POINCARÉ-LELONG APPROACH TO UNIVERSALITY AND SCALING OF CORRELATIONS BETWEEN ZEROS

PAVEL BLEHER, BERNARD SHIFFMAN, AND STEVE ZELDITCH

Abstract. This note is concerned with the scaling limit as N → ∞ of n-point correlations between zeros of random holomorphic polynomials of degree N in m variables. More generally we study correlations between zeros of holomorphic sections of powers L^{N} of any positive holomorphic line bundle L over a compact Kähler manifold. Distances are rescaled so that the average density of zeros is independent of N. Our main result is that the scaling limits of the correlation functions and, more generally, of the “correlation forms” are universal, i.e. independent of the bundle L, manifold M or point on M.

Introduction

This note is a companion to our article [BSZ], in which we study the correlations between the zeros of a random holomorphic section s ∈ H^{0}(M, L^{N}) of a power L^{N} of a positive line bundle L → M over a compact m-dimensional complex manifold M. Since the hypersurface volume of the zeros of a section of L^{N} in a ball U around a given point z_{0} is ∼ NVol (U), we rescale U → \sqrt{N}U to get a density of zeros independent of N. After expanding U this way, all manifolds and line bundles appear asymptotically alike, and it is natural to ask if the local statistics of zeros are universal, i.e. independent of L, M, ω and z_{0}. To define our statistics, we first provide H^{0}(M, L^{N}) with a natural Gaussian measure (see §§1.1–1.2). The local statistics are measured by the scaled n-point zero correlation forms K⃗_{n}^{N}(\frac{z^{1}}{\sqrt{N}} , …, \frac{z^{n}}{\sqrt{N}}), z^{j} ∈ \sqrt{N} U (see §1.3). They are smooth off the diagonal, and their norms define scaled zero correlation measures K̃_{n}^{N}(\frac{z^{1}}{\sqrt{N}} , …, \frac{z^{n}}{\sqrt{N}}). (The scaled correlation forms extend to all of

(\sqrt{N} U)^{n} as currents of order 0.) The correlation measures K̃_{n}^{N} give the expected value of the product of the volumes of the zero set in n domains in M; see equation (12). In [BSZ], we used geometric probability methods and a (universal) scaled Szegö kernel to prove that there exist universal limits as N → ∞ of these correlation measures and more generally of the correlations between simultaneous zeros of k ≤ m sections. Here we use a complex analytic approach based on the Poincaré-Lelong formula for the currents of integration over the zero set of a section, together with the scaled Szegö kernel from [BSZ], to prove universality for the correlation forms. This approach, although limited to the hypersurface case, allows for a result on the level of forms and a somewhat simpler proof.

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.

Our universality theorem is as follows:

Main Theorem. There is a universal current K⃗_{n}^{∞} ∈ D′^{(m−1)n,(m−1)n}(ℂ^{mn}) such that the following holds: suppose that (L, h) is a positive Hermitian line bundle on an m-dimensional compact complex manifold M, and let K⃗_{n}^{N} be the n-point zero correlation current on M^{n}. Suppose z^{0} ∈ M and choose local holomorphic coordinates in M about z^{0} such that Θ_{h}|_{z^{0}} = ∂ ¯∂|z|^{2}. Then

K⃗_{n}^{N} (\frac{z^{1}}{\sqrt{N}} , …, \frac{z^{n}}{\sqrt{N}} ) = K⃗_{n}^{∞}(z^{1}, …, z^{n}) + O (\frac{1}{\sqrt{N}} ) .

Furthermore, K⃗_{n}^{∞} is a smooth form on the off-diagonal domain G_{n}^{m} consisting of n-tuples of distinct points z^{1}, …, z^{n} in ℂ^{m}. The error term has k^{th} order derivatives ≤ \frac{C_{A,k}}{\sqrt{N}} on each compact subset A ⊂ G_{n}^{m}, ∀k ≥ 0.

Our method leads to integral formulae for these universal limit forms, although the details rapidly become complicated as the number n of points increase. For the case m = 2, we carry out the calculation in complete detail in dimension one and also use the method to obtain an explicit formula for the scaling limit pair correlation measures in all dimensions (Theorems 4.1 and 4.2). In particular, our formula gives the scaling limit pair correlations for SU(m + 1)-polynomials (which are sections of powers of the hyperplane bundle over complex projective space ℂℙ^{m}). The universal formula in dimension one agrees, as it must, with that of Bogomolny-Bohigas-Leboeuf [BBL] and Hannay [Ha] in the case of random SU(2)-polynomials. Similar formulas for correlations of zeros of real polynomials were given in [BD].

Before we get started on the proof, the following heuristic remark on correlation measures and forms may be helpful. Although the definition of the correlation measures is formulated in terms of expected volumes of the zero set (or in the one-dimensional case, expected numbers of zeros) in domains of M, it can also be given a probabilistic interpretation: the probability that the zero divisor of a random section s simultaneously intersects balls of radius ε around z^{1}, …, z^{n}, respectively, is ≈ cε^{2n}K_{n}^{N}(z^{1}, …, z^{n}) (where K_{n}^{N} is the correlation function given by K_{n}^{N}dV_{M} = K̃_{n}^{N}). The correlation form K⃗_{n}^{N} gives a more refined probability that the zero divisor of s has tangent hyperplanes close to n fixed complex hyperplanes in T_{M}.

Acknowledgment. The first draft of this paper was completed while the third author was visiting the Erwin Schrodinger Institute in July 1998. He wishes to thank that institution for its hospitality and financial support.

1. Notation

We summarize here the notation from complex analysis that we will need in the proof. This notation is the same as in [SZ] and [BSZ], except that different normalizations for the metric and volume form are used in [SZ].

1.1. Complex geometry. We denote by (L, h) → M a holomorphic line bundle with smooth Hermitian metric h whose curvature form

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

is a positive (1,1)-form. Here, e_{L} is a local non-vanishing holomorphic section of L over an open set U ⊂ M, and ∥e_{L}∥_{h} = h(e_{L}, e_{L})^{1/2} is the h-norm of e_{L}. As in [BSZ], we give M the Hermitian metric corresponding to the Kähler form ω = \frac{\sqrt{−1}}{2}Θ_{h} and the induced Riemannian volume form

dV_{M} = \frac{1}{m!}ω^{m} . (2)

We denote by H^{0}(M, L^{N}) the space of holomorphic sections of L^{N} = L ⊗ ⋯ ⊗ L. 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}.

For a holomorphic section s ∈ H^{0}(M, L^{N}), we let Z_{s} denote the current of integration over the zero divisor of s:

(Z_{s}, φ) = ∫_{Z_{s}} φ , φ ∈ D^{m−1,m−1}(M) .

The Poincaré-Lelong formula (see e.g., [GH]) expresses the integration current of a holomorphic section s = ge_{L}^{⊗N} in the form:

Z_{s} = \frac{i}{π} ∂ ¯∂log |g| = \frac{i}{π} ∂ ¯∂log ∥s∥_{h^{N}} + Nω . (4)

We also denote by |Z_{s}| the Riemannian (2m − 2)-volume along the regular points of Z_{s}, regarded as a measure on M:

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

i.e., |Z_{s}| is the “total variation measure” of the current of integration over Z_{s}:

|Z_{s}| = Z_{s} ∧ \frac{1}{(m−1)!}ω^{m−1} . (6)

1.2. Random sections and Gaussian measures. We now give H^{0}(M, L^{N}) the complex Gaussian probability measure

dμ(s) = \frac{1}{π^{d_{N}}} e^{−|c|^{2}}dc , s = ∑_{j=1}^{d_{N}} c_{j}S_{j}^{N} , (7)

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: E φ = ∫ φdμ.

We then regard the currents Z_{s} (resp. measures |Z_{s}|), as current-valued (resp. measurevalued) random variables on the probability space (H^{0}(M, L^{N}), dμ); i.e., for each test form (resp. function) φ, (Z_{s}, φ) (resp. (|Z_{s}|, φ)) is a complex-valued random variable.

Since the zero current Z_{s} is unchanged when s is multiplied by an element of ℂ^{∗}, our results are the same if we instead regard Z_{s} as a random variable on the unit sphere SH^{0}(M, L^{N}) with Haar probability measure. We prefer to use Gaussian measures in order to facilitate our computations.

1.3. Correlation currents and measures. The n-point correlation current of the zeros is the current on M^{n} = M × M × ⋯ × M (n times) given by

K⃗_{n}^{N}(z^{1}, …, z^{n}) := E (Z_{s}(z^{1}) ⊗ Z_{s}(z^{2}) ⊗ ⋯ ⊗ Z_{s}(z^{n})) (8)

in the sense that for any test form φ_{1}(z^{1})⊗⋯⊗φ_{n}(z^{n}) ∈ D^{m−1,m−1}(M)⊗⋯⊗D^{m−1,m−1}(M),

(K⃗_{n}^{N}(z^{1}, …, z^{n}), φ_{1}(z^{1}) ⊗ ⋯ ⊗ φ_{n}(z^{n})) = E [(Z_{s}, φ_{1})(Z_{s}, φ_{2}) ⋯ (Z_{s}, φ_{n})] . (9)

In a similar way we define the n-point correlation measures K̃_{n}^{N} as the “total variation measures” of the n-point correlation currents:

K̃_{n}^{N}(z^{1}, …, z^{n}) = K⃗_{n}^{N}(z^{1}, …, z^{n}) ∧ \frac{1}{(m − 1)!}ω_{z^{1}}^{m−1} ∧ ⋯ ∧ \frac{1}{(m − 1)!}ω_{z^{n}}^{m−1} . (10)

By (6) and (10), we have

(K̃_{n}^{N}(z^{1}, …, z^{n}), φ_{1}(z^{1}) … φ_{n}(z^{n})) = E [(|Z_{s}|, φ_{1})(|Z_{s}|, φ_{2}) ⋯ (|Z_{s}|, φ_{n})] (11)

where φ_{j} ∈ C^{0}(M). Equivalently,

K̃_{n}^{N}(U_{1} × ⋯ × U_{n}) = E [∏_{j=1}^{n} Vol(Z_{s} ∩ U_{j})] , U_{1}, …, U_{n} ⊂ M . (12)

Remark: In the case of pair correlation on a Riemann surface (n = 2, dim M = 1), the correlation measures take the form

K⃗_{2}^{N}(z, w) = [Δ] ∧ (K⃗_{1}^{N}(z) ⊗ 1) + κ^{N}(z, w)ω_{z} ⊗ ω_{w} (N ≫ 0)

where [Δ] denotes the current of integration along the diagonal Δ = {(z, z)} ⊂ M × M, and κ^{N} ∈ C^{∞}(M × M).

1.4. Szegö kernels. As in [Ze, SZ, BSZ] and elsewhere, we analyze the N → ∞ limit by lifting it to a principal S^{1} bundle π : X → M. Let us recall how this goes.

We denote by L^{∗} the dual line bundle to L, and define X as the circle bundle X = {λ ∈ L^{∗} : ∥λ∥_{h^{∗}} = 1}, where h^{∗} is the norm on L^{∗} dual to h. We can view X as 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} . (13)

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, where we use 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) . (14)

We let r_{θ}x = e^{iθ}x (x ∈ X) denote the S^{1} action on X. The action r_{θ} commutes with the Cauchy-Riemann operator ¯∂_{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_{N} of L^{N} determines an equivariant function ŝ_{N} on X:

ŝ_{N}(z, λ) = (λ^{⊗N}, s_{N}(z)) , (z, λ) ∈ X; (15)

then ŝ_{N}(r_{θ}x) = e^{iNθ}s_{N}(x). The map s ↦ ŝ is a unitary equivalence between H^{0}(M, L^{⊗N}) and H_{N}^{2}(X).

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) . (16)

It can be given as

Π_{N}(x, y) = ∑_{j=1}^{d_{N}} Ŝ_{j}^{N}(x)\overline{Ŝ_{j}^{N}(y)} , (17)

where S_{1}^{N}, …, S_{d_{N}}^{N} form an orthonormal basis of H^{0}(M, L^{N}).

2. Scaling

In order that we may study the local nature of the random variable Z_{s}, we fix a point z^{0} ∈ M and choose a holomorphic coordinate chart Ψ : Ω, 0 → U, z_{0} (Ω ⊂ ℂ^{m}, U ⊂ M) such that

Ψ^{∗}ω_{z^{0}} = \frac{i}{2} ∑_{j=1}^{m} dz_{j} ∧ dz̄_{j}|_{0} . (18)

For example, if L is the hyperplane section bundle O(1) over ℂℙ^{m} with the Fubini-Study metric h_{FS}, and z_{0} = (1 : 0 : ⋯: 0), then the coordinate chart

Ψ : ℂ^{m} → U = {w ∈ ℂℙ^{m} : w_{0} ≠ 0} , Ψ(z) = (1 : z_{1} : ⋯: z_{m})

(i.e., z_{j} = w_{j}/w_{0}) satisfies (18).

To simplify notation, we identify U with Ω. For a current T ∈ D′^{p,q}(Ω), we write

T (\frac{z}{\sqrt{N}} ) = (τ\sqrt{N})_{∗} T ∈ D′^{p,q}(\sqrt{N}Ω) (τ_{λ}(z) = λz) .

(In particular, if T = ∑ T_{jk}(z)dz_{j} ∧ dz̄_{k}, then T (\frac{z}{\sqrt{N}}) = \frac{1}{N} ∑ T_{jk}(\frac{z}{\sqrt{N}})dz_{j} ∧ dz̄_{k}.) We define the rescaled zero current of s ∈ H^{0}(M, L^{N}) by

Ẑ_{s}^{N}(z) := Z_{s} (\frac{z}{\sqrt{N}} ) . (19)

The scaled n-point correlation currents are then defined by:

E (Ẑ_{s}^{N}(z^{1}) ⊗ Ẑ_{s}^{N}(z^{2}) ⊗ ⋯ ⊗ Ẑ_{s}^{N}(z^{n})) = K⃗_{n}^{N} (\frac{z^{1}}{\sqrt{N}} , …, \frac{z^{n}}{\sqrt{N}} ) ∈ D′^{n,n}(M^{n}). (20)

Following the approach of [SZ], we fix an orthonormal basis {S_{j}^{N}} of H^{0}(M, L^{N}) and write S_{j}^{N} = f_{j}^{N}e_{L}^{⊗N} over U. Any section in H^{0}(M, L^{N}) may then be written as s = ∑_{j=1}^{d_{N}} c_{j}f_{j}^{N}e_{L}^{N}. To simplify the notation we let f^{N} = (f_{1}^{N}, …, f_{d_{N}}^{N}) : U → ℂ^{d_{N}} and we put

∑_{j=1}^{d_{N}} c_{j}f_{j} = c · f^{N} .

Hence

Z_{s} = \frac{\sqrt{−1}}{π}∂ ¯∂log |c · f^{N}| , Ẑ_{s}^{N} = \frac{\sqrt{−1}}{π}∂_{z} ¯∂_{z} log |c · f^{N}(\frac{z}{\sqrt{N}} )| (21)

and therefore

Ẑ_{s}(z^{1}) ⊗ ⋯ ⊗ Ẑ_{s}(z^{n}) = (\frac{i}{π} )^{n} ∂_{z^{1}} ¯∂_{z^{1}} ⋯ ∂_{z^{n}} ¯∂_{z^{n}} [log |c · f^{N}(\frac{z^{1}}{\sqrt{N}})| ⋯ log |c · f^{N}(\frac{z^{n}}{\sqrt{N}})|] . (22)

We then can write the rescaled correlation currents in the form

\begin{matrix}K⃗_{n}^{N} (\frac{z^{1}}{\sqrt{N}} , …, \frac{z^{n}}{\sqrt{N}} ) = E (Ẑ_{s}(z^{1}) ⊗ ⋯ ⊗ Ẑ_{s}(z^{n})) \\ = (\frac{i}{π} )^{n} ∂_{z^{1}} ¯∂_{z^{1}} ⋯ ∂_{z^{n}} ¯∂_{z^{n}} ∫_{ℂ^{d_{N}}} log |c · f^{N} (\frac{z^{1}}{\sqrt{N}} )| ⋯ log |c · f^{N} (\frac{z^{n}}{\sqrt{N}} )| \frac{e^{−|c|^{2}}}{π^{d_{N}}} dc . \end{matrix} (23)

2.1. Scaling limit of the Szegö kernel. The asymptotics of the Szegö kernel along the diagonal were given by [Ti] and [Ze]:

\frac{π^{m}}{N^{m}} Π_{N}(x, x) = 1 + O(N^{−1}) . (24)

For our proof of the Main Theorem, we need the following lemma from [BSZ], which gives the ‘near-diagonal’ asymptotics of the Szegö kernel.

Lemma 2.1. Let z^{0} ∈ M and choose local coordinates {z^{j}} in a neighborhood of z_{0} so that z^{0} = 0 and Θ_{h}(z_{0}) = ∑ dz^{j} ∧ dz̄^{j}. Then

\frac{π^{m}}{N^{m}} Π_{N}(\frac{z}{\sqrt{N}} , \frac{θ}{N}; \frac{w}{\sqrt{N}} , \frac{φ}{N}) = e^{i2π(θ−φ)+iℑ(z·w̄)− \frac{1}{2} |z−w|^{2}} + O(N^{−1/2}) .

Here, (z, θ) denotes the point e^{iθ}∥e_{L}(z)∥_{h}e_{L}^{∗}(z) ∈ X, and similarly for (w, φ). In (24) and Lemma 2.1, the expression O(N^{α}) means a term with k^{th} order derivatives ≤ C_{k}N^{α}, for all k ≥ 0. Lemma 2.1 says that the Szegö kernel has a universal scaling limit. In fact, its scaling limit is the first Szegö kernel of the reduced Heisenberg group; see [BSZ].

3. Universality

All the ideas of the proof of the Main Theorem occur in the simplest case n = 2. So first we prove universality in that case and then extend the proof to general n.

Thus, our first object is to prove that the large N limit of the rescaled pair correlation current (from (23) with n = 2)

\begin{matrix}K⃗_{2}^{N} (\frac{z}{\sqrt{N}} , \frac{w}{\sqrt{N}} ) = E (Ẑ_{s}^{N}(z) ⊗ Ẑ_{s}^{N}(w)) \\ = \frac{−1}{π^{2}}∂_{z} ¯∂_{z}∂_{w} ¯∂_{w} ∫_{ℂ^{d_{N}}} log |c · f^{N}(\frac{z}{\sqrt{N}})| log |c · f^{N}(\frac{w}{\sqrt{N}})| \frac{e^{−|c|^{2}}}{π^{d_{N}}} dc \end{matrix} (25)

is universal.

As in [SZ], we write f^{N} = |f^{N}|u^{N} and expand the integrand in (25):

log |c · f^{N}(\frac{z}{\sqrt{N}})| log |c · f^{N}(\frac{w}{\sqrt{N}})| = log |f^{N}(\frac{z}{\sqrt{N}})| log |f^{N}(\frac{w}{\sqrt{N}})|

+ log |f^{N}(\frac{z}{\sqrt{N}})| log |c · u^{N}(\frac{w}{\sqrt{N}})|

+ log |f^{N}(\frac{w}{\sqrt{N}})| log |c · u^{N}(\frac{z}{\sqrt{N}})|

+ log |c · u^{N}(\frac{z}{\sqrt{N}})| log |c · u^{N}(\frac{w}{\sqrt{N}})| . (26)

Let us denote the terms resulting from this expansion by E_{1}, E_{2}, E_{3}, E_{4}, respectively. In particular,

E_{1} = \frac{−1}{π^{2}}∂_{z} ¯∂_{z}∂_{w} ¯∂_{w} [log |f^{N}(\frac{z}{\sqrt{N}} , \frac{z}{\sqrt{N}})| log |f^{N}(\frac{w}{\sqrt{N}} , \frac{w}{\sqrt{N}})|] . (27)

By (15), Ŝ_{j}^{N}(z, θ) = e^{iNθ}∥e_{L}(z)∥_{h}^{N}f_{j}^{N}(z), where (z, θ) are the coordinates in X given in §2.1. By (17),

Π_{N}(z, w) = ∥e_{L}(z)∥_{h}^{N}∥e_{L}(w)∥_{h}^{N}⟨f^{N}(z), f^{N}(w)⟩ , (28)

where we write Π_{N}(z, w) = Π_{N}(z, 0; w, 0). Since Π_{N}(z, z)^{1/2} = ∥e_{L}(z)∥_{h}^{N}|f^{N}(z)|, each factor in (27) has the form \frac{1}{2} log Π_{N}(\frac{z}{\sqrt{N}} , \frac{z}{\sqrt{N}}) − N log ∥e_{L}(\frac{z}{\sqrt{N}})∥_{h}. By (24), log Π_{N}(\frac{z}{\sqrt{N}} , \frac{z}{\sqrt{N}}) → 0 as N → ∞. On the other hand

−iN∂_{z} ¯∂_{z} log ∥e_{L}(\frac{z}{\sqrt{N}})∥_{h} = ω(\frac{z}{\sqrt{N}}) .

Hence the first term converges to the normalized Euclidean (double) Kähler form:

E_{1} = \frac{i}{2π} ∂ ¯∂|z|^{2} ∧ \frac{i}{2π} ∂ ¯∂|w|^{2} + O(\frac{1}{N}) . (29)

The middle two terms vanish since the integrals in E_{2} and E_{3} are independent of w and z respectively (see [SZ, §3.2]). The “interesting term” is therefore

E_{4} = \frac{−1}{π^{2}}∂_{z} ¯∂_{z}∂_{w} ¯∂_{w} ∫_{ℂ^{d_{N}}} log |c · u^{N}(\frac{z}{\sqrt{N}})| log |c · u^{N}(\frac{w}{\sqrt{N}})|\frac{e^{−|c|^{2}}}{π^{d_{N}}} dc . (30)

To evaluate E_{4}, we consider the integral

G_{2}^{N}(x^{1}, x^{2}) := ∫_{ℂ^{d_{N}}} log |c · x^{1}| log |c · x^{2}|\frac{e^{−|c|^{2}}}{π^{d_{N}}} dc (x^{1}, x^{2} ∈ ℂ^{d_{N}}) (31)

with x^{1} = u^{N}(\frac{z}{\sqrt{N}}), x^{2} = u^{N}(\frac{w}{\sqrt{N}}). To simplify it, we construct a Hermitian orthonormal basis {e_{1}, …, e_{d_{N}} } for ℂ^{d_{N}} such that x^{1} = e_{1} and

x^{2} = ξ_{1}e_{1} + ξ_{2}e_{2}, ξ_{1} = ⟨x^{2}, x^{1}⟩, ξ_{2} = \sqrt{1 − |ξ_{1}|^{2}}. (32)

This is possible because we can always multiply e_{2} by a phase e^{iθ} so that ξ_{2} is positive real. We then make a unitary change of variables to express the integral in the {e_{j}} coordinates. Since the Gaussian is U(d_{N})-invariant, (31) simplifies to

G_{2}^{N}(x^{1}, x^{2}) = G_{2}(ξ_{1}, ξ_{2}) = \frac{1}{π^{2}} ∫_{ℂ^{2}} e^{−(|c_{1}|^{2}+|c_{2}|^{2})} log |ξ_{1}| log |c_{1}ξ_{1} + c_{2}ξ_{2}|dc_{1}dc_{2} (33)

(where we used the fact that the Gaussian integral in each c_{j}, j ≥ 3 equals one by construction). By performing a rotation of the c_{1} variable, we may replace ξ_{1} with |ξ_{1}| and replace G_{2}(ξ_{1}, ξ_{2}) with

(34) G(cos θ) := G_{2}(cos θ, sin θ) ,

where cos θ = |ξ_{1}| = |⟨x^{1}, x^{2}⟩|, 0 ≤ θ ≤ π/2. Hence (30) becomes

E_{4} = \frac{−1}{π^{2}}∂_{z} ¯∂_{z}∂_{w} ¯∂_{w}G(cos θ_{N}) , cos θ_{N} = |⟨u^{N}(\frac{z}{\sqrt{N}}), u^{N}(\frac{w}{\sqrt{N}})⟩| . (35)

By the universal scaling formula for the Szegö kernel (Lemma 2.1) and (28), we have

cos θ_{N} = \frac{|Π_{N}(z, w)|}{Π_{N}(z, z)^{1/2}Π_{N}(w, w)^{1/2}} = e^{− \frac{1}{2} |z−w|^{2}} + O(N^{− \frac{1}{2}} ) . (36)

Thus we get the universal formula:

K⃗_{2}^{∞}(z, w) = \frac{i}{2π} ∂ ¯∂|z|^{2} ∧ \frac{i}{2π} ∂ ¯∂|w|^{2} + \frac{−1}{π^{2}}∂_{z} ¯∂_{z}∂_{w} ¯∂_{w}G(e^{− \frac{1}{2} |z−w|^{2}}). (37)

This completes the proof for the pair correlation case n = 2. (Notice that the formula has the same form in all dimensions.)

The proof for general n is similar. We again write f^{N} = |f^{N}|u^{N} and expand the integrand in (23):

log |c · f^{N}(\frac{z^{1}}{\sqrt{N}})| log |c · f^{N}(\frac{z^{2}}{\sqrt{N}})| ⋯ log |c · f^{N}(\frac{z^{n}}{\sqrt{N}})|
= log |f^{N}(\frac{z^{1}}{\sqrt{N}})| log |f^{N}(\frac{z^{2}}{\sqrt{N}})| ⋯ log |f^{N}(\frac{z^{n}}{\sqrt{N}})|
+ log |f^{N}(\frac{z^{1}}{\sqrt{N}})| log |f^{N}(\frac{z^{2}}{\sqrt{N}})| ⋯ log |f^{N}(\frac{z^{n−1}}{\sqrt{N}})| log |c · u^{N}(\frac{z^{n}}{\sqrt{N}})|
+ ⋯
+ log |c · u^{N}(\frac{z^{1}}{\sqrt{N}})| log |c · u^{N}(\frac{z^{2}}{\sqrt{N}})| ⋯ log |c · u^{N}(\frac{z^{n}}{\sqrt{N}})| .

We denote the terms resulting from this expansion by E_{1}, …, E_{2^{n}}, respectively. As before, the first term converges to the normalized Euclidean “n-fold” Kähler form:

E_{1} = \frac{i}{2π} ∂ ¯∂|z^{1}|^{2} ∧ ⋯ ∧ \frac{i}{2π} ∂ ¯∂|z^{n}|^{2} + O(\frac{1}{N}) .

The E_{2^{n}} term is obtained from the function

(38) G_{n}^{N}(x^{1}, x^{2}, …, x^{n}) := ∫_{ℂ^{d_{N}}} log |c · x^{1}| log |c · x^{2}| ⋯ log |c · x^{n}|\frac{e^{−|c|^{2}}}{π^{d_{N}}} dc ,

x^{1}, x^{2}, …, x^{n} ∈ ℂ^{d_{N}}. Precisely, we substitute

x^{j} = u^{N}(\frac{z^{j}}{\sqrt{N}}) (39)

in (38) and apply the operator (\frac{i}{π})^{n} ∂_{z^{1}} ¯∂_{z^{1}} ⋯ ∂_{z^{n}} ¯∂_{z^{n}}. As above, we define a special Hermitian orthonormal basis {e_{1}, …, e_{n}} for the n-dimensional complex subspace spanned by {x_{1}, …, x_{n}}. We put:

x^{1} = e_{1}
x^{2} = ξ_{21}e_{1} + ξ_{22}e_{2}  ξ_{22} = \sqrt{1 − |ξ_{21}|^{2}}
⋮
x^{n} = ξ_{n1}e_{1} + ⋯ + ξ_{nn}e_{n}  ξ_{nn} = \sqrt{1 − ∑_{j≤n−1} |ξ_{nj}|^{2}}.

Such a basis exists because we can always multiply e_{j} by a phase e^{iθ} so that the last component ξ_{jj} is positive real. We complete {e_{j}} to a basis of ℂ^{d_{N}}, and we now let c_{j} denote coordinates relative to this basis. As above, we rewrite the Gaussian integral in these coordinates. After integrating out the variables {c_{n+1}, …, c_{d_{N}} }, (38) simplifies to the n-dimensional complex Gaussian integral

G_{n}^{N}(x^{1}, …, x^{n}) = G_{n}(ξ_{21}, ξ_{22}, …, ξ_{nn})
= \frac{1}{π^{n}} ∫_{ℂ^{n}} e^{−|c|^{2}} log |c_{1}| log |c_{1}ξ_{21} + c_{2}ξ_{22}| ⋯ log |c_{1}ξ_{n1} + … c_{n}ξ_{nn}|dc .

(40)

Note that the variables ξ_{jk} depend on N; we write ξ_{jk} = ξ_{jk}^{N} when we need to indicate this dependence.

To prove universality, we observe that the ξ_{jk} are universal algebraic functions of the inner products ⟨x^{a}, x^{b}⟩. Indeed,

ξ_{j1}ξ̄_{k1} + ⋯ + ξ_{jk}ξ̄_{kk} = ⟨x^{j}, x^{k}⟩ , 1 ≤ k ≤ j ≤ n , (41)

where we set ξ_{11} = 1. These algebraic functions are obtained by induction (lexicographically) using (41). (The triangular matrix (ξ_{jk}) is just the inverse of the matrix describing the Gram-Schmidt process.)

By (39), it follows that the ξ_{jk}^{N} are universal algebraic functions of the variables

⟨u^{N}(\frac{z^{j}}{\sqrt{N}}), u^{N}(\frac{z^{k}}{\sqrt{N}})⟩ = \frac{Π_{N}(\frac{z^{j}}{\sqrt{N}} , \frac{z^{k}}{\sqrt{N}})}{Π_{N}(\frac{z^{j}}{\sqrt{N}} , \frac{z^{j}}{\sqrt{N}})^{1/2}Π_{N}(\frac{z^{k}}{\sqrt{N}} , \frac{z^{k}}{\sqrt{N}})^{1/2}} = e^{iℑ(z^{j}·z̄^{k})− \frac{1}{2} |z^{j}−z^{k}|^{2}} + O(\frac{1}{\sqrt{N}}) .

We note here that

(42) |x^{1} ∧ ⋯ ∧ x^{n}|^{2} = det(⟨x^{j}, x^{k}⟩) → det (e^{iℑ(z^{j}·z̄^{k})− \frac{1}{2} |z^{j}−z^{k}|^{2}}) = e^{− ∑ |z^{j}|^{2}} det (e^{z_{j}·z̄_{k}}) .

When the z_{j} are distinct (i.e., (z^{1}, …, z^{n}) ∈ G_{n}^{m}), the limit determinant in (42) is nonzero (see [BSZ]) and thus ξ_{jk}^{N} = ξ_{jk}^{∞} + O(\frac{1}{\sqrt{N}}), where the ξ_{jk}^{∞} are universal real-analytic functions of z ∈ G_{n}^{m}. We conclude that the E_{2^{n}} term converges to a universal current:

E_{2^{n}} = (\frac{i}{π} )^{n} ∂_{z^{1}} ¯∂_{z^{1}} ⋯ ∂_{z^{n}} ¯∂_{z^{n}}G_{n}(ξ_{21}^{∞}, …, ξ_{kk}^{∞}) + O(\frac{1}{\sqrt{N}}) .

Consider now a general term E_{a}. Suppose without loss of generality that E_{a} comes from

log |c · u^{N}(\frac{z^{1}}{\sqrt{N}})| ⋯ log |c · u^{N}(\frac{z^{k}}{\sqrt{N}})| log |f^{N}(\frac{z^{k+1}}{\sqrt{N}})| ⋯ log |f^{N}(\frac{z^{n}}{\sqrt{N}})| .

As above we obtain

E_{a} = (\frac{i}{π} )^{k} ∂_{z^{1}} ¯∂_{z^{1}} ⋯ ∂_{z^{k}} ¯∂_{z^{k}}G_{k}(ξ_{21}^{∞}, …, ξ_{kk}^{∞}) ∧ \frac{i}{2π} ∂ ¯∂|z^{k+1}|^{2} ∧ ⋯ ∧ \frac{i}{2π} ∂ ¯∂|z^{n}|^{2} + O(\frac{1}{\sqrt{N}}) .

Hence this term also approaches a universal current. (As in the pair correlation case, terms with only one u^{N} vanish.)

4. Explicit formulae

We now calculate explicitly the limit pair correlation measures K̃_{2}^{∞}(z, w).

4.1. Preliminaries. The first step is to compute ΔG(e^{− \frac{1}{2} r^{2}}), where Δ is the Euclidean Laplacian on ℂ^{m} and r = |ζ| (ζ ∈ ℂ^{m}). To begin this computation, we write a_{j} = r_{j}e^{iφ_{j}} and then rewrite (33)–(34) as

(43) G(cos θ) = \frac{2}{π} ∫_{0}^{∞} ∫_{0}^{∞} ∫_{0}^{2π} r_{1}r_{2}e^{−(r_{1}^{2}+r_{2}^{2})} log r_{1} log |r_{1} cos θ + r_{2}e^{iφ} sin θ|dφdr_{1}dr_{2} .

We now evaluate the inner integral by Jensen’s formula, which gives

(44) ∫_{0}^{2π} log |r_{1} cos θ + r_{2} sin θe^{iφ}|dφ = \{ \begin{matrix}2π log(r_{1} cos θ) & for r_{2} sin θ ≤ r_{1} cos θ \\ 2π log(r_{2} sin θ) & for r_{2} sin θ ≥ r_{1} cos θ \end{matrix}

Hence

(45) G(cos θ) = 4 ∫_{0}^{∞} ∫_{0}^{∞} r_{1}r_{2}e^{−(r_{1}^{2}+r_{2}^{2})} log r_{1} log max(r_{1} cos θ, r_{2} sin θ)dr_{1}dr_{2}.

Now change variables again with r_{1} = ρ cos φ, r_{2} = ρ sin φ to get

G(cos θ) = 4 ∫_{0}^{∞} ∫_{0}^{π/2} ρ^{3}e^{−ρ^{2}} log(ρ cos φ) log max(ρ cos φ cos θ, ρ sin φ sin θ) cos φ sin φdφdρ . (46)

12 PAVEL BLEHER, BERNARD SHIFFMAN, AND STEVE ZELDITCH

Since

log max(ρ cos φ cos θ, ρ sin φ sin θ) = log(ρ cos φ cos θ) + log^{+}(tan φ tan θ) , we can write G = G_{1} + G_{2}, where

(47) G_{1}(cos θ) = 4 ∫_{0}^{∞} ∫_{0}^{π/2} ρ^{3}e^{−ρ^{2}} log(ρ cos φ) log(ρ cos φ cos θ) cos φ sin φdφdρ
(48) G_{2}(cos θ) = 4 ∫_{0}^{∞} ∫_{π/2−θ}^{π/2} ρ^{3}e^{−ρ^{2}} log(ρ cos φ) log(tan φ tan θ) cos φ sin φdφdρ .

From (47), G_{1}(cos θ) = C_{1} + C_{2} log cos θ and thus

G_{1}(e^{− \frac{1}{2} r^{2}}) = C_{1} − \frac{1}{2}C_{2}r^{2} ,

so that

ΔG_{1}(e^{− \frac{1}{2} r^{2}}) = (\frac{d^{2}}{dr^{2}} + \frac{2m − 1}{r} \frac{d}{dr}) (C_{1} − \frac{1}{2}C_{2}r^{2}) = −2mC_{2} . (49)

We now evaluate ΔG_{2}(e^{− \frac{1}{2} r^{2}}). Since the integrand in (48) vanishes when φ = π/2 − θ, we have

\frac{d}{dr}G_{2}(cos θ) = 4 (\frac{d}{dr} log tan θ) ∫_{0}^{∞} ∫_{π/2−θ}^{π/2} ρ^{3}e^{−ρ^{2}} log(ρ cos φ) cos φ sin φdφdρ .

Substituting tan^{2} θ = e^{r^{2}} − 1, we have

\frac{d}{dr} log tan θ = \frac{r}{1 − e^{−r^{2}}} .

Thus

\frac{d}{dr}G_{2}(e^{− \frac{1}{2} r^{2}}) = \frac{4r}{1 − e^{−r^{2}}} (I_{1} + I_{2}) ,

where

I_{1} = ∫_{0}^{∞} ∫_{π/2−θ}^{π/2} ρ^{3}e^{−ρ^{2}}(log ρ) cos φ sin φdφdρ = C sin^{2} θ = C(1 − e^{−r^{2}}) ,
I_{2} = ∫_{0}^{∞} ∫_{π/2−θ}^{π/2} ρ^{3}e^{−ρ^{2}}(log cos φ) cos φ sin φdφdρ .

We compute

I_{2} = \frac{1}{2} ∫_{π/2−θ}^{π/2}(log cos φ) cos φ sin φdφ = \frac{1}{2} ∫_{0}^{sin θ} t log tdt
= \frac{1}{8}(sin^{2} θ log sin^{2} θ − sin^{2} θ) = \frac{1}{8}(1 − e^{−r^{2}}) [log(1 − e^{−r^{2}}) − 1]

Thus

\frac{d}{dr}G_{2}(e^{− \frac{1}{2} r^{2}}) = \frac{r}{2} log(1 − e^{−r^{2}}) + C′r . (50)

Hence by (49) and (50),

ΔG(e^{− \frac{1}{2} r^{2}}) = −2mC_{2} + (\frac{d}{dr} + \frac{2m − 1}{r}) (\frac{r}{2} log(1 − e^{−r^{2}}) + C′r)
(51)  = m log(1 − e^{−r^{2}}) + \frac{r^{2}}{e^{r^{2}} − 1} + C′′ .

4.2. Pair correlation in dimension 1. In dimension one, the pair correlation form is the same as the pair correlation measure. We first give our universal formula in the onedimensional case. Our formula agrees with that of Bogomolny-Bohigas-Leboeuf [BBL] and Hannay [Ha] for SU(2) polynomials.

Theorem 4.1. Suppose dim M = 1. Then

K⃗_{2}^{N}(\frac{z}{\sqrt{N}} , \frac{w}{\sqrt{N}}) → K⃗_{2}^{∞}(z, w) = [πδ_{0}(z − w) + H(\frac{1}{2}|z − w|^{2})] \frac{i}{2π} ∂ ¯∂|z|^{2} ∧ \frac{i}{2π} ∂ ¯∂|w|^{2} ,

where

H(t) = \frac{(sinh^{2} t + t^{2}) cosh t − 2t sinh t}{sinh^{3} t} = t − \frac{2}{9}t^{3} + \frac{2}{45}t^{5} + O(t^{7}) .

Proof. Making the change of variables ζ = z − w, we have by (37),

E (Ẑ^{N}(z) ⊗ Ẑ^{N}(w)) → \frac{i}{2π} ∂ ¯∂|z|^{2} ∧ \frac{i}{2π} ∂ ¯∂|w|^{2} − \frac{1}{π^{2}} ∂_{z} ¯∂_{z}∂_{w} ¯∂_{w}G(e^{− \frac{1}{2} |z−w|^{2}})

= [1 + 4\frac{∂^{2}}{∂z ¯∂z} \frac{∂^{2}}{∂w ¯∂w}G(e^{− \frac{1}{2} |z−w|^{2}})] \frac{i}{2π} ∂ ¯∂|z|^{2} ∧ \frac{i}{2π} ∂ ¯∂|w|^{2}

= [1 + 4 (\frac{∂^{2}}{∂ζ ¯∂ζ} )^{2} G(e^{− \frac{1}{2} |ζ|^{2}})] \frac{i}{2π} ∂ ¯∂|z|^{2} ∧ \frac{i}{2π} ∂ ¯∂|w|^{2}

= [1 + \frac{1}{4}Δ^{2}G(e^{− \frac{1}{2} r^{2}})] \frac{i}{2π} ∂ ¯∂|z|^{2} ∧ \frac{i}{2π} ∂ ¯∂|w|^{2}

By (51) with m = 1, we have

Δ^{2}G(e^{− \frac{1}{2} r^{2}}) = (\frac{d^{2}}{dr^{2}} + \frac{1}{r} \frac{d}{dr}) [log(1 − e^{−r^{2}}) + \frac{r^{2}}{e^{r^{2}} − 1}]

= 4πδ_{0} + \frac{8(e^{r^{2}} − 1)^{2} − 16r^{2}e^{r^{2}}(e^{r^{2}} − 1) + 4r^{4}e^{r^{2}}(e^{r^{2}} + 1)}{(e^{r^{2}} − 1)^{3}} .

Finally,

[1 + \frac{1}{4}Δ^{2}G(e^{− \frac{1}{2} r^{2}})] = πδ_{0} + \frac{(e^{r^{2}} + 1)(e^{r^{2}} − 1)^{2} − 4r^{2}e^{r^{2}}(e^{r^{2}} − 1) + r^{4}e^{r^{2}}(e^{r^{2}} + 1)}{(e^{r^{2}} − 1)^{3}}

= πδ_{0} + \frac{(sinh^{2} \frac{1}{2}r^{2} + \frac{1}{4}r^{4}) cosh \frac{1}{2}r^{2} − r^{2} sinh \frac{1}{2}r^{2}}{sinh^{3} \frac{1}{2}r^{2}} .

4.3. Pair correlation in higher dimensions. The limit pair correlation measure is given by

K̃_{2}^{∞}(z, w) = lim_{N→∞} N^{2(m−1)}K̃_{2}^{N}(\frac{z}{\sqrt{N}} , \frac{w}{\sqrt{N}})

= K⃗_{2}^{∞}(z, w) ∧ \frac{1}{(m − 1)!} (\frac{i}{2}∂ ¯∂|z|^{2})^{m−1} ∧ \frac{1}{(m − 1)!} (\frac{i}{2}∂ ¯∂|w|^{2})^{m−1} .

(The scaling N^{2(m−1)} comes from the fact that Nω(\frac{z}{\sqrt{N}}) = N(τ\sqrt{N})_{∗}ω → \frac{i}{2}∂ ¯∂|z|^{2}.) We now

compute K̃_{2}^{∞} for the case of a manifold of general dimension m > 1. It is convenient to express this measure in terms of the expected density of zeros

K̃_{1}^{∞}(z) = lim_{N→∞} N^{m−1}K̃_{1}^{N}(\frac{z}{\sqrt{N}}) = \frac{m}{π}dV_{ℂ^{m}} = \frac{1}{π(m − 1)!} (\frac{i}{2}∂ ¯∂|z|^{2})^{m} . (52)

We have the following explicit universal formula for the limit pair correlation measure. In particular, it gives the scaling limit pair correlation for the zeros of SU(m + 1)-polynomials.

Theorem 4.2. Suppose dim M = m > 1. Then

K̃_{2}^{∞}(z, w) = [γ_{m}(\frac{1}{2}|z − w|^{2})] K̃_{1}^{∞}(z) ∧ K̃_{1}^{∞}(w) ,

where

γ_{m}(t) = \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}
= \frac{(m − 1)}{2m}t^{−1} + \frac{m − 1}{2m} + \frac{(m + 2)(m + 1)}{6m^{2}}t

−\frac{(m + 4)(m + 3)}{90m^{2}}t^{3} + \frac{(m + 6) (m + 5)}{945m^{2}}t^{5} + O(t^{7}) .

Proof. By (37) and (51), again writing ζ = z − w (except this time ζ ∈ ℂ^{m}),

K̃_{2}^{∞}(z, w) = [1 + \frac{4}{m^{2}} ∑_{j,k=1}^{m} \frac{∂^{2}}{∂z_{j}∂z̄_{j}} \frac{∂^{2}}{∂w_{k}∂w̄_{k}} G(e^{− \frac{1}{2} |z−w|^{2}})] K̃_{1}^{∞}(z) ∧ K̃_{1}^{∞}(w)

= [1 + \frac{1}{4m^{2}} Δ_{ζ}^{2}G(e^{− \frac{1}{2} |ζ|^{2}})] K̃_{1}^{∞}(z) ∧ K̃_{1}^{∞}(w)

= [1 + \frac{1}{4m^{2}} (\frac{d^{2}}{dr^{2}} + \frac{2m − 1}{r} \frac{d}{dr}) (m log(1 − e^{−r^{2}}) + \frac{r^{2}}{e^{r^{2}} − 1})] K̃_{1}^{∞}(z) ∧ K̃_{1}^{∞}(w) . (53)

Computing the Laplacian in (53) leads to the stated formula.

Note that if we substitute m = 1 in the expression for γ_{m}(t), we obtain Hannay’s function H(t). However for the case m > 1, the limit measure is absolutely continuous on ℂ^{m} × ℂ^{m}, whereas in the one-dimensional case, there is a self-correlation delta measure.

References

[BD] P. Bleher and X. Di, Correlations between zeros of a random polynomial, J. Stat. Phys. 88 (1997), 269–305.

[BSZ] P. Bleher, B. Shiffman and S.Zelditch, Universality and scaling of correlations between zeros on complex manifolds (preprint 1999).

[BBL] E. Bogomolny, O. Bohigas, and P. Leboeuf, Quantum chaotic dynamics and random polynomials, J. Stat. Phys. 85 (1996), 639–679.

[GH] P. Griffiths and J. Harris, Principles of Algebraic Geometry, Wiley-Interscience, N.Y. (1978).

[Ha] J. H. Hannay, Chaotic analytic zero points: exact statistics for those of a random spin state, J. Phys. A: Math. Gen. 29 (1996), 101–105.

[SZ] B. Shiffman and S. Zelditch, Distribution of zeros of random and quantum chaotic sections of positive line bundles, Commun. Math. Phys. 200 (1999), 661–683.

[Ti] G. Tian, On a set of polarized Kähler metrics on algebraic manifolds, J. Diff. Geometry 32 (1990), 99–130.

[Ze] S. Zelditch, Szegö kernels and a theorem of Tian, Int. Math. Res. Notices 6 (1998), 317–331.

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