). Below we will select such a basis of left invariant vector fields.

H^{m} is a strictly convex CR manifold which may be embedded in ℂ^{m+1} as the boundary of a strictly pseudoconvex domain, namely the upper half space U^{m} := {z ∈ ℂ^{m+1} : ℑz_{m+1} > \frac{1}{2} ∑_{j=1}^{m} |z_{j}|^{2}\}. The boundary of U^{m} equals ∂U^{m} = \{z ∈ ℂ^{m+1} : ℑz_{m+1} = \frac{1}{2} ∑_{j=1}^{m} |z_{j}|^{2}\}. H^{m} acts simply transitively on ∂U^{m} (cf. [St], XII), and we get an identification of H^{m} with ∂U^{m} by:

[ζ, t] → (ζ, t + i|ζ|^{2}) ∈ ∂U^{m}.

The Szegö projector of H^{m} is the operator Π : L^{2}(H^{m}) → H^{2}(H^{m}) of orthogonal projection onto boundary values of holomorphic functions on U^{m} which lie in L^{2}. The kernel of Π is given by (cf. [St], XII §2 (29))

Π(x, y) = K(y^{−1}x), K(x) = −C_{m} \frac{∂}{∂t}[t + i|ζ|^{2}]^{−m} ∈ D′(H^{m}) .

The linear model for the principal S^{1} bundle described in §1.2 is the so-called reduced Heisenberg group H_{red}^{m} = H^{m}/{(0, 2πk) : k ∈ ℤ} = ℂ^{m} × S^{1} with group law

(ζ, e^{it}) · (η, e^{is}) = (ζ + η, e^{i(t+s+ℑ(ζ·η̄))}).

It is the principal S^{1} bundle over ℂ^{m} associated to the line bundle L_{H} = ℂ^{m} ×ℂ. The metric on L_{H} with curvature Θ = ∂ ¯∂|z|^{2} is given by setting h_{H}(z) = e^{−|z|^{2}}; i.e., |f|_{h_{H}} = |f|e^{−|z|^{2}/2}.

The reduced Heisenberg group H_{red}^{m} may be viewed as the boundary of the dual disc bundle D ⊂ L_{H}^{∗} and hence is a strictly pseudoconvex CR manifold.

It seems most natural to approach the analysis of the Szegö kernels on H_{red}^{m} from the representation-theoretic point of view. Let us begin with the case N = 1. We thus consider the space V_{1} ⊂ L^{2}(H_{red}^{m}) of functions f satisfying \frac{1}{i} \frac{∂}{∂θ}f = f, which forms a (reducible) representation of H_{red}^{m} with central character e^{iθ}. By the Stone-von Neumann theorem there exists a unique (up to equivalence) representation (V_{1}, ρ_{1}) with this character and by the Plancherel theorem, V_{1} ≅ V_{1} ⊗ V_{1}^{∗}.

The space of CR functions in V_{1} is an irreducible invariant subspace. Here, by CR functions we mean the functions satisfying the left-invariant Cauchy-Riemann equations Z̄_{q}^{L}f = 0 on H_{red}^{m}. Here, {Z̄_{q}^{L}} denotes a basis of the left-invariant anti-holomorphic vector fields on H_{red}^{m}. Let us recall their definition: we first equip H_{red}^{m} with its left-invariant contact form α^{L} = ∑_{q}(u_{q}dv_{q} − v_{q}du_{q}) + dθ (ζ = u + iv). The left-invariant CR holomorphic (resp. anti-holomorphic) vector fields Z_{q}^{L} (resp. Z̄_{q}^{L}) are the horizontal lifts of the vector fields \frac{∂}{∂z_{q}} (resp. \frac{∂}{∂z̄_{q}}) with respect to α^{L}. They span the left-invariant CR structure of H_{red}^{m} and the Z_{q}^{L} obviously have the form Z_{q}^{L} = \frac{∂}{∂z_{q}} + A\frac{∂}{∂θ} where the coefficient A is determined by the condition α^{L}(Z_{q}^{L}) = 0. An easy calculation gives:

Z_{q}^{L} = \frac{∂}{∂z_{q}} + \frac{i}{2}z̄_{q} \frac{∂}{∂θ}, Z̄_{q}^{L} = \frac{∂}{∂z̄_{q}} − \frac{i}{2}z_{q} \frac{∂}{∂θ}.

The vector fields \{\frac{∂}{∂θ}, Z_{q}^{L}, Z̄_{q}^{L}\} span the Lie algebra of H_{red}^{m} and satisfy the canonical commutation relations above.

We then define the Hardy space H^{2}(H_{red}^{m}) of CR holomorphic functions, i.e. solutions of Z̄_{q}^{L}f = 0, which lie in L^{2}(H_{red}^{m}). We also put H_{1}^{2} = V_{1}∩H^{2}(H_{red}^{m}). The group H_{red}^{m} acts by left translation on H_{1}^{2}. The generators of this representation are the right-invariant vector fields Z_{q}^{R}, Z̄_{q}^{R} together with \frac{∂}{∂θ}. They are horizontal with respect to the right-invariant contact form α^{R} = ∑_{q}(u_{q}dv_{q} − v_{q}du_{q}) − dθ and are given by:

Z_{q}^{R} = \frac{∂}{∂z_{q}} − \frac{i}{2}z̄_{q} \frac{∂}{∂θ}, Z̄_{q}^{R} = \frac{∂}{∂z̄_{q}} + \frac{i}{2}z_{q} \frac{∂}{∂θ} .

In physics terminology, Z_{q}^{R} is known as an annihilation operator and Z̄_{q}^{R} is a creation operator.

The representation H_{1}^{2} is irreducible and may be identified with the Bargmann-Fock space of entire holomorphic functions on ℂ^{m} which are square integrable relative to e^{−|z|^{2}} (or equivalently, holomorphic sections of the trivial line bundle L_{H} = ℂ^{m} × ℂ mentioned above, with hermitian metric h_{H} = e^{−|z|^{2}}). The identification goes as follows: the function φ_{0}(z, θ) := e^{iθ}e^{−|z|^{2}/2} is CR holomorphic and is also the ground state for the right invariant “annihilation operator;” i.e., it satisfies

Z̄_{q}^{L}φ_{0}(z, θ) = 0 = Z_{q}^{R}φ_{0}(z, θ) .

Any element F (z, θ) of H_{1}^{2} may be written in the form F (z, θ) = f(z)φ_{0}. Then Z̄_{q}^{L}F = (\frac{∂}{∂z̄_{q}} f)φ_{0}, so that F is CR if and only if f is holomorphic. Moreover, F ∈ L^{2}(H_{red}^{m}) if and only if f is square integrable relative to e^{−|z|^{2}}.

The Szegö kernel Π_{1}^{H}(z, θ, w, φ) of H_{red}^{m} is by definition the orthogonal projection from L(H_{red}^{m}) to H_{1}^{2}. As will be seen below, Π_{1}^{H}(z, θ, w, φ) = \frac{1}{π_{m}} e^{i(θ−φ)}e^{(z·w̄− \frac{1}{2} |z|^{2}− \frac{1}{2} |w|^{2})}, which is the left translate of φ_{0} by (−w, −φ). In the physics terminology it is the coherent state associated to the phase space point w.

So far we have set N = 1, but the story is very similar for any N. We define H_{N}^{2} as the space of square- integrable CR functions transforming by e^{iNθ} under the central S^{1}. By the Stone-von Neumann theorem there is a unique irreducible V_{N} with this central character. The main difference to the case N = 1 is that H_{N}^{2} is of multiplicity N^{m}. The Szegö kernel Π_{N}^{H}(x, y) is the orthogonal projection to H_{N}^{2} and is given by the dilate of Π_{1}^{H}. Thus,

Π_{N}^{H}(x, y) = \frac{1}{π^{m}} N^{m}e^{iN(θ−φ)}e^{N(z·w̄− \frac{1}{2} |z|^{2}− \frac{1}{2} |w|^{2})}.

To prove these formulae for the Szegö kernels, we observe that the reduced Szegö kernels are obtained by projecting the Szegö kernel on H^{m} to H_{red}^{m} as an automorphic kernel, i.e.

Π^{H}(x, y) = ∑_{n∈ℤ}Π(x, y · (0, 2πn)).

Let us write x = (z, θ), y = (w, φ). Then the N-th Fourier component Π_{N}^{H}(x, y) of Π^{H}, i.e. the projection onto square integrable holomorphic sections of L^{N}, is given by:

Π_{N}^{H}(x, y) = ∫_{ℝ} e^{−iNt}Π(e^{it}x, y)dt = ∫_{ℝ} e^{−iNt}K(e^{it}y^{−1}x)dt
= ∫_{ℝ} e^{−iNt}K(z − w, e^{i(θ−φ+t+ℑ(z·w̄)})dt .

Here we abbreviated the element (0, e^{it}) by e^{it}. Change variables t ↦ t − θ + φ − ℑ(z · w̄) to get

\begin{matrix}Π_{N}^{H}(x, y) = e^{iN(θ−φ)}e^{iNℑ(z·w̄}) ∫_{ℝ} e^{−iNt}K(z − w, t)dt \\ = e^{iN(θ−φ)}e^{iNℑ(z·w̄)}K̂_{t}(z − w, N) \end{matrix}

where K̂_{t} is the Fourier transform of K with respect to the t variable. By [St, p. 585], the full ℝ^{2m} × ℝ Fourier transform of K is given by K̂(z, N) = C′_{m}e^{−|z|^{2}/2N}, so by taking the inverse Fourier transform in the z variable we get the Fourier transform just in the t variable:

Π_{N}^{H}(x, y) = \frac{1}{π^{m}} N^{m}e^{iN(θ−φ)}e^{iNℑ(z·w̄)}e^{− \frac{1}{2} N|z−w|^{2}}. (21)

(Our constant factor \frac{1}{π^{m}} in (21) is determined by the condition that Π_{N}^{H} is an orthogonal projection.)

In our study of the correlation functions, we will need explicit formulae for the horizontal derivatives of the Szegö kernel. The left-invariant derivatives are given by

N^{−m}Z_{q}^{L}Π_{N}^{H}(z, θ; w, φ) = N(w̄_{q} − z̄_{q})Π_{N}^{H}(z, θ; w, φ) ,

(22)

N^{−m}Z_{q}^{L}W̄_{q′}^{L}Π_{N}^{H}(z, θ; w, φ) = N^{2}(z_{q′} − w_{q′})(w̄_{q} − z̄_{q})Π_{N}^{H}(z, θ; w, φ) + Nδ_{qq′}Π_{N}^{H}(z, θ; w, φ) .

Comparing the definitions of the horizontal vector fields with (14), using h_{H} = e^{−|z|^{2}}, we see that d_{z_{q}}^{H} = Z_{q}^{L}, as expected, since α^{L} agrees with the contact form α for L_{H} (as defined in §1.2). We will see later that our formulas for computing correlations are valid with any connection, and thus it is sometimes useful to also consider the right invariant derivatives:

N^{−m}Z_{q}^{R}Π_{N}^{H}(z, θ; w, φ) = Nw̄_{q}Π_{N}^{H}(z, θ; w, φ) ,

(23)

N^{−m}Z_{q}^{R}W̄_{q′}^{R}Π_{N}^{H}(z, θ; w, φ) = N^{2}z_{q′} w̄_{q}Π_{N}^{H}(z, θ; w, φ) + Nδ_{qq′}Π_{N}^{H}(z, θ; w, φ) .

Remark: Recall that the metric on O(N) → ℂℙ^{m} is given by h^{N}(z) = (1 + |z|^{2})^{−N} using the coordinates and local frame from Example 1.3.1. Since

h^{N}(z/\sqrt{N}) → h_{H}(z) ,

the Heisenberg bundle can be regarded as the scaling limit of O(N). (Of course, in the same way L_{H} is the scaling limit of L^{N}, for any positive line bundle L → M.)

2. Correlation functions

This section begins with a generalization to arbitrary dimension and codimension a formula of [Han] and [BD] for the “correlation density function” in the one-dimensional case. In fact, our formula (Theorem 2.1) applies to a general class of probability spaces of k-tuples of (real or complex) functions. We then specialize to the case where the space of sections has a Gaussian measure. Finally, we show how the correlations of the zeros of k-tuples of sections of the N-th power of a holomorphic line bundle are given by a rational function in the Szegö kernel Π_{N} and its derivatives (Theorem 2.4).

2.1. General formula for zero correlations. For our general setting, we let (V, h) be a Hermitian holomorphic vector bundle on an m-dimensional Hermitian complex manifold (M, g). (Here, we make no curvature assumptions.) Suppose that S is a finite dimensional subspace of the space H^{0}(M, V) of global holomorphic sections of V, and let dμ be a probability measure on S given by a semi-positive C^{0} volume form that is strictly positive in a neighborhood of 0 ∈ S. (We shall later apply our results to the case where V = L^{N} ⊕⋯⊕L^{N}, for a holomorphic line bundle L over a compact complex manifold M, and S = H^{0}(M, V) with a Gaussian measure dμ. Our formulation involving general vector bundles allows us to reduce the study of n-point correlations to the case n = 1, i.e., to expected densities of zeros.)

As in the introduction, we introduce the punctured product

M_{n} = \{(z^{1}, …, z^{n}) ∈ \underbrace{M × ⋯ × M}_{n} : z^{p} ≠ z^{q} for p ≠ q\} ,

and we write

s(z) = (s(z^{1}), …, s(z^{n})) , ∇s(z) = (∇s(z^{1}), …, ∇s(z^{n})) , z = (z^{1}, …, z^{n}) ∈ M_{n} ,

where ∇s(ζ) ∈ T_{ζ}^{∗} ⊗ V_{ζ} is the covariant derivative with respect to the Hermitian connection on V. We define the map

J : M_{n} × S → [(ℂ ⊕ T_{M}^{∗}) ⊗ V ]^{n} , J (z, s) = (s(z), ∇s(z)) ;

i.e., J (z, s) is the 1-jet of s at z ∈ M_{n}.

We write g = ℜ ∑ g_{qq′}dz_{q} ⊗ dz̄_{q′}, h_{jj′} = h(e_{j}, e_{j,}), where {z_{1}, …, z_{m}} are local coordinates in M and {e_{1}, …, e_{k}} is a local frame in V (m = dim M, k = rank V). We let G = det(g_{qq′}), H = det(h_{jj′}). We let

dζ = \frac{1}{m!}ω_{ζ}^{m} = G(ζ) ∏_{j=1}^{m} dℜζ_{j}dℑζ_{j} , ζ ∈ M

denote Riemannian volume in M, and we write

x^{p} = ∑_{j} b_{j}^{p}e_{j}(z^{p}), dx^{p} = H(z^{p}) ∏_{j} dℜb_{j}^{p}dℑb_{j}^{p} x^{p} ∈ V_{z^{p}} , (24)

ξ^{p} = ∑_{j,q} a_{jq}^{p}dz_{q} ⊗ e_{j}|_{z^{p}}, dξ^{p} = G(z^{p})^{−k}H(z^{p})^{m} ∏_{j,q} dℜa_{jq}^{p}dℑa_{jq}^{p} ξ_{j} ∈ (T_{M}^{∗} ⊗ V)_{z^{p}} .

The quantities dx^{p}, dξ^{p} are the intrinsic volume measures on V_{z^{p}} and (T_{M}^{∗} ⊗V)_{z^{p}}, respectively, induced by the metrics g, h.

Definition: Suppose that J is surjective. We define the n-point density D_{n}(x, ξ, z)dxdξdz of μ by

J_{∗}(dz × dμ) = D_{n}(x, ξ, z)dxdξdz , x = (x^{1}, …, x^{n}) ∈ V_{z^{1}} × ⋯ × V_{z^{n}} , (25)

ξ = (ξ^{1}, …, ξ^{n}) ∈ (T_{M}^{∗} ⊗ V)_{z^{1}} × ⋯ × (T_{M}^{∗} ⊗ V)_{z^{n}} , z = (z^{1}, …, z^{n}) ∈ M_{n} ,
dx = dx^{1} ⋯ dx^{n} , dξ = dξ^{1} ⋯ dξ^{n} , dz = dz^{1} ⋯ dz^{n} .

In this case, for each z ∈ M_{n}, the (vector-valued) random variable (s(z), ∇s(z)) has (joint) probability distribution D_{n}(x, ξ, z)dxdξ.

Remark: If we let n = 1 and fix a point z ∈ M, then the measure D(x, ξ, z)dxdξ is intrinsically defined as a measure on the space J_{z}^{1}(M, V) of 1-jets of sections of V at z. Taking a section to its 1-jet at z defines a map J_{z} : S → J_{z}^{1}(M, V) and hence induces a measure J_{z∗}μ on J_{z}^{1}(M, V) independently of any choices of connections, coordinates or metrics. Similarly for n > 1, D(x, ξ, z)dxdξ is an intrinsic measure on ∏_{p=1}^{n} J_{z_{p}}^{1}(M, V).

For a vector-valued 1-form ξ ∈ T_{M,z}^{∗} ⊗ V_{z} = Hom(T_{M,z}, V_{z}), we let ξ^{∗} ∈ Hom(V_{z}, T_{M,z}) denote the adjoint to ξ (i.e., ⟨ξ^{∗}v, t⟩ = ⟨v, ξt⟩), and we consider the endomorphism ξξ^{∗} ∈ Hom(V_{z}, V_{z}). In terms of local frames, if

ξ = ∑_{j} ξ_{j} ⊗ e_{j} = ∑_{j,q} a_{jq}dz_{j} ⊗ e_{j} ,

then

ξ^{∗} = ∑_{j,q} α_{jq} \frac{∂}{∂z_{q}} ⊗ e_{j}^{∗} , α_{jq} = ∑_{j′,q′} h_{jj′}γ_{q′q}ā_{j′q′} ,

where (γ_{qq′}) = (g_{qq′})^{−1}; hence we have

ξξ^{∗} = ∑_{j,j′,j′′,q,q′} h_{j′j′′}a_{jq}γ_{q′q}ā_{j′′q′} e_{j} ⊗ e_{j′}^{∗} . (26)

Its determinant is given by

(27) det(ξξ^{∗}) = H det (∑_{q,q′} a_{jq}γ_{q′q}ā_{j′q′})_{1≤j,j′≤k} = H det⟨ξ_{j}, ξ_{j′}⟩ = H∥ξ_{1} ∧ ⋯ ∧ ξ_{k}∥^{2} .

Remark: The measure det(ξξ^{∗})D(0, ξ, z)dξdz will play a fundamental role in our study of correlation functions. We observe here that it depends only on the metric ω on M, and in the case where the zero sets are points (k = m), it is independent of the choice of metric on M as well. Indeed, as mentioned in the previous remark, D(x, ξ, z)dxdξ is well-defined on J_{z}^{1}(M, V). The conditional density D(0, ξ, z)dξ equals J_{z∗}μ/dx|_{x=0} and thus depends only on the choice of volume forms dx^{p} on V_{z^{p}}. Since dz/dx transforms in the opposite way to det ξξ^{∗} it follows that det(ξξ^{∗})D(0, ξ, z)dξdz is an invariantly defined measure on (T_{M}^{∗} ⊗V)^{n}.

Recall that for s ∈ S so that codim Z_{s} = k, we let |Z_{s}| denote Riemannian (2m − 2k)- volume along the regular points of Z_{s}, regarded as a measure on M.

Definition: For s ∈ S so that codim Z_{s} = k, we consider the product measure on M_{n},

|Z_{s}|^{n} = ( \underbrace{|Z_{s}| × ⋯ × |Z_{s}|}_{n} ) .

Its expectation E |Z_{s}|^{n} is called the n-point zero correlation measure.

We shall use the following general formula to compute the correlations of zeros and to show universality of the scaling limit:

Theorem 2.1. Let M, V, S, dμ be as above, and suppose that J is surjective and the volumes |Z_{s}| are locally uniformly bounded above. Then

E |Z_{s}|^{n} = K_{n}(z)dz , K_{n}(z) = ∫ dξ D_{n}(0, ξ; z) ∏_{p=1}^{n} det (ξ^{p}ξ^{p∗}) . (28)

The function K_{n}(z^{1}, …, z^{n}), which is continuous on M_{n} is called the n-point zero correlation function. For k < m, (28) holds on all of the n-fold product M × ⋯ × M, including the diagonal, and K_{n} is locally integrable on M × ⋯ × M (and is infinite on the diagonal). In the case k = m, when the zero sets are discrete, the zero correlation measure on M ×⋯×M is the sum of the absolutely continuous measure K_{n}(z)dz plus a measure supported on the diagonal.

Proof of Theorem 2.1: Consider the Hermitian vector bundle V_{n} = ⨁_{p=1}^{n} π_{p}^{∗}V ⟶ M_{n}, where π_{p} : M_{n} → M denotes the projection onto the p-th factor. By replacing V → M with V_{n} → M_{n} and s ∈ H^{0}(M, V) with

s̃(z^{1}, …, z^{n}) = (s(z^{1}), …, s(z^{n})) ∈ H^{0}(M_{n}, V_{n}) ,

and noting that T_{M_{n},z} = ∏_{p} T_{M,z^{p}} and |Z_{s}|^{n} = |Zs̃|, we can assume without loss of generality that n = 1.

It follows from the above remarks that D(0, ξ; z) does not depend on the choice of connection on V. We can also verify this in terms of local coordinates: write s = ∑ b_{j}e_{j}, ∇s = ∑ a_{jq}dz_{q} ⊗ e_{j} as in (25); we have a_{jq} = \frac{∂b_{j}}{∂z_{q}} + ∑_{k} b_{k}θ_{jq}^{k}. Then if we write a_{jq}^{0} = \frac{∂b_{j}}{∂z_{q}}, we have

\frac{∂(a_{jq}, b_{j})}{∂(a_{jq}^{0}, b_{j})} = 1 .

Hence D(0, ξ; z) is unchanged if we substitute the (local) flat connection given by a_{jq}^{0}.

We now restrict to a coordinate neighborhood U ⊂ M where V has a local frame {e_{j}}. By hypothesis, we can suppose that the e_{j} are restrictions of sections in S. We write s = ∑ s_{j}e_{j}, and by the above we may assume that ∇s = ∑ ds_{j} ⊗ e_{j}. We use the notation

|||ξ||| = \sqrt{det(ξξ^{∗})} , for ξ ∈ T_{M,z}^{∗} ⊗ V_{z} = Hom(T_{M,z}, V_{z}) .

Then by (27),

|||∇s|||^{2} = H∥ds_{1} ∧ ⋯ ∧ ds_{k}∥^{2} = ∥Ψ∥ ,

where Ψ is the (k, k)-form on U given by:

Ψ = H (\frac{i}{2}ds_{1} ∧ \overline{ds_{1}}) ∧ ⋯ ∧ (\frac{i}{2}ds_{k} ∧ \overline{ds_{k}}) .

Thus, by the Leray formula,

|Z_{s}| = ∥ds_{1} ∧ ⋯ ∧ ds_{k}∥^{2} \frac{dz}{\frac{i}{2}ds_{1} ∧ ds̄_{1} ⋯ ∧ \frac{i}{2}ds_{k} ∧ ds̄_{k}} |_{Z_{s}} = |||∇s|||^{2} \frac{dz}{Ψ} |_{Z_{s}} , (29)

UNIVERSALITY AND SCALING OF CORRELATIONS BETWEEN ZEROS 19

Define the measure λ on M × S by (λ, φ) = ∫_{S} (|Z_{s}|, φ(z, s)) dμ(s) . (30) Then

π_{∗}λ = E |Z_{s}|^{n} ,

where π : M × S → M is the projection. Hence,

λ = ∫_{S} dμ(s) |Z_{s}| = ∫_{S} dμ(s) (|||∇s|||^{2} \frac{dz}{Ψ} )|_{Z_{s}} . (31)

For (almost all) x ∈ ℂ^{k}, let I(s, x) be the measure on U given by

(I(s, x), φ) = ∫_{s(z)=∑ x_{j}e_{j}(z)} φ(z)dVol_{(2m−2k)n}(z) = ∫_{s(z)=∑ x_{j}e_{j}(z)} |||∇s|||^{2} \frac{dz}{Ψ} φ(z) , φ ∈ C^{0}(U) ,

where the second equality is by (29) applied to s − ∑ x_{j}e_{j}(z). Then ∫ I(s, x)dx = |||∇s(z)|||^{2}dz . (32) Now let λ_{x} be the measure on U given by

(λ_{x}, φ) = ∫_{S}(I(s, x), φ)dμ(s) .

Claim: The map x ↦ (λ_{x}, φ) is continuous. To prove this claim, we first note that the hypothesis that |Z_{s}| is locally uniformly bounded implies that (I(s, x), φ) ≤ C < +∞ uniformly in s, x. Thus we can assume without loss of generality that μ has compact support in S. By hypothesis, the map

σ : U × S → ℂ^{k}, σ(z, s) = (s_{1}(z), …, s_{k}(z))

is a submersion. We can now write λ_{x} as a fiber integral of a compactly supported C^{0} form:

λ_{x} = \frac{1}{(m − k)!} ∫_{σ^{−1}(x)} φ(z)ω^{m−k}(z) ∧ dμ(s) ,

and thus λ_{x} is continuous, verifying the claim.

We note that λ_{0} = λ|_{U}. Hence, to complete the proof, we must show that π_{∗}λ_{0} = K_{1}(z)dz|_{U} .

By (25) and (32), for a test function φ(x, ξ, z),

∫ φ(x, ξ, z)|||ξ|||^{2}D_{1}(x, ξ, z)dxdξdz = ∫ dμ(s) ∫ φ(J (z, s))|||∇s(z)|||^{2}dz
= ∫ dx ∫ (I(s, x), φ ∘ J)dμ(s)
= ∫ (λ_{x}, φ ∘ J)dx .

By choosing φ(x, ξ, z) = ρ_{ε}(x)ψ(z), where ρ_{ε} is an approximate identity, and letting ε → 0, we conclude that

∫ ψ(z)K_{1}(z)dz = ∫ ψ(z)|||ξ|||^{2}D_{1}(0, ξ, z)dξdz = (λ_{0}, ψ(z)) .

We note the following analogous formula for real manifolds:

Theorem 2.2. Let V be a C^{∞} real vector bundle over a C^{∞} Riemannian manifold M, and let μ be a probability measure on a finite dimensional vector space S of C^{∞} sections of V given by a semi-positive volume form that is strictly positive at 0. Suppose that the volumes |Z_{s}| are locally uniformly bounded above. Let D_{n}(x, ξ, z)dxdξdz denote the n-point density of μ. Then

E |Z_{s}|^{n} = K_{n}(z)dz , K_{n}(z) = ∫ dξ D_{k}(0, ξ, z) ∏_{p=1}^{n} \sqrt{det(ξ^{p}ξ^{p∗})} . (33)

The proof is similar to that of Theorem 2.1, except that (29) is replaced by the Leray formula

|Z_{s}| = ∥ds_{1} ∧ ⋯ ∧ ds_{k}∥ \frac{dζ}{ds_{1} ∧ ⋯ ∧ ds_{k}} |_{Z_{s}} (34)

in the real case.

2.2. Formula for Gaussian densities. We now specialize our formula from Theorem 2.1 to the case where μ is a Gaussian measure. Fix z = (z^{1}, …, z^{n}) ∈ M_{n} and choose local coordinates {z_{q}^{p}} and local frames {e_{j}^{p}} near z^{p}, p = 1, …, n. We consider the random variables b_{j}^{p}, a_{jq}^{p} given by

s(z^{p}) = ∑_{j=1}^{k} b_{j}^{p}e_{j}^{p}, ∇s(z^{p}) = ∑_{j=1}^{k} ∑_{q=1}^{m} a_{jq}^{p}dz_{q}^{p} ⊗ e_{j}^{p}, p = 1, …, n. (35)

By (4)–(5) and (24)–(25) the n-point density

D_{n}(x, ξ, z)dxdξdz = D_{n} [∏_{p=1}^{n} G(z^{p})^{−k}H(z^{p})^{m}] dbdadz

is given by:

(36)  D_{n}(b, a; z) = \frac{exp⟨−Δ_{n}^{−1}v, v⟩}{π^{kn(1+m)} det Δ_{n}} ,  v = (\begin{matrix}b \\ a \end{matrix}) ,

where

Δ_{n} = ( \begin{matrix}A_{n} & B_{n} \\ B_{n}^{∗} & C_{n} \end{matrix} ) (37)

A_{n} = (A_{j′p′}^{jp}) = ( E b_{j}^{p}b̄_{j′}^{p′} ), B_{n} = (B_{j′p′q′}^{jp}) = ( E b_{j}^{p}ā_{j′q′}^{p′} ), C_{n} = (C_{j′p′q′}^{jpq}) = ( E a_{jq}^{p}ā_{j′q′}^{p′} );

j, j′ = 1, …, k; p, p′ = 1, …, n; q, q′ = 1, …, m.

(We note that A_{n}, B_{n}, C_{n} are kn×kn, kn×knm, knm×knm matrices, respectively; j, p, q index the rows, and j′, p′, q′ index the columns.)

The function D_{n}(0, a; z) is a Gaussian function, but it is not normalized as a probability density. It can be represented as

D_{n}(0, a; z) = Z_{n}(z)D_{Λ_{n}}(a; z), (38)

where

(39) D_{Λ_{n}}(a; z) = \frac{1}{π^{knm} det Λ_{n}} exp (−⟨Λ_{n}^{−1}a, a⟩)

is the Gaussian density with covariance matrix

Λ_{n} = C_{n} − B_{n}^{∗}A_{n}^{−1}B_{n} = (C_{j′p′q′}^{jpq} − ∑_{j_{1},p_{1},j_{2},p_{2}} B̄_{jpq}^{j_{1}p_{1}}Γ_{j_{2}p_{2}}^{j_{1}p_{1}}B_{j′p′q′}^{j_{2}p_{2}}) (Γ = A_{n}^{−1}) (40)

and

(41) Z_{n}(z) = \frac{det Λ_{n}}{π^{kn} det Δ_{n}} = \frac{1}{π^{kn} det A_{n}} .

This reduces formula (28) to

K_{n}(z) = \frac{1}{π^{kn} det A_{n}} ⟨∏_{p=1}^{n} det (a^{p∗}γ^{p}a^{p})⟩_{Λ_{n}} (42)

where ⟨·⟩_{Λ_{n}} stands for averaging with respect to the Gaussian density D_{Λ_{n}}(a; z), and (γ_{qq′}^{p}) = (g_{qq′}^{p})^{−1}, g_{qq′}^{p} = g_{qq′}(z^{p}).

2.3. Densities and the Szegö kernel. We return to our positive Hermitian line bundle (L, h) on a compact complex manifold M with Kähler form ω = \frac{i}{2}Θ_{h}. We now apply formulas (37)–(42) to the vector bundle

V = \underbrace{L^{N} ⊕ ⋯ ⊕ L^{N}}_{k}

and space of sections

S = H^{0}(M, V) = H^{0}(M, L^{N})^{k}

with the Gaussian measure μ = ν_{N} × ⋯ × ν_{N}, where ν_{N} is the standard Gaussian measure on H^{0}(M, L^{N}) given by (4). We denote the resulting n-point density by D_{nk}^{N}, and we also write Δ_{n} = Δ_{nk}^{N}, A_{n} = A_{nk}^{N}, etc.

As above, we fix z = (z^{1}, …, z^{n}) ∈ M_{n} and choose local coordinates {z_{q}^{p}} near z^{p}, p = 1, …, n. We also choose local frames {e_{L}^{p}} for L near the points z^{p} so that

∥e_{L}^{p}(z^{p})∥_{h} = 1 .

For s ∈ S, we write

s(z^{p}) = ( s_{1}(z^{p}) ; ⋮ ; s_{k}(z^{p}) ) = ( b_{1}^{p} ; ⋮ ; b_{k}^{p} ) (e_{L}^{p}(z^{p}))^{⊗N} , (43)

∇_{N}s_{j}(z^{p}) = ∑_{q=1}^{m} a_{jq}^{p}dz_{q}^{p} ⊗ (e_{L}^{p}(z^{p}))^{⊗N} . (44)

Since the s_{j} are independent and have identical distributions, we have

A_{nk}^{N} = (A_{j′p′}^{jp}) = (δ_{jj′}E (b_{1}^{p}b̄_{1}^{p′})) , B_{nk}^{N} = (B_{j′p′q′}^{jp}) = (δ_{jj′}E (b_{1}^{p}ā_{1q′}^{p′})) , (45)

C_{nk}^{N} = (C_{j′p′q′}^{jpq}) = (δ_{jj′}E (a_{1q}^{p}ā_{1q′}^{p′})) .

We write

s_{1} = ∑_{α=1}^{d_{N}} c_{α}S_{α}^{N} = (∑_{α=1}^{d_{N}} c_{α}f_{α}^{p}) (e_{L}^{p})^{⊗N} ,

where {S_{α}^{N}} is an orthonormal basis for H^{0}(M, L^{N}). Using the local coordinates (z^{p}, θ) in X as described in §1.1, we have by (45) and (13) (noting that h(z^{p}) = 0 by the above choice of local frames),

A_{j′p′}^{jp} = δ_{jj′} ∑_{α,β=1}^{d_{N}} E (c_{α}c̄_{β})f_{α}^{p}(z^{p})\overline{f_{β}^{p′}(z^{p′})} = δ_{jj′} ∑_{α=1}^{d_{N}} f_{α}^{p}(z^{p})\overline{f_{α}^{p′}(z^{p′})} = δ_{jj′}Π_{N}(z^{p}, 0; z^{p′}, 0) . (46)

Similarly,

B_{j′p′q′}^{jp} = δ_{jj′} ∑_{α=1}^{d_{N}} f_{α}^{p}(z^{p})\overline{f_{α;q′}^{p′}(z^{p′})} = δ_{jj′}dH ; w̄_{q′}Π_{N}(z^{p}, 0; z^{p′}, 0) , (47)

C_{j′p′q′}^{jpq} = δ_{jj′} ∑_{α=1}^{d_{N}} f_{α;q}^{p}(z^{p})\overline{f_{α;q′}^{p′}(z^{p′})} = δ_{jj′}d_{z_{q}}^{H}dH ; w̄_{q′}Π_{N}(z^{p}, 0; z^{p′}, 0) . (48)

Lemma 2.3. There is a positive integer N_{0} = N_{0}(M, n) such that

det (Π_{N}(z^{p}, 0; z^{p′}, 0))_{1≤p,p′≤n} ≠ 0 ,

for distinct points z^{1}, …, z^{n} of M and for all N ≥ N_{0}.

Proof. It is a well-known consequence of the Kodaira Vanishing Theorem (see for example, [GH]) that we can find N_{0} such that if N ≥ N_{0} and x_{1}, …, x_{n} ∈ M with x_{p} ≠ x_{1} for 2 ≤ p ≤ n, then there is a section s ∈ H^{0}(M, L^{N}) with s(x_{1}) ≠ 0 and s(x_{p}) = 0 for 2 ≤ p ≤ n.

We write Ã_{pp′} = Π_{N}(z^{p}, 0; z^{p′}, 0). Suppose on the contrary that det(Ã_{pp′}) = 0, and chose a nonzero vector v = (v_{1}, …, v_{n}) such that ∑_{p} v_{p}Ã_{pp′} = 0. Then recalling (11), we have

0 = ∑_{p,p′} v_{p}Ã_{pp′}v̄_{p′} = ∑_{p,p′,α} v_{p}Ŝ_{α}^{N}(z^{p}, 0)\overline{Ŝ_{α}^{N}(z^{p′}, 0)v_{p′}} = ∑_{α=1}^{d_{N}} |x_{α}|^{2} , (49)

where x_{α} = ∑_{p} v_{p}Ŝ_{α}^{N}(z^{p}, 0). Since the S_{α}^{N} span H^{0}(M, L^{N}), it follows that for all s ∈ H^{0}(M, L^{N}), we have ∑_{p} v_{p}ŝ(z^{p}) = 0. But this contradicts the fact that, choosing p_{0} with v_{p_{0}} ≠ 0, we can find a section s ∈ H^{0}(M, L^{N}) with s(z^{p_{0}}) ≠ 0 and s(z^{p}) = 0 for p ≠ p_{0}.

Thus we see that the n-point correlation functions depend only on the Szegö kernel, as follows:

Theorem 2.4. Let (L, h) be a positive Hermitian line bundle on an m-dimensional compact complex manifold M with Kähler form ω = \frac{i}{2}Θ_{h}, let S = H^{0}(M, L^{N})^{k} (k ≥ 1), and give S the standard Gaussian measure μ described above. Let n ≥ 1 and suppose that N is sufficiently large so that J is surjective. Let z = (z^{1}, …, z^{n}) ∈ M_{n} and choose local coordinates (ζ_{1}, …, ζ_{m}) at each point z^{p} such that Θ_{h}(z^{p}) = ∑_{q} dζ_{q} ∧ dζ̄_{q}, 1 ≤ p ≤ n. Then the n-point correlation K_{nk}^{N}(z) is given by a universal rational function, homogeneous of degree 0, in the values of Π_{N} and its first and second derivatives at the points (z^{p}, z^{p′}). Specifically,

(50) K_{nk}^{N}(z) = \frac{P_{nkm}(Π_{N}(z^{p}, z^{p′}), dH ; w̄_{q}Π_{N}(z^{p}, z^{p′}), d_{z_{q}}^{H}Π_{N}(z^{p}, z^{p′}), d_{z_{q}}^{H}dH ; w̄_{q′}Π_{N}(z^{p}, z^{p′}))}{π^{kn} [det (Π_{N}(z^{p}, z^{p′}))_{1≤p,p′≤n}]^{k(n+1)}}

(1 ≤ p, p′ ≤ n, 1 ≤ q, q′ ≤ m), where P_{nkm} is a universal homogeneous polynomial of degree kn(n + 1) with integer coefficients depending only on n, k, m.

Proof. The n-point zero correlation K_{nk}^{N}(z) is given by equation (42) with γ_{qq′}^{p} = δ_{qq′}. By the Wick formula ([Si, (I.13)]), the expectation

⟨∏_{p=1}^{n} det (a^{p∗}a^{p})⟩_{Λ_{n}}

in (42) is a homogeneous polynomial (over ℤ) of degree kn in the coefficients of Λ_{n}. By (40) and (46), the coefficients of det (Π_{N}(z^{p}, z^{p′}))Λ_{n} are homogeneous polynomials of degree n+1 in the coefficients of A_{n}, B_{n}, C_{n}. The conclusion then follows from (46)–(48).

Remark: In the statement of Theorem 2.4, we wrote Π_{N}(z, w) for Π_{N}(z, θ; w, φ). Since the expression is homogeneous of degree 0, it is independent of θ and φ. Alternately, we could regard Π_{N}(z, w) as functions on M × M having values in L_{z} ⊗ \overline{L_{w}} (replacing the horizontal derivatives with the corresponding covariant derivatives); again the degree 0 homogeneity makes the expression a scalar. Furthermore, since Theorem 2.1 is valid for all connections, we can replace the horizontal derivatives in (50) with the derivatives with respect to an arbitrary connection.

2.4. Zero correlation for SU(m + 1)-polynomials. In this section, we use our methods to describe the zero correlation functions for SU(m + 1)-polynomials. We do not carry out the computations in complete detail, since we are primarily interested in the scaling limits, which we shall compute in §4.

The SU(m + 1)-polynomials are random homogeneous polynomials of degree N > 0 on ℂ^{m+1},

s(z) = s(z_{0}, z_{1}, …, z_{m}) = ∑_{|J|=N} \sqrt{N!/J!} c_{J}z^{J}, z^{J} = z_{0}^{j_{0}} ⋯ z_{m}^{j_{m}}, J! = j_{0}! ⋯ j_{m}!, (51)

where the coefficients c_{J} are complex independent Gaussian random variables with mean 0 and variance 1:

E c_{J} = 0;  E c_{J} \overline{c_{K}} = δ_{JK}, δ_{JK} = δ_{j_{0}k_{0}} … δ_{j_{m}k_{m}};  E c_{J}c_{K} = 0. (52)

Then s(z) is a Gaussian random polynomial on ℂ^{N+1} with first and second moments given by

E s(z) = 0;  E s(z)\overline{s(w)} = ⟨z, w⟩^{N} = (∑_{q=0}^{m} z_{q}\overline{w_{q}})^{N} ;  E s(z)s(w) = 0. (53)

This implies that the probability distribution of s(z) is invariant with respect to the map s(z) → s(Uz) for all U ∈ SU(m + 1).

Let (S_{N}, μ_{N}) denote the Gaussian probability space of independent k-tuples (k ≤ m) of SU(m + 1)-polynomials of degree N. For s = (s_{1}, …, s_{k}) ∈ S_{N}, the zero set

Z_{s} = {z : s_{1}(z) = ⋯ = s_{k}(z) = 0} .

is an algebraic variety in the complex projective space ℂℙ^{m}. We will assume that ℂℙ^{m} is supplied with the Fubini-Study Hermitian metric ω, which is SU(m + 1)-invariant. In the affine coordinates z = (1, z_{1}, …, z_{m}),

ω = \frac{\sqrt{−1}}{2}∂ ¯∂log (1 + ∑ |z_{q}|^{2}) = \frac{\sqrt{−1}}{2} [\frac{∑ dz_{q} ∧ \overline{dz_{q}}}{1 + ∑ |z_{q}|^{2}} − \frac{(∑ \overline{z_{q}}dz_{q}) ∧ (∑ z_{q}\overline{dz_{q}})}{(1 + ∑ |z_{q}|^{2})^{2}}] ; (54)

i.e.,

ω = \frac{\sqrt{−1}}{2} ∑ g_{qq′}dz_{q} ∧ dz_{q′} , g_{qq′} = \frac{(1 + |z|^{2})δ_{qq′} − z̄_{q}z_{q′}}{(1 + |z|^{2})^{2}} . (55)

To simplify our computations, we consider only points z^{p} with finite affine coordinates, z^{p} = (1, z_{1}^{p}, …, z_{m}^{p}), p = 1, …, n, and we regard the SU(m+1)-polynomials s_{j} as polynomials of degree ≤ N on ℂ^{m}; i.e., we regard the s_{j} as sections of the trivial line bundle on ℂ^{m} with the flat metric h = 1 (so that the covariant derivatives coincide with the usual derivatives of functions).

As above, we consider the random variables

b_{j}^{p} = s_{j}(z^{p}), a_{jq}^{p} = \frac{∂s_{j}}{∂z_{q}} (z^{p}) ,

and we denote their joint distribution by

\begin{aligned}D_{nk}^{N}(b, a; z)db da, & b = (b^{1}, …, b^{n}) , b^{p} = (b_{j}^{p})_{j=1,...,k}; \\ & a = (a^{1}, …, a^{n}) , a^{p} = (a_{jq}^{p})_{j=1,...,k; q=1,...,m} . \end{aligned} (56)

(Here, the n-point density is with respect to Lebesgue measure db = ∏ db_{j}^{p}db̄_{j}^{p}, da = ∏ da_{jq}^{p}dā_{jq}^{p}.) We assume that N > nm to ensure that μ_{N} possesses a continuous n-point density. Since μ_{N} is Gaussian, the density D_{nk}^{N}(b, a; z) is Gaussian as well, and it is described by the covariance matrix

Δ_{nk}^{N} = ( \begin{matrix}A_{nk}^{N} & B_{nk}^{N} \\ B_{nk}^{N∗} & C_{nk}^{N} \end{matrix} ) (57)

where

\begin{matrix}A_{nk}^{N} = ( E s_{j}(z^{p})\overline{s_{j′}(z^{p′})} ) , \\ B_{nk}^{N} = ( E s_{j}(z^{p})\overline{\frac{∂s_{j′}}{∂z_{q′}} (z^{p′})} ) , \\ C_{nk}^{N} = ( E \frac{∂s_{j}}{∂z_{q}} (z^{p})\overline{\frac{∂s_{j′}}{∂z_{q′}} (z^{p′})} ); \\ j, j′ = 1, …, k; p, p′ = 1, …, n; q, q′ = 1, …, m. \end{matrix} (58)

By (58) and (53),

\begin{aligned}A_{nk}^{N} = (δ_{jj′}S_{N}(z^{p}, z^{p′})) , & S_{N}(z, w) = (1 + ∑_{r=1}^{m} z_{r}\overline{w_{r}})^{N} , \\ B_{nk}^{N} = (δ_{jj′}S_{Nq′}(z^{p}, z^{p′})) , & S_{Nq′}(z, w) = Nz_{q′} (1 + ∑_{r=1}^{m} z_{r}\overline{w_{r}})^{N−1} , \\ C_{nk}^{N} = (δ_{jj′}S_{Nqq′}(z^{p}, z^{p′})) , & S_{Nqq′}(z, w) = N(N − 1)\overline{w_{q}}z_{q′} (1 + ∑_{r=1}^{m} z_{r}\overline{w_{r}})^{N−2} \\ & +δ_{qq′}N (1 + ∑_{r=1}^{m} z_{r}\overline{w_{r}})^{N−1} \end{aligned} (59)

The n-point zero correlation functions K_{nk}^{N} for the SU(m + 1)-polynomial k-tuples S_{N} can be computed by substituting (59) into formulas (40) and (42). (Alternately, we can compute the zero correlation functions with respect to the Euclidean volume on ℂ^{m} by setting γ^{p} =Id in (42).)

Remark: Note that the one-point correlation function, or the zero-density function, is constant, since it is invariant with respect to the group SU(m + 1). Indeed, by Bézout’s theorem and (7),

(60) |Z_{s}|(1) = Vol(V_{s}) = ∫_{V_{s}} \frac{1}{(m−k)!}ω^{m−k} = Ω_{2m−2k} deg V_{s} = Ω_{2m−2k}N^{k} ,

where

Ω_{2ℓ} = Vol ℂℙ^{ℓ} = \frac{π^{ℓ}}{ℓ!} . (61)

Hence,

(62) K_{1k}^{N}(z) = \frac{Vol Z_{s}}{Vol ℂℙ^{m}} = \frac{N^{k}Ω_{2m−2k}}{Ω_{2m}} = \frac{N^{k}m!}{(m − k)!π^{k}} .

We can also use our formulas to compute K_{1k}^{N} directly: By (59),

A_{1k}^{N} = (δ_{jj′}(1 + |z|^{2})^{N}) , B_{1k}^{N} = (δ_{jj′}Nz_{q′}(1 + |z|^{2})^{N−1}) , (63)

C_{1k}^{N} = (δ_{jj′}N[(N − 1)z̄_{q}z_{q′} + (1 + |z|^{2})δ_{qq′}](1 + |z|^{2})^{N−2}) .

Hence by (40),

Λ_{1k}^{N} = (δ_{jj′}N[(1 + |z|^{2})δ_{qq′} − z̄_{q}z_{q′}](1 + |z|^{2})^{N−2}) = (δ_{jj′}N(1 + |z|^{2})^{N}g_{qq′}(z)) . (64)

In the hypersurface case (k = 1), we compute

K_{11}^{N} = \frac{1}{π(1 + |z|^{2})^{N}} ⟨ ∑_{q,q′=1}^{m} ā_{1q}γ_{qq′}a_{1q′}⟩|_{Λ_{11}^{N}} = \frac{N}{π} ∑_{q,q′=1}^{m} γ_{qq′}g_{q′q} = \frac{Nm}{π} ,

as expected. For k > 1, we have Λ_{1k}^{N}(0) = NI where I is the unit matrix, and (42) yields

K_{1k}^{N}(0) = \frac{N^{k}}{π^{k}} ⟨det (∑_{q=1}^{m} ā_{jq}a_{j′q})_{j,j′=1,...,k}⟩_{I} = \frac{N^{k}m!}{(m − k)!π^{k}} ,

which agrees with (62).

3. Universality and scaling

Our goal is to derive scaling limits of the n-point correlations between the zeros of random k-tuples of sections of powers of a positive line bundle over a complex manifold. We expect the scaling limits to exist and to be universal in the sense that they should depend only on the dimensions of the algebraic variety of zeros and the manifold. Our plan is the following. We first describe scaling in the Heisenberg model, which we use to provide the universal scaling limit for the Szegö kernel (Theorem 3.1). Together with Theorem 2.4, this demonstrates the universality of the scaling-limit zero correlation in the case of powers of any positive line bundle on any complex manifold.

3.1. Scaling of the Szegö kernel in the Heisenberg group. Our model for scaling is the Szegö kernel for the reduced Heisenberg group described in §1.3.2. Recall that for the simply-connected Heisenberg group H^{m}, the scaling operators (or Heisenberg dilations)

δ_{r}(ζ, t) = (rζ, r^{2}t) , r ∈ ℝ^{+}

are automorphisms of H^{m} ([Fo] [St]). Since the Szegö kernel Π of H^{m} is the unique selfadjoint holomorphic reproducing kernel, it follows that it must be invariant (up to a multiple) under these automorphisms. In fact, one has ([St, p. 538]):

Π(δ_{r}x, δ_{r}y) = r^{−2m−2}Π(x, y) (65)

The condition for a dilation δ_{r} to descend to the quotient group H_{red}^{m} is that r^{2}ℤ ⊂ ℤ, or equivalently, r = \sqrt{N} with N ∈ ℤ^{+}. Note however that δ\sqrt{N} is not an automorphism of H_{red}^{m} and there is no well-defined dilation by \sqrt{N}^{−1}.

The scaling identity (65) descends to H_{red}^{m} in the form

Π_{N}^{H}(x, y) = N^{m}Π_{1}^{H}(δ\sqrt{N} x, δ\sqrt{N} y) (66)

with

Π_{1}^{H}(x, y) = \frac{1}{π^{m}} e^{i(θ−φ)}e^{iℑ(z·w̄)}e^{− \frac{1}{2} |z−w|^{2}} , x = (z, θ) , y = (w, φ) . (67)

(Recall (21).) Informally, we may say that the scaling limit of Π_{N}^{H} equals Π_{1}^{H}. Since scaling by \sqrt{N}^{−1} is not well-defined on H_{red}^{m} it is more correct to say that Π_{N}^{H} is the \sqrt{N} scaling of the scaling limit kernel.

3.2. Scaling limit of a general Szegö kernel. We now show that Π_{1}^{H} is the scaling limit of the N-th Szegö kernel Π_{N} of an arbitrary positive line bundle L → M in the sense of the following “near-diagonal asymptotic estimate for the Szegö kernel.”

Theorem 3.1. Let z_{0} ∈ M and choose local coordinates in a neighborhood of z_{0} so that Θ_{h}(z_{0}) = ∑ dz_{j} ∧ dz̄_{j}. Then

N^{−m}Π_{N}(z_{0} + \frac{u}{\sqrt{N}} , \frac{θ}{N}; z_{0} + \frac{v}{\sqrt{N}} , \frac{φ}{N}) = \frac{1}{π^{m}} e^{i(θ−φ)+iℑ(u·v̄)− \frac{1}{2} |u−v|^{2}} + O(N^{−1/2})
= Π_{1}^{H}(u, θ; v, φ) + O(N^{−1/2}) .

To prove Theorem 3.1, we need to recall the Boutet de Monvel-Sjostrand parametrix construction:

Theorem 3.2. [BS, Th. 1.5 and §2.c] Let Π(x, y) be the Szegö kernel of the boundary X of a bounded strictly pseudo-convex domain Ω in a complex manifold L. Then: there exists a symbol s ∈ S^{n}(X × X × ℝ^{+}) of the type

s(x, y, t) ∼ ∑_{k=0}^{∞} t^{n−k}s_{k}(x, y)

so that

Π(x, y) = ∫_{0}^{∞} e^{itψ(x,y)}s(x, y, t)dt

where the phase ψ ∈ C^{∞}(X × X) is determined by the following properties:

• ψ(x, x) = \frac{1}{i}ρ(x) where ρ is the defining function of X.

• ¯∂_{x}ψ and ∂_{y}ψ vanish to infinite order along the diagonal.

• ψ(x, y) = −ψ̄(y, x).

The integral is defined as a complex oscillatory integral and is regularized by taking the principal value (see [BS]). The phase is determined only up to a function which vanishes to infinite order at x = y and its Taylor expansion at the diagonal is given by

ψ(x + u, x + v) = \frac{1}{i} ∑ \frac{∂^{α+β}ρ}{∂z^{α}∂z̄^{β}} (x)\frac{u^{α}}{α!} \frac{v̄^{β}}{β!} . (68)

The Szegö kernels Π_{N} are Fourier coefficients of Π and hence may be expressed as:

Π_{N}(x, y) = ∫_{0}^{∞} ∫_{0}^{2π} e^{−iNθ}e^{itψ(r_{θ}x,y)}s(r_{θ}x, y, t)dθdt (69) where r_{θ} denotes the S^{1} action on X. Changing variables t ↦ Nt gives

Π_{N}(x, y) = N ∫_{0}^{∞} ∫_{0}^{2π} e^{iN(−θ+tψ(r_{θ}x,y))}s(r_{θ}x, y, tN)dθdt . (70)

We now fix z_{0} and consider the asymptotics of

\begin{matrix}Π_{N}(z_{0} + \frac{u}{\sqrt{N}} , 0; z_{0} + \frac{v}{\sqrt{N}} , 0) \\ = N ∫_{0}^{∞} ∫_{0}^{2π} e^{iN(−θ+tψ(z_{0}+ \frac{u}{\sqrt{N}} ,θ;z_{0}+ \frac{v}{\sqrt{N}} ,0))}s(z_{0} + \frac{u}{\sqrt{N}} , θ; z_{0} + \frac{v}{\sqrt{N}} , 0), tN)dθdt . \end{matrix} (71)

In our setting the phase takes the following concrete form: We let h(z, w̄) be the almost analytic function on M × M satisfying h(z, z̄) = ∥e_{L}∥_{h}^{−2}(z). The function h(z, w̄) is defined by

h(z_{0} + u, z̄_{0} + v̄) = ∑ \frac{∂^{α+β}h(z, z̄)}{∂z^{α}∂z̄^{β}}(z_{0})\frac{u^{α}}{α!} \frac{v̄^{β}}{β!} . (72)

We consider the complex manifold Y = L^{∗} and we let (z, λ) denote the coordinates of ξ ∈ Y given by ξ = λ(e_{L}^{∗})_{z}. In the associated coordinates (x, y) = (z, λ, w, μ) on Y × Y, we have:

ρ(z, λ) = 1 − h(z, z̄)|λ|^{2}, ψ(z, λ, w, μ) = \frac{1}{i}(1 − h(z, w̄)λμ̄) . (73)

We consider Ω = {ρ < 0} and X = ∂Ω = {ρ = 0}. We may assume without loss of generality that h(z, w̄) = h̄(w, z̄) since h(z, z̄) is real so we could replace h by \frac{1}{2}h(z, w̄) + \frac{1}{2} h̄(w, z̄). On X we have h(z, z̄)|λ|^{2} = 1 so we may write λ = h(z, z̄)^{− \frac{1}{2}} e^{iφ}, and similarly for μ. So for (x, y) = (z, φ, w, φ′) ∈ X × X we have

ψ(z, φ, w, φ′) = \frac{1}{i} [1 − \frac{h(z, w̄)}{\sqrt{h(z, z̄)}\sqrt{h(w, w̄)}}e^{i(φ−φ′)}] . (74)

It follows that

\begin{matrix}ψ(z_{0} + \frac{u}{\sqrt{N}} , θ; z_{0} + \frac{v}{\sqrt{N}} , 0) \\ = \frac{1}{i} [1 − \frac{h(z_{0} + \frac{u}{\sqrt{N}} , z̄_{0} + \frac{v̄}{\sqrt{N}})}{\sqrt{h(z_{0} + \frac{u}{\sqrt{N}} , z̄_{0} + \frac{ū}{\sqrt{N}})}\sqrt{h(z_{0} + \frac{v}{\sqrt{N}} , z̄_{0} + \frac{v̄}{\sqrt{N}})}}e^{iθ}] . \end{matrix} (75)

We now assume that e_{L} is a normal frame centered at z_{0}. By definition, this means that

h(z_{0}) = 1, ∂h|_{z_{0}} = ¯∂h|_{z_{0}} = 0. (76)

We furthermore assume that our coordinates {z_{j}} are chosen so that the Levi form of h is the identity at z_{0}:

\frac{∂^{2}h}{∂z^{α}∂z^{β}} (z_{0}, z̄_{0}) = δ_{αβ} . (77)

(This is equivalent to specifying that ω(z_{0}) = \frac{i}{2} ∑_{j} dz_{j} ∧ dz̄_{j}.) Then by (72),

h(z_{0} + \frac{u}{\sqrt{N}} , z̄_{0} + \frac{v̄}{\sqrt{N}}) = \frac{1}{N} u · v̄ + O(N^{−3/2}). (78)

Now let us return to the phase. It is given by

t [1 − \frac{h(z_{0} + \frac{u}{\sqrt{N}} , z̄_{0} + \frac{v̄}{\sqrt{N}})}{h(z_{0} + \frac{u}{\sqrt{N}} , z̄_{0} + \frac{ū}{\sqrt{N}})^{\frac{1}{2}} h(z_{0} + \frac{v}{\sqrt{N}} , z̄_{0} + \frac{v̄}{\sqrt{N}})^{\frac{1}{2}}} e^{iθ}] − iθ. (79)

By (78), the phase (79) has the form:

(t[1 − e^{iθ}] − iθ) + \frac{t}{N} [u · v̄ − \frac{1}{2}|u|^{2} − \frac{1}{2}|v|^{2}] e^{iθ} + O(N^{−3/2}). (80)

It is now evident that Π_{N}(z_{0} + \frac{u}{\sqrt{N}} , 0; z_{0} + \frac{v}{\sqrt{N}} , 0) is given by an oscillatory integral with phase (t[1 − e^{iθ}] − iθ); the latter two terms can be absorbed into the amplitude.

Thus we have:

\begin{matrix}Π_{N}(z_{0} + \frac{u}{\sqrt{N}} , 0; z_{0} + \frac{v}{\sqrt{N}} , 0) \\ = N ∫_{0}^{∞} ∫_{0}^{2π} e^{iN(t[1−e^{iθ}]−iθ)}e^{t[u·v̄− \frac{1}{2} |u|^{2}− \frac{1}{2} |v|^{2}]+O(N^{−1/2})}s(z_{0} + \frac{u}{\sqrt{N}} , θ; z_{0} + \frac{v}{\sqrt{N}} , 0), tN)dθdt. \end{matrix} (81)

We may then evaluate the integral asymptotically by the stationary phase method as in [Ze]. The phase is precisely the same as occurs in Π_{N}(x, x), and as discussed in [Ze], the single critical point occurs at t = 1, θ = 0. We may also Taylor-expand the amplitude to determine its contribution to the asymptote. Precisely as in the calculation of the stationary phase expansion in [Ze], we get:

Π_{N}(z_{0} + \frac{u}{\sqrt{N}} , 0; z_{0} + \frac{v}{\sqrt{N}} , 0) = \frac{N^{m}}{π^{m}} e^{u·v̄− \frac{1}{2} |u|^{2}− \frac{1}{2} |v|^{2}} + O(N^{m− \frac{1}{2}} ). (82)

Finally, we note that

u · v̄ − \frac{1}{2}|u|^{2} − \frac{1}{2}|v|^{2} = −\frac{1}{2}|u − v|^{2} + iℑ(u · v̄) ,

which completes the proof of Theorem 3.1.

3.3. Universality of the scaling limit of correlations of zeros. We are now ready to pass to the scaling limit as N → ∞ of the correlation functions of sections of powers L^{N} of our line bundle. To explain this notion, let us consider the case k = m where the zeros are (almost surely) discrete. An m-tuple of sections of L^{N} will have N^{m} times as many zeros as m-tuples of sections of L. Hence we must expand our neighborhood (or contract our “yardstick”) by a factor of N^{m/2}. Let z^{0} ∈ M and choose a coordinate neighborhood U ∈ M with coordinates \{z_{j}\} for which z^{0} = 0 and ω(z^{0}) = \frac{i}{2} ∑_{q} dz_{q} ∧ dz̄_{q}. We define the n-point scaling limit zero correlation function

K_{nkm}^{∞}(z) = lim_{N→∞} \frac{1}{N^{nk}} K_{nk}^{N} (\frac{z}{\sqrt{N}} ) , z = (z^{1}, …, z^{n}) ∈ (ℂ^{m})_{n} .

We show below (Theorem 3.4) that this limit exists and that K_{nkm}^{∞} is universal by passing to the limit in Theorem 2.4, using Theorem 3.1. First, we need the following fact:

Lemma 3.3. Let z^{1}, …, z^{n} be distinct points of ℂ^{m}. Then

det (Π_{1}^{H}(z^{p}, 0; z^{p′}, 0)) = e^{− ∑ |z^{p}|^{2}} det (e^{z^{p}·z̄^{p′}}) ≠ 0 .

Proof. We consider the first Szegö projector on the reduced Heisenberg group

(83) Π_{1}^{H} : L^{2}(H_{red}^{m}) → H_{1}^{2}(H_{red}^{m}) ≈ L^{2}(ℂ^{m}, e^{−|z|^{2}}) ∩ O(ℂ^{m}) ,

where

L^{2}(ℂ^{m}, e^{−|z|^{2}}) = {f ∈ L_{loc}^{2}(ℂ^{m}) : _{∫ℂ^{m}}|f|^{2}e^{−|z|^{2}}dz < +∞} .

(See the remark at the end of §1.3.2.) Its kernel can be written in the form

Π_{1}^{H}(z, θ; w, φ) = e^{i(θ−φ)} ∑_{α=1}^{∞} f_{α}(z)\overline{f_{α}(w)} , (84)

where the f_{α} form a complete orthonormal basis for L^{2}(ℂ^{m}, e^{−|z|^{2}}) ∩ O(ℂ^{m}). (E.g., {f_{α}} can be taken to be the set of monomials {c_{j_{1}···j_{m}}z_{1}^{j_{1}} ⋯ z_{m}^{j_{m}}}. In fact, Π_{1}^{H}(z, 0; w, 0)) is just a “weighted Bergman kernel” on ℂ^{m}.) We now mimic the proof of Lemma 2.3, except this time we have an infinite sum over the index α; this sum converges uniformly on bounded sets in ℂ^{m} × ℂ^{m} since the sup norm over a bounded set is dominated by the Gaussian-weighted L^{2} norm (by the same argument as in the case of the ordinary Bergman kernel on a bounded domain). We then obtain a nonzero vector (v_{1}, …, v_{n}) ∈ ℂ^{m} such that ∑_{p} v_{p}f_{α}(z^{p}) = 0 for all α. But then ∑_{p} v_{p}f(z^{p}) = 0 for all polynomials f on ℂ^{m}, a contradiction.

We can now show the universality of the scaling limit of the zero correlation functions:

Theorem 3.4. Let (L, h) be a positive Hermitian line bundle on an m-dimensional compact complex manifold M with Kähler form ω = \frac{i}{2}Θ_{h}, let S = H^{0}(M, L^{N})^{k} (k ≥ 1), and give S the standard Gaussian measure μ. Then

\frac{1}{N^{nk}} K_{nk}^{N} (\frac{z^{1}}{\sqrt{N}} , …, \frac{z^{n}}{\sqrt{N}} ) = K_{nkm}^{∞}(z^{1}, …, z^{n}) + O (\frac{1}{\sqrt{N}} ) ,

where K_{nkm}^{∞}(z^{1}, …, z^{n}) is given by a universal rational function in the quantities z_{q}^{p}, z̄_{q}^{p}, e^{z^{p}·z̄^{p′}}, and the error term has ℓ^{th} order derivatives ≤ \frac{C_{S,ℓ}}{\sqrt{N}} on each compact subset S ⊂ (ℂ^{m})_{n}, for all ℓ ≥ 0.

Proof. By taking the scaling limit of (50), we obtain

(85) K_{nkm}^{∞}(z) = \frac{P_{nkm}(Π_{1}^{H}(z^{p}, z^{p′}), dH ; w̄_{q}Π_{1}^{H}(z^{p}, z^{p′}), d_{z_{q}}^{H}Π_{1}^{H}(z^{p}, z^{p′}), d_{z_{q}}^{H}dH ; w̄_{q′}Π_{1}^{H}(z^{p}, z^{p′}))}{π^{kn} [det (Π_{1}^{H}(z^{p}, z^{p′}))_{1≤p,p′≤n}]^{k(n+1)}} .

Indeed, since the coefficients of Λ_{n} are either of degree 1 in the coefficients of C_{n} or of degree 2 in the coefficients of B_{n}, we see by the proof of Theorem 2.4, using (23), (47)–(48) and Theorem 3.1, that the leading term of the asymptotic expansion of K_{nk}^{N} is N^{nk} times the right side of (85). The bound on the error term follows from Theorem 3.1 and Lemma 3.3.

Substituting into (85) the values of Π_{1}^{H}(z^{p}, z^{p′}) and its horizontal derivatives obtained from (22) (with N = 1) and (67) and canceling out common factors of e^{−|z^{p}|^{2}/2} and π, we obtain

K_{nkm}^{∞}(z) = \frac{P_{nkm}(e^{z^{p}·z̄^{p′}}, (z_{q}^{p} − z_{q}^{p′})e^{z^{p}·z̄^{p′}}, (z̄_{q}^{p′} − z̄_{q}^{p})e^{z^{p}·z̄^{p′}}, [(z_{q′}^{p} − z_{q′}^{p′})(z̄_{q}^{p′} − z̄_{q}^{p}) + δ_{qq′}]e^{z^{p}·z̄^{p′}})}{π^{kn} [det (e^{z^{p}·z̄^{p′}})_{1≤p,p′≤n}]^{k(n+1)}}

(86)

= \frac{Q_{nkm}(z_{q}^{p}, z̄_{q}^{p}, e^{z^{p}·z̄^{p′}})}{π^{kn} [det (e^{z^{p}·z̄^{p′}})]^{k(n+1)}} ,

where Q_{nkm} is a universal polynomial (homogeneous of degree k(n+1) in each of the variables e^{z^{p}·z̄^{p′}} and with integer coefficients).

Remark: As we remarked previously, formula (85) is valid for any connection, so we can replace the left invariant vector fields with their right-invariant counterparts to obtain

K_{nkm}^{∞}(z) = \frac{P_{nkm}(e^{z^{p}·z̄^{p′}}, z_{q}^{p}e^{z^{p}·z̄^{p′}}, z̄_{q}^{p′}e^{z^{p}·z̄^{p′}}, (z_{q′}^{p}z̄_{q}^{p′} + δ_{qq′})e^{z^{p}·z̄^{p′}})}{π^{kn} [det (e^{z^{p}·z̄^{p′}})_{1≤p,p′≤n}]^{k(n+1)}} . (87)

4. Formulas for the scaling limit zero correlation function

We now apply the formulas from §§2.2–2.3 to transform (87) into explicit formulas for K_{nkm}^{∞}. We use the right-invariant connection α^{R} so that d_{z_{q}}^{H} = Z_{q}^{R}. Indeed, by the proofs of Theorems 2.4 and 3.4 (which use formulas (40), (42), (46)–(48)), formula (87) becomes

K_{nkm}^{∞}(z^{1}, …, z^{n}) = \frac{1}{π^{kn} det A_{nkm}} ⟨∏_{p=1}^{n} det(a^{p}a^{p∗})⟩_{Λ_{nkm}} , (88)

where

Λ_{nkm} = C_{nkm} − B_{nkm}^{∗}A_{nkm}^{−1}B_{nkm} (89)

with

A_{nkm} = (δ_{jj′}S(z^{p}, z^{p′})) ,  S(z, w) = exp (∑_{r=1}^{m} z_{r}\overline{w_{r}}) ,

B_{nkm} = (δ_{jj′}S_{q′}(z^{p}, z^{p′})) ,  S_{q′}(z, w) = z_{q′} exp (∑_{r=1}^{m} z_{r}\overline{w_{r}}) ,

C_{nkm} = (δ_{jj′}S_{qq′}(z^{p}, z^{p′})) ,  S_{qq′}(z, w) = (δ_{qq′} + \overline{w_{q}}z_{q′}) exp (∑_{r=1}^{m} z_{r}\overline{w_{r}})
j, j′ = 1, …, k; p, p′ = 1, …, n; q, q′ = 1, …, m.

(90)

The metric tensor g^{p} in (42) becomes a unit tensor in the scaling limit, so there is no γ^{p} on the right in (88).

Because Π_{1}^{H} is invariant with respect to unitary transformations and equivariant with respect to translations (i.e., Π_{1}^{H}(z + u, w + u) = e^{iℑ(z·ū)}e^{−iℑ(w·ū)}Π_{1}^{H}(z, w)), the scaling limit zero correlation K_{nkm}^{∞} is invariant with respect to the group of isometric transformations— unitary transformations and translations—of ℂ^{m}.

In particular, the limit one-point zero correlation, or the zero-density function, is constant, since it is invariant under translation. Indeed by (90), A_{1km} = e^{|z|^{2}}I_{k} and Λ_{1km} = e^{|z|^{2}}I_{km}, where I_{k}, resp. I_{km}, denotes the unit k × k, resp. (km) × (km), matrix. Thus by (88) and the Wick formula,

K_{1km}^{∞}(z) = \frac{1}{π^{k}e^{k|z|^{2}}} ⟨det (∑_{q=1}^{m} ā_{jq}a_{j′q})_{j,j′=1,...,k}⟩_{e^{|z|^{2}}I_{km}} = \frac{m!}{π^{k}(m − k)!} . (91)

Thus we define the normalized n-point scaling limit zero correlation function

K̃_{nkm}^{∞}(z) = (K_{1km}^{∞})^{−n}K_{nkm}^{∞}(z) = (\frac{π^{k}(m − k)!}{m!})^{n} K_{nkm}^{∞}(z) . (92)

Remark: These formulas also follow from §2.4. For example, equation (91) is a consequence of (62) since

K_{1km}^{∞}(z) = \frac{1}{N^{k}} K_{1k}^{N}(z) .

Furthermore, using the notation of §2.4, we observe that

lim_{N→∞} S_{N} (\frac{z}{\sqrt{N}} , \frac{w}{\sqrt{N}} ) = lim_{N→∞} (1 + N^{−1} ∑_{r=1}^{m} z_{r}\overline{w_{r}})^{N} = S(z, w) ,

lim_{N→∞} N^{−1/2}S_{Nq′} (\frac{z}{\sqrt{N}} , \frac{w}{\sqrt{N}} ) = S_{q′}(z, w) ,

lim_{N→∞} N^{−1}S_{Nqq′} (\frac{z}{\sqrt{N}} , \frac{w}{\sqrt{N}} ) = S_{qq′}(z, w) .

(93)

Equations (93) provide an alternate derivation of (90).

4.1. Decay of correlations. Explicit formulas for the correlation functions K̃_{nkm}^{∞} can be obtained from (88), (90) and the Wick formula. We shall illustrate these computations for the cases n = 2, k = 1, 2 in §§4.2–4.3 below. We now note that the limit correlations are “short range” in the following sense:

Theorem 4.1. The correlation functions satisfy the estimate

K̃_{nkm}^{∞}(z^{1}, …, z^{n}) = 1 + O(r^{4}e^{−r^{2}}) as r → ∞ , r = min_{p≠p′} |z^{p} − z^{p′}| .

Proof. We use formula (85), which comes from (88)–(89) as in the proof of Theorem 3.4. To determine the matrices A, B, C, we let d_{z_{q}}^{H} = Z_{q}^{L}, dH ; w̄_{q} = W̄_{q}^{L} (instead of the right-invariant vector fields we used above). Recalling (22), we have:

A_{j′p′}^{jp} = δ_{jj′}A_{p′}^{p} ,  A_{p′}^{p} = π^{m}Π_{1}^{H}(z^{p}, 0; z^{p′}, 0) ,
(94)  B_{j′p′q′}^{jp} = δ_{jj′}(z_{q′}^{p} − z_{q′}^{p′})A_{p′}^{p} ,
C_{j′p′q′}^{jpq} = δ_{jj′}(δ_{qq′} + (z̄_{q}^{p′} − z̄_{q}^{p})(z_{q′}^{p} − z_{q′}^{p′}))A_{p′}^{p} .

By (67),

A_{p′}^{p} = \{ \begin{matrix}1 & p = p′ \\ O(e^{−r^{2}/2}) & p ≠ p′ \end{matrix} ,
B = O(re^{−r^{2}/2}) ,
C = I + O(r^{2}e^{−r^{2}/2}) ,  C_{jpq}^{jpq} = 1 .

Recalling (40), we have

Λ = I + O(r^{2}e^{−r^{2}/2}) , Λ_{jpq}^{jpq} = 1 + O(r^{2}e^{−r^{2}}) . (95)

We now apply formula (88); note that the Wick formula involves terms that are products of diagonal elements of Λ, and products that contain at least two off-diagonal elements of Λ. The former terms are of the form 1 + O(r^{2}e^{−r^{2}}), and the latter are O(r^{4}e^{−r^{2}}). Similarly, det A = 1 + O(r^{4}e^{−r^{2}}). The desired estimate then follows from (92).

We shall see from our computations of the pair correlation below that Theorem 4.1 is sharp. The theorem can be extended to estimates of the connected correlation functions (called also truncated correlation functions, cluster functions, or cumulants), as follows. The n-point connected correlation function is defined as (see, e.g., [GJ, p. 286])

T̃_{nkm}^{∞}(z^{1}, …, z^{n}) = ∑_{G}(−1)^{l+1}(l − 1)! ∏_{j=1}^{l} K̃_{n_{j}km}^{∞}(z^{i_{1}}, …, z^{i_{n_{j}}}), (96)

where the sum is taken over all partitions G = (G_{1}, …, G_{l}) of the set (1, …, n) and G_{j} = (i_{1}, …, i_{n_{j}}). In particular, recalling that K̃_{1km}^{∞} ≡ 1,

T̃_{1km}^{∞}(z^{1}) = K̃_{1km}^{∞}(z^{1}) = 1 ,
T̃_{2km}^{∞}(z^{1}, z^{2}) = K̃_{2km}^{∞}(z^{1}, z^{2}) − K̃_{1km}^{∞}(z^{1})K̃_{1km}^{∞}(z^{2}) = K̃_{2km}^{∞}(z^{1}, z^{2}) − 1 ,
T̃_{3km}^{∞}(z^{1}, z^{2}, z^{3}) = K̃_{3km}^{∞}(z^{1}, z^{2}, z^{3}) − K̃_{2km}^{∞}(z^{1}, z^{2})K̃_{1km}^{∞}(z^{3}) − K̃_{2km}^{∞}(z^{1}, z^{3})K̃_{1km}^{∞}(z^{2})
− K̃_{2km}^{∞}(z^{2}, z^{3})K̃_{1km}^{∞}(z^{1}) + 2K̃_{1km}^{∞}(z^{1})K̃_{1km}^{∞}(z^{2})K̃_{1km}^{∞}(z^{3})
= K̃_{3km}^{∞}(z^{1}, z^{2}, z^{3}) − K̃_{2km}^{∞}(z^{1}, z^{2}) − K̃_{2km}^{∞}(z^{1}, z^{3}) − K̃_{2km}^{∞}(z^{2}, z^{3}) + 2 ,

and so on. The inverse of (96) is

K̃_{nkm}^{∞}(z^{1}, …, z^{n}) = ∑_{G} ∏_{j=1}^{l} T̃_{n_{j}km}^{∞}(z^{i_{1}}, …, z^{i_{n_{j}}}) (97)

(Moebius’ theorem). The advantage of the connected correlation functions is that they go to zero if at least one of the distances |z^{i} − z^{j}| goes to infinity (see Corollary 4.3 below). In our case the connected correlation functions can be estimated as follows. Define

d(z^{1}, …, z^{n}) = max_{G} ∏_{l∈L} |z^{i(l)} − z^{f(l)}|^{2}e^{−|z^{i(l)}−z^{f(l)}|^{2}/2}. (98)

where the maximum is taken over all oriented connected graphs G = (V, L, i, f) such that V = (z^{1}, …, z^{n}) and for every vertex z^{j} ∈ V there exist at least two edges emanating from z^{j}. Here V denotes the set of vertices of G, L the set of edges, and i(l) and f(l) stand for the initial and final vertices of the edge l, respectively. Observe that the maximum in (98) is achieved at some graph G, because te^{−t/2} ≤ 2/e < 1 and therefore the product in (98) is less or equal (2/e)^{|L|} which goes to zero as |L| → ∞.

Theorem 4.2. The connected correlation functions satisfy the estimate

T̃_{nkm}^{∞}(z^{1}, …, z^{n}) = O(d(z^{1}, …, z^{n})) as max_{p,q} |z^{p} − z^{q}| → ∞ ,

provided that min_{p,q} |z^{p} − z^{q}| ≥ c > 0.

This theorem implies Theorem 4.1 because of the inversion formula (97). To prove the theorem let us remark that we can rewrite (88) (using the Wick theorem) as a sum over Feynman diagrams. Namely, for the normalized correlation functions K̃_{nkm}^{∞}(z^{1}, …, z^{n}) we have that

K̃_{nkm}^{∞}(z^{1}, …, z^{n}) = \frac{[(m − k)!/m!]^{n}}{det A_{nkm}} ∑_{F} A_{F}(z^{1}, …, z^{n}) , (99)

where the sum is taken over all graphs F = (V, L, i, f) (Feynman diagrams) such that V = (z^{1}, …, z^{n}) and the edges l ∈ L connect the paired variables a_{jq}^{i(l)}, a_{j′q′}^{f(l)∗} in a given term of the Wick sum for K̃_{nkm}^{∞}(z^{1}, …, z^{n}). The function A_{F}(z^{1}, …, z^{n}) is a sum over all terms in the Wick sum with a fixed Feynman diagram F. In other words, to get A_{F}(z^{1}, …, z^{n}) we fix pairings (a_{jq}^{p}, a_{j′q′}^{p′∗}) prescribed by F and sum up in the Wick formula over all indices j, q at every a^{p}. A remarkable property of the connected correlation functions is that they are represented by the sum over connected Feynman diagrams (see, e.g., [GJ]),

T̃_{nkm}^{∞}(z^{1}, …, z^{n}) = \frac{[(m − k)!/m!]^{n}}{det A_{nkm}} ∑_{F}^{conn}A_{F}(z^{1}, …, z^{n}) . (100)

Since det A_{nkm} ≥ c_{1} > 0 and |Λ_{j′p′q′}^{jpq}| ≤ c_{2} < +∞ when min_{p,q} |z^{p} − z^{q}| ≥ c > 0, we conclude from (40), (94) and (67) that for all connected Feynman diagrams F,

A_{F}(z^{1}, …, z^{n}) = O(d) , (101)

where d = d(z^{1}, …, z^{n}) is defined in (98). Summing up over F, we prove Theorem 4.2.

Corollary 4.3. The connected correlation functions satisfy the estimate

T̃_{nkm}^{∞}(z^{1}, …, z^{n}) = O(e^{−R^{2}/2n}) as R → ∞ , R = max_{p,q} |z^{p} − z^{q}| ,

provided that min_{p,q} |z^{p} − z^{q}| ≥ c > 0.

4.2. Hypersurface pair correlation. We now give an explicit formula [(110)] for pair correlations in codimension 1 (k = 1, n = 2). The case m = 1 of this formula coincides, as it must, with the formula given by [Han] and [BBL] for the universal scaling limit pair correlation for SU(2) polynomials. In another paper [BSZ], we gave a different proof of (110) using the Poincaré-Lelong formula.

Since the scaling-limit pair correlation function K_{2km}^{∞}(z^{1}, z^{2}) is invariant with respect to the group of isometries of ℂ^{m}, it depends only on the distance r = |z^{1} − z^{2}|, so we can set z^{1} = 0 and z^{2} = (r, 0, …, 0). To simplify notation, we shall henceforth write A = A_{2km}, B = B_{2km}, C = C_{2km}, Λ = Λ_{2km}.

In this case, (90) reduces to

\begin{matrix}A = ( \begin{matrix}1 & 1 \\ 1 & e^{r^{2}} \end{matrix} ); \\ B = (B_{p′q}^{p}); & (B_{p′1}^{p}) = ( \begin{matrix}0 & 0 \\ r & re^{r^{2}} \end{matrix} ); & (B_{p′q}^{p}) = ( \begin{matrix}0 & 0 \\ 0 & 0 \end{matrix} ) , q ≥ 2; \\ C = (C_{p′q′}^{pq}); & (C_{p′1}^{p1}) = ( \begin{matrix}1 & 1 \\ 1 & (1 + r^{2})e^{r^{2}} \end{matrix} ); & (C_{p′q′}^{pq}) = δ_{qq′} ( \begin{matrix}1 & 1 \\ 1 & e^{r^{2}} \end{matrix} ) , q, q′ ≥ 2. \end{matrix} (102)

The matrix Λ = (Λ_{p′q′}^{pq}) = C − B^{∗}A^{−1}B (103) is given by

Λ_{p′1}^{p1} = ( \begin{matrix}\frac{e^{u} − 1 − u}{e^{u} − 1} & \frac{e^{u} − 1 − ue^{u}}{e^{u} − 1} \\ \frac{e^{u} − 1 − ue^{u}}{e^{u} − 1} & \frac{e^{2u} − e^{u} − ue^{u}}{e^{u} − 1} \end{matrix} ); Λ_{p′q′}^{pq} = δ_{qq′} ( \begin{matrix}1 & 1 \\ 1 & e^{u} \end{matrix} ) , q, q′ ≥ 2 , (104)

where u = r^{2} = |z^{1} − z^{2}|^{2}. By (88), (92) and the formula for A in (102), we have

K̃_{21m}^{∞}(z^{1}, z^{2}) = \frac{1}{m^{2}(e^{u} − 1)} ⟨(∑_{q=1}^{m} a_{q}^{1}\overline{a_{q}^{1}}) (∑_{q′=1}^{m} a_{q′}^{2}\overline{a_{q′}^{2}})⟩_{Λ} (105)

By the Wick formula (see for example, [Si, (I.13)]),

K̃_{21m}^{∞}(z^{1}, z^{2}) = \frac{1}{m^{2}(e^{u} − 1)} [(∑_{q=1}^{m}⟨a_{q}^{1}\overline{a_{q}^{1}}⟩_{Λ_{2}}) (∑_{q′=1}^{m}⟨a_{q′}^{2}\overline{a_{q′}^{2}}⟩_{Λ_{2}}) + ∑_{q,q′=1}^{m}⟨a_{q}^{1}\overline{a_{q′}^{2}}⟩_{Λ_{2}}⟨\overline{a_{q}^{1}}a_{q′}^{2}⟩_{Λ_{2}}]
= \frac{1}{m^{2}(e^{u} − 1)} [(∑_{q=1}^{m} Λ_{1q}^{1q}) (∑_{q′=1}^{m} Λ_{2q′}^{2q′}) + ∑_{q,q′=1}^{m} Λ_{2q′}^{1q}Λ_{1q}^{2q′}] .

(106)

Substituting the values of Λ_{p′q′}^{pq} given by (104), we obtain

\begin{aligned}K̃_{21m}^{∞}(z^{1}, z^{2}) & = & \frac{1}{m^{2}(e^{u} − 1)} [(\frac{e^{u} − 1 − u}{e^{u} − 1} + m − 1) (\frac{e^{2u} − e^{u} − ue^{u}}{e^{u} − 1} + (m − 1)e^{u}) \\ & & + (\frac{e^{u} − 1 − ue^{u}}{e^{u} − 1} )^{2} + (m − 1)] ,  u = |z^{1} − z^{2}|^{2} . \end{aligned} (107)

After simplification,

K̃_{21m}^{∞}(z^{1}, z^{2}) = \frac{u^{2}(e^{2u} + e^{u}) − 2u(e^{2u} − e^{u}) + m^{2}(e^{u} − 1)^{2}e^{u} + m(e^{u} − 1)^{2}}{m^{2}(e^{u} − 1)^{3}} . (108)

Putting u = 2t and writing

K̃_{21m}^{∞}(z^{1}, z^{2}) = κ_{1m}(|z^{1} − z^{2}|) , (109)

we then obtain

(110) κ_{1m}(r) = \frac{[\frac{1}{2}(m^{2} + m) sinh^{2} t + t^{2}] cosh t − (m + 1)t sinh t}{m^{2} sinh^{3} t} + \frac{(m − 1)}{2m}, t = \frac{r^{2}}{2} .

The case m = 1 of formula (110) was obtained by Bogomolny-Bohigas-Leboeuf [BBL] and Hannay [Han].

As r → ∞,

κ_{1m}(r) = 1 + \frac{r^{4} − 2(m^{2} + 1)r^{2} + m(3m + 1)}{m^{2}}e^{−r^{2}} + O(r^{4}e^{−2r^{2}}) . (111)

The following expansion of the correlation function was obtained from (110) using Maple^{TM}:

κ_{1m} = \frac{m − 1}{2m}t^{−1} + \frac{m − 1}{2m} + \frac{1}{6} \frac{(m + 2) (m + 1)}{m^{2}}t − \frac{1}{90} \frac{(m + 4) (m + 3)}{m^{2}}t^{3}

+\frac{1}{945} \frac{(m + 6) (m + 5)}{m^{2}}t^{5} − \frac{1}{9450} \frac{(m + 8) (m + 7)}{m^{2}}t^{7}

+\frac{1}{93555} \frac{(m + 10) (m + 9)}{m^{2}}t^{9} − \frac{691}{638512875} \frac{(m + 12) (m + 11)}{m^{2}}t^{11}

+\frac{2}{18243225} \frac{(m + 14) (m + 13)}{m^{2}}t^{13} − · · · .

In particular, in the one-dimensional case we have

κ_{11}(r) = \frac{1}{2}r^{2} − \frac{1}{36}r^{6} + \frac{1}{720}r^{10} − \frac{1}{16800}r^{14} + · · · . (112)

4.3. Pair correlation in higher codimension. Next we compute the two-point correlation functions for the case k = 2. For k > 1, we have

A = (A_{j′p′}^{jp}) = (δ_{jj′}A_{p′}^{p}) , B = (B_{j′p′q′}^{jp}) = (δ_{jj′}B_{p′q′}^{p}) , C = (C_{j′p′q′}^{jpq}) = (δ_{jj′}C_{p′q′}^{pq}) , (113)

where A_{p′}^{p}, B_{p′q′}^{p}, C_{p′q′}^{pq} are given by (102). It follows that

Λ = (Λ_{j′p′q′}^{jpq}) = (δ_{jj′}Λ_{p′q′}^{pq}) , (114)

where Λ_{p′q′}^{pq} is given by (104).

By (88),

K_{2km}^{∞}(z^{1}, z^{2}) = \frac{1}{π^{2k}(e^{u} − 1)^{k}} ⟨det |a_{j}^{1}\overline{a_{j′}^{1}}|_{j,j′=1,...,k} det |a_{j}^{2}\overline{a_{j′}^{2}}|_{j,j′=1,...,k}⟩_{Λ} , a_{j}^{p}\overline{a_{j′}^{p}} = ∑_{q=1}^{m} a_{jq}^{p}\overline{a_{j′q}^{p}} , (115)

where u = r^{2} = |z^{1} − z^{2}|^{2} as before. Observe that the random variables a_{jq}^{p} and \overline{a_{j′q′}^{p′}} are independent if either j ≠ j′ or q ≠ q′.

UNIVERSALITY AND SCALING OF CORRELATIONS BETWEEN ZEROS 39

Recalling (92), we write K̃_{2km}^{∞}(z^{1}, z^{2}) = κ_{km}(|z^{1} − z^{2}|) . (116) When k = 2, (115) reduces to the following

κ_{2m}(r) = \frac{⟨[(a_{1}^{1}\overline{a_{1}^{1}})(a_{2}^{1}\overline{a_{2}^{1}}) − (a_{1}^{1}\overline{a_{2}^{1}})(a_{2}^{1}\overline{a_{1}^{1}})] [(a_{1}^{2}\overline{a_{1}^{2}})(a_{2}^{2}\overline{a_{2}^{2}}) − (a_{1}^{2}\overline{a_{2}^{2}})(a_{2}^{2}\overline{a_{1}^{2}})]⟩_{Λ}}{m^{2}(m − 1)^{2}(e^{u} − 1)^{2}} . (117)

By the Wick formula,

κ_{2m}(r) = \frac{d_{11} − d_{21} − d_{12} + d_{22}}{m^{2}(m − 1)^{2}(e^{u} − 1)^{2}} , (118)

where

d_{11} = ⟨(a_{1}^{1}\overline{a_{1}^{1}})(a_{2}^{1}\overline{a_{2}^{1}})(a_{1}^{2}\overline{a_{1}^{2}})(a_{2}^{2}\overline{a_{2}^{2}})⟩_{Λ} = ∑_{α,β,γ,δ} ⟨a_{1α}^{1}\overline{a_{1α}^{1}}a_{2β}^{1}\overline{a_{2β}^{1}}a_{1γ}^{2}\overline{a_{1γ}^{2}}a_{2γ}^{2}\overline{a_{2γ}^{2}}⟩_{Λ}
= ∑_{α,β,γ,δ} Λ_{1α}^{1α}Λ_{1β}^{1β}Λ_{2γ}^{2γ}Λ_{2δ}^{2δ} + 2 ∑_{α,β,γ} Λ_{1α}^{1α}Λ_{2β}^{1β}Λ_{1β}^{2β}Λ_{2γ}^{2γ} + ∑_{α,β} Λ_{2α}^{1α}Λ_{1α}^{2α}Λ_{2β}^{1β}Λ_{1β}^{2β}
= [(∑_{q} Λ_{1q}^{1q}) (∑_{q} Λ_{2q}^{2q}) + ∑_{q} Λ_{2q}^{1q}Λ_{1q}^{2q}]^{2} ;

(119)

similarly,

d_{12} = ⟨(a_{1}^{1}\overline{a_{1}^{1}})(a_{2}^{1}\overline{a_{2}^{1}})(a_{1}^{2}\overline{a_{2}^{2}})(a_{2}^{2}\overline{a_{1}^{2}})⟩_{Λ}
= (∑_{q} [Λ_{2q}^{2q}]^{2}) (∑_{q} Λ_{1q}^{1q})^{2} + 2 (∑_{q} Λ_{2q}^{2q}Λ_{1q}^{2q}Λ_{2q}^{1q}) (∑_{q} Λ_{2q}^{1q}) + ∑_{q} [Λ_{1q}^{2q}Λ_{1q}^{1q}]^{2} ,

(120)

d_{21} = ⟨(a_{1}^{1}\overline{a_{2}^{1}})(a_{2}^{1}\overline{a_{1}^{1}})(a_{1}^{2}\overline{a_{1}^{2}})(a_{2}^{2}\overline{a_{2}^{2}})⟩_{Λ}
= (∑_{q} [Λ_{1q}^{1q}]^{2}) (∑_{q} Λ_{2q}^{2q})^{2} + 2 (∑_{q} Λ_{1q}^{1q}Λ_{1q}^{2q}Λ_{2q}^{1q}) (∑_{q} Λ_{2q}^{2q}) + ∑_{q} [Λ_{1q}^{2q}Λ_{2q}^{1q}]^{2} ,

(121)

d_{22} = ⟨(a_{1}^{1}\overline{a_{2}^{1}})(a_{2}^{1}\overline{a_{1}^{1}})(a_{1}^{2}\overline{a_{2}^{2}})(a_{2}^{2}\overline{a_{1}^{2}})⟩_{Λ}
= (∑_{q} [Λ_{1q}^{1q}]^{2}) (∑_{q} [Λ_{2q}^{2q}]^{2}) + 2 ∑_{q} Λ_{1q}^{1q}Λ_{2q}^{2q}Λ_{1q}^{2q}Λ_{2q}^{1q} + (∑_{q} Λ_{1q}^{2q}Λ_{2q}^{1q})^{2} .

(122)

Substituting the values of the matrix elements of Λ we then obtain

\begin{aligned}κ_{2m}(r) & = \frac{(m^{2} − m)e^{2u} + 2(m − 1)e^{u} + 2}{(e^{u} − 1)^{2}m(m − 1)} − \frac{4ue^{u}[(m − 1)e^{u} + 1](m + 1)}{(e^{u} − 1)^{3}(m − 1)m^{2}} \\ & +\frac{2u^{2}e^{u}[(m − 1)e^{2u} + 2me^{u} + 1]}{(e^{u} − 1)^{4}(m − 1)m^{2}}, u = r^{2}. \end{aligned} (123)

As r → ∞,

κ_{2m} = 1 + \frac{2[r^{4} − 2(m + 1)r^{2} + m(m + 1)]e^{−r^{2}}}{m^{2}} + O(r^{4}e^{−2r^{2}}) . (124)

As r → 0,

\begin{aligned}κ_{2m}(r) & = \frac{m − 2}{m} r^{−4} + \frac{m − 2}{m} r^{−2} + \frac{5m^{2} − 7m + 12}{12(m − 1)m} + \frac{(m − 2)(m + 2)(m + 1)}{12(m − 1)m^{2}} r^{2} \\ & +\frac{(m + 3)(m + 2)}{240(m − 1)m} r^{4} − \frac{(m − 2)(m + 4)(m + 3)}{720(m − 1)m^{2}} r^{6} + … \end{aligned} (125)

When m = 2 the asymptotics reduce to

κ_{22}(r) = \frac{3}{4} + \frac{r^{4}}{24} − \frac{r^{8}}{288} + \frac{r^{12}}{4800} − \frac{r^{16}}{96768} + …, (126)

and in this case κ_{22} is a series in r^{4}.

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Department of Mathematical Sciences, IUPUI, Indianapolis, IN 46202, USA

  E-mail address: bleher@math.iupui.edu

Department of Mathematics, Johns Hopkins University, Baltimore, MD 21218, USA

  E-mail address: shiffman@math.jhu.edu

Department of Mathematics, Johns Hopkins University, Baltimore, MD 21218, USA

  E-mail address: zel@math.jhu.edu