% LATEX 2e % Poincare-Lelong approach to universality and scaling of % correlations between zeros % Pavel Bleher, Bernard Shiffman, Steve Zelditch \documentclass[12pt]{amsart} \headheight=8.55pt %6.15pt \textheight=8.0in \textwidth=6.5in \oddsidemargin=0in \evensidemargin=0in \topmargin=0in \newcommand{\kahler}{K\"ahler\ } \newcommand{\sqrtn}{\sqrt{N}} \newcommand{\wt}{\widetilde} \newcommand{\wh}{\widehat} \newcommand{\PP}{{\mathbb P}} \newcommand{\R}{{\mathbb R}} \newcommand{\C}{{\mathbb C}} \newcommand{\Q}{{\mathbb Q}} \newcommand{\Z}{{\mathbb Z}} \newcommand{\B}{{\mathbb B}} \newcommand{\CP}{\C\PP} \renewcommand{\d}{\partial} \newcommand{\dbar}{\bar\partial} \newcommand{\ddbar}{\partial\dbar} \newcommand{\U}{{\rm U}} \newcommand{\E}{{\mathbf E}\,} \renewcommand{\H}{{\mathbf H}} \newcommand{\Enu}{{\mathbf E}_\nu\,} \newcommand{\half}{{\frac{1}{2}}} \newcommand{\vol}{{\operatorname{Vol}}} \newcommand{\codim}{{\operatorname{codim\,}}} \newcommand{\SU}{{\operatorname{SU}}} \newcommand{\FS}{{{\operatorname{FS}}}} \newcommand{\corr}{{\operatorname{Corr}}} \newcommand{\supp}{{\operatorname{Supp\,}}} \renewcommand{\phi}{\varphi} \newcommand{\eqd}{\buildrel {\operatorname{def}}\over =} \newcommand{\go}{\mathfrak} \newcommand{\ccal}{\mathcal{C}} \newcommand{\dcal}{\mathcal{D}} \newcommand{\ecal}{\mathcal{E}} \newcommand{\gcal}{\mathcal{G}} \newcommand{\lcal}{\mathcal{L}} \newcommand{\pcal}{\mathcal{P}} \newcommand{\ocal}{\mathcal{O}} \newcommand{\scal}{\mathcal{S}} \newcommand{\ucal}{\mathcal{U}} \newcommand{\zcal}{\mathcal{Z}} \newcommand{\al}{\alpha} \newcommand{\be}{\beta} \newcommand{\ga}{\gamma} \newcommand{\Ga}{\Gamma} \newcommand{\La}{\Lambda} \newcommand{\la}{\lambda} \newcommand{\ep}{\varepsilon} \newcommand{\de}{\delta} \newcommand{\De}{\Delta} \newcommand{\om}{\omega} \newcommand{\di}{\displaystyle} \newcommand{\bbb}{|\!|\!|} \newtheorem{theo}{{\sc Theorem}}[section] \newtheorem{cor}[theo]{{\sc Corollary}} \newtheorem{lem}[theo]{{\sc Lemma}} \newtheorem{prop}[theo]{{\sc Proposition}} \newenvironment{example}{\medskip\noindent{\it Example:\/} }{\medskip} \newenvironment{rem}{\medskip\noindent{\it Remark:\/} }{\medskip} \newenvironment{defin}{\medskip\noindent{\it Definition:\/} }{\medskip} \newenvironment{claim}{\medskip\noindent{\it Claim:\/} }{\medskip} \title[Poincar\'e-Lelong approach to correlations between zeros] {Poincar\'e-Lelong approach to universality and scaling of correlations between zeros} \author{Pavel Bleher} \address{Department of Mathematical Sciences, IUPUI, Indianapolis, IN 46202, USA} \email{bleher@math.iupui.edu} \author{Bernard Shiffman} \address{Department of Mathematics, Johns Hopkins University, Baltimore, MD 21218, USA} \email{shiffman@math.jhu.edu} \author{Steve Zelditch} \address{Department of Mathematics, Johns Hopkins University, Baltimore, MD 21218, USA} \email{zel@math.jhu.edu} \thanks{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.} %\date{March 4, 1999} \begin{document} \begin{abstract} This note is concerned with the scaling limit as $N \to \infty$ 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 \kahler 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 $M$. \end{abstract} \maketitle \section*{Introduction} This note is a companion to our article \cite{BSZ}, in which we study the correlations between the zeros of a random holomorphic section $s \in H^0(M, L^N)$ of a power $L^N$ of a positive line bundle $L \to M$ 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 $\sim N {\rm Vol\,}(U)$, we rescale $U \to \sqrt{N} U$ 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 {\it local statistics} of zeros are universal, i.e. independent of $L, M, \omega$ and $z_0$. To define our statistics, we first provide $H^0(M,L^N)$ with a natural Gaussian measure (see \S\S\ref{cxgeom}--\ref{random}). The local statistics are measured by the scaled {\it $n$-point zero correlation forms} $\vec K_{n}^N(\frac{z^1}{\sqrtn},\dots,\frac{z^n}{\sqrtn})$, $z^j\in \sqrtn\,U$ (see \S\ref{CC}). They are smooth off the diagonal, and their norms define scaled zero correlation measures $\wt K_{n}^N(\frac{z^1}{\sqrtn},\dots,\frac{z^n}{\sqrtn})$. (The scaled correlation forms extend to all of $(\sqrtn\,U)^n$ as currents of order 0.) The correlation measures $\wt K_{n}^N$ give the expected value of the product of the volumes of the zero set in $n$ domains in $M$; see equation (\ref{defcormeas2}). In \cite{BSZ}, we used geometric probability methods and a (universal) scaled Szeg\"o kernel to prove that there exist universal limits as $N \to \infty$ of these correlation measures and more generally of the correlations between simultaneous zeros of $k\le m$ sections. Here we use a complex analytic approach based on the Poincar\'e-Lelong formula for the currents of integration over the zero set of a section, together with the scaled Szeg\"o kernel from \cite{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. Our universality theorem is as follows: \bigskip %\pagebreak \noindent{\sc Main Theorem.} {\it There is a universal current $\vec K_{n}^{\infty}\in\dcal'{}^{(m-1)n,(m-1)n}(\C^{mn})$ 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 $\vec K_{n}^N$ be the $n$-point zero correlation current on $M^n$. Suppose $z^0\in M$ and choose local holomorphic coordinates in $M$ about $z^0$ such that $\Theta_h|_{z^0} =\ddbar|z|^2$. Then $$\vec K_{n}^N\left(\frac{z^1}{\sqrt{N}}, \dots,\frac{z^n}{\sqrt{N}}\right) =\vec K_{n}^{\infty}(z^1,\dots,z^n)+ O\left(\frac{1}{\sqrtn}\right)\,.$$ Furthermore, $\vec K_{n}^{\infty}$ is a smooth form on the off-diagonal domain $\gcal^m_n$ consisting of $n$-tuples of distinct points $z^1,\dots,z^n$ in $\C^m$. The error term has $k^{th}$ order derivatives $\le \frac{C_{A,k}}{\sqrtn}$ on each compact subset $A\subset \gcal^m_n$, $\forall k\ge 0$. } \bigskip 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 \ref{Hformula} and \ref{mformula}). 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 \cite{BBL} and Hannay \cite{H} in the case of random $\SU(2)$-polynomials. Similar formulas for correlations of zeros of real polynomials were given in \cite{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 $\ep$ around $z^{1}, \dots, z^n$, respectively, is $\approx c\ep^{2n} K_{n}^N(z^1, \dots,z^n)\;$ (where $K_n^N$ is the correlation {\it function\/} given by $K_n^N dV_M= \wt K_{n}^N$). The correlation form $\vec K_{n}^N$ gives a more refined probability that the zero divisor of $s$ has tangent hyperplanes close to $n$ fixed complex hyperplanes in $T_M$. \medskip \noindent{\bf 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. \section{Notation}\label{notation} We summarize here the notation from complex analysis that we will need in the proof. This notation is the same as in \cite{SZ} and \cite{BSZ}, except that different normalizations for the metric and volume form are used in \cite{SZ}. \subsection{Complex geometry}\label{cxgeom} We denote by $(L, h) \to M$ a holomorphic line bundle with smooth Hermitian metric $h$ whose curvature form \begin{equation}\label{curvature}\Theta_h=-\ddbar \log\|e_L\|_h^2\;,\end{equation} is a positive (1,1)-form. Here, $e_L$ is a local non-vanishing holomorphic section of $L$ over an open set $U\subset M$, and $\|e_L\|_h=h(e_L,e_L)^{1/2}$ is the $h$-norm of $e_L$. As in \cite{BSZ}, we give $M$ the Hermitian metric corresponding to the \kahler form $\omega=\frac{\sqrt{-1}}{2}\Theta_h$ and the induced Riemannian volume form \begin{equation}\label{dV} dV_M= \frac{1}{m!} \omega^m\;.\end{equation} We denote by $H^0(M, L^{N})$ the space of holomorphic sections of $L^N=L\otimes\cdots\otimes L$. The metric $h$ induces Hermitian metrics $h^N$ on $L^N$ given by $\|s^{\otimes N}\|_{h^N}=\|s\|_h^N$. We give $H^0(M,L^N)$ the Hermitian inner product \begin{equation}\label{inner}\langle s_1, s_2 \rangle = \int_M h^N(s_1, s_2)dV_M \quad\quad (s_1, s_2 \in H^0(M,L^N)\,)\;,\end{equation} and we write $|s|=\langle s,s \rangle^{1/2}$. For a holomorphic section $s\in H^0(M, L^N)$, we let $Z_s$ denote the current of integration over the zero divisor of $s$: $$(Z_s,\phi)=\int_{Z_s}\phi\,,\quad \phi\in\dcal^{m-1,m-1}(M)\,.$$ The Poincar\'e-Lelong formula (see e.g., \cite{GH}) expresses the integration current of a holomorphic section $s=ge_L^{\otimes N}$ in the form: \begin{equation} \label{PL} Z_s = \frac{i}{\pi} \ddbar \log |g| = \frac{i}{\pi} \ddbar \log \|s\|_{h^N} + N\omega\;.\end{equation} We also denote by $|Z_s|$ the Riemannian $(2m-2)$-volume along the regular points of $Z_s$, regarded as a measure on $M$: \begin{equation}\label{volmeasure} (|Z_s|,\phi)=\int_{Z_s^{\rm reg}}\phi d\vol_{2m-2}=\frac{1}{(m-1)!}\int_{Z_s^{\rm reg}}\phi \omega^{m-1}\,;\end{equation} i.e., $|Z_s|$ is the ``total variation measure'' of the current of integration over $Z_s$: \begin{equation}\label{volmeasure2}|Z_s|= Z_s\wedge{\textstyle \frac{1}{(m-1)!}}\omega^{m-1}\,.\end{equation} \subsection{Random sections and Gaussian measures}\label{random} We now give $H^0(M,L^N)$ the complex Gaussian probability measure \begin{equation}\label{gaussian}d\mu(s)=\frac{1}{\pi^{d_N}}e^{- |c|^2}dc\,,\qquad s=\sum_{j=1}^{d_N}c_jS_j^N\,,\end{equation} 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 $\Re c_j, \Im c_j$ ($j=1,\dots,d_N$) are independent random variables with mean 0 and variance $\half$; i.e., $$\E c_j = 0,\quad \E c_j c_k = 0,\quad \E c_j \bar c_k = \de_{jk}\,.$$ Here and throughout this paper, $\E$ denotes expectation: $\E\phi = \int \phi d\mu$. We then regard the currents $Z_s$ (resp.\ measures $|Z_s|$), as current-valued (resp.\ measure-valued) random variables on the probability space $(H^0(M,L^N), d\mu)$; i.e., for each test form (resp.\ function) $\phi$, $(Z_s,\phi)$ (resp.\ $(|Z_s|,\phi)$) 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. \subsection{Correlation currents and measures}\label{CC} The $n$-point correlation current of the zeros is the current on $M^n = M \times M \times \cdots \times M$ ($n$ times) given by \begin{equation} \vec K_n^N(z^1, \dots, z^n):= \E(Z_s(z^1) \otimes Z_s(z^2) \otimes \dots \otimes Z_s(z^n)) \end{equation} in the sense that for any test form $\phi_1(z^1) \otimes \dots \otimes \phi_n(z^n) \in \dcal^{m-1,m-1}(M) \otimes \dots \otimes \dcal^{m-1,m-1}(M)$, \begin{equation} \big(\vec K_n^N(z^1, \dots, z^n), \phi_1(z^1) \otimes \dots \otimes \phi_n(z^n)\big) = \E \left[\big( Z_s,\phi_1\big)\big( Z_s,\phi_2\big)\cdots \big(Z_s, \phi_n\big)\right]. \end{equation} In a similar way we define the $n$-point correlation measures $\wt K_n^N$ as the ``total variation measures" of the $n$-point correlation currents: \begin{equation}\wt K_n^N(z^1, \dots, z^n)=\vec K_n^N(z^1, \dots, z^n)\wedge \frac{1}{(m-1)!}\om_{z^1}^{m-1}\wedge\cdots\wedge \frac{1}{(m-1)!}\om_{z^n}^{m-1} \,.\label{defcormeas}\end{equation} By (\ref{volmeasure2}) and (\ref{defcormeas}), we have \begin{equation}\big(\wt K_n^N(z^1, \dots, z^n), \phi_1(z^1) \dots \phi_n(z^n)\big) = \E \big[(|Z_s|, \phi_1)(|Z_s|, \phi_2)\cdots (|Z_s|, \phi_n)\big] \end{equation} where $\phi_j \in \ccal^0(M)$. Equivalently, \begin{equation}\wt K_n^N(U_1\times\cdots\times U_n) = \E\left[\prod_{j=1}^n \vol(Z_s\cap U_j)\right]\,, \qquad U_1,\dots,U_n\subset M\,. \label{defcormeas2}\end{equation} \begin{rem} In the case of pair correlation on a Riemann surface ($n=2, \dim M=1$), the correlation measures take the form $$ \vec K^N_2(z,w) = [\Delta]\wedge (\vec K^N_1(z) \otimes 1) + \kappa^N(z,w)\omega_z\otimes \omega_w \quad (N\gg 0)$$ where $[\Delta]$ denotes the current of integration along the diagonal $\Delta=\{(z,z)\}\subset M\times M$, and $\kappa^N\in\ccal^\infty(M\times M)$. \end{rem} \subsection{Szeg\"o kernels} As in \cite{Z,SZ,BSZ} and elsewhere, we analyze the $N \to \infty$ limit by lifting it to a principal $S^1$ bundle $\pi: X \to M$. Let us recall how this goes. We denote by $L^*$ the dual line bundle to $L$, and define $X$ as the circle bundle $X=\{\la \in L^* : \|\la\|_{h^*}= 1\}$, where $h^*$ is the norm on $L^*$ dual to $h$. We can view $X$ as the boundary of the disc bundle $D = \{\la \in L^* : \rho(\la)>0\}$, where $\rho(\la)=1-\|\la\|^2_{h^*}$. The disc bundle $D$ is strictly pseudoconvex in $L^*$, since $\Theta_h$ is positive, and hence $X$ inherits the structure of a strictly pseudoconvex CR manifold. Associated to $X$ is the contact form $\al= -i\partial\rho|_X=i\dbar\rho|_X$. We also give $X$ the volume form \begin{equation}\label{dvx}dV_X=\frac{1}{m!}\al\wedge (d\al)^m=\al\wedge\pi^*dV_M\,.\end{equation} The setting for our analysis of the Szeg\"o kernel is the Hardy space $H^2(X) \subset \lcal^2(X)$ of square integrable CR functions on $X$, where we use the inner product \begin{equation}\label{unitary} \langle F_1, F_2\rangle =\frac{1}{2\pi}\int_X F_1\overline{F_2}dV_X\,,\quad F_1,F_2\in\lcal^2(X)\,.\end{equation} We let $r_{\theta}x =e^{i\theta} x$ ($x\in X$) denote the $S^1$ action on $X$. The action $r_\theta$ commutes with the Cauchy-Riemann operator $\bar{\partial}_b$; hence $H^2(X) = \bigoplus_{N = 0}^{\infty} H^2_N(X)$, where $$H^2_N(X) = \{ F \in H^2(X): F(r_{\theta}x) = e^{i N \theta} F(x) \}\,.$$ A section $s_N$ of $L^{N}$ determines an equivariant function $\hat{s}_N$ on $X$: \begin{equation}\label{snhat}\hat{s}_N(z, \lambda) = \left( \lambda^{\otimes N}, s_N(z) \right)\,,\quad (z,\la)\in X\,;\end{equation} then $\hat s_N(r_\theta x) = e^{iN\theta} s_N(x)$. The map $s\mapsto \hat{s}$ is a unitary equivalence between $H^0(M, L^{\otimes N})$ and $H^2_N(X)$. We let $\Pi_N : \lcal^2(X) \rightarrow H^2_N(X)$ denote the orthogonal projection. The Szeg\"o kernel $\Pi_N(x,y)$ is defined by \begin{equation} \Pi_N F(x) = \int_X \Pi_N(x,y) F(y) dV_X (y)\,, \quad F\in\lcal^2(X)\,. \end{equation} It can be given as \begin{equation}\label{Szego}\Pi_N(x,y)=\sum_{j=1}^{d_N}\wh S_j^N(x)\overline{\wh S_j^N(y)}\,,\end{equation} where $S_1^N,\dots,S_{d_N}^N$ form an orthonormal basis of $H^0(M,L^N)$. \section{Scaling} In order that we may study the local nature of the random variable $Z_s$, we fix a point $z^0\in M$ and choose a holomorphic coordinate chart $\Psi:\Omega,0\to U,z_0$ ($\Omega\subset \C^m,\ U\subset M$) such that \begin{equation}\label{coord} \Psi^*\omega_{z^0} =\left.\frac{i}{ 2}\sum_{j=1}^m dz_j\wedge d\bar z_j\right|_0\;.\end{equation} For example, if $L$ is the hyperplane section bundle $\ocal(1)$ over $\C\PP^m$ with the Fubini-Study metric $h_\FS$, and $z_0=(1:0:\dots:0)$, then the coordinate chart $$\Psi:\C^m\to U=\{w\in\CP^m:w_0\neq 0\}\,,\quad \Psi(z)=(1:z_1:\dots :z_m)$$ (i.e., $z_j=w_j/w_0$) satisfies (\ref{coord}). To simplify notation, we identify $U$ with $\Omega$. For a current $T\in\dcal'{}^{p,q}(\Omega)$, we write $$T\left(\frac{z}{\sqrtn}\right)=\left(\tau_{\sqrtn}\right)_* T\in \dcal'{}^{p,q}(\sqrtn\Omega)\qquad (\tau_\la(z)=\la z)\,.$$ (In particular, if $T=\sum T_{jk}(z)dz_j\wedge d\bar z_k$, then $T(\frac{z}{\sqrtn})= \frac{1}{N}\sum T_{jk}(\frac{z}{\sqrtn})dz_j\wedge d\bar z_k$.) We define the {\it rescaled zero current} of $s \in H^0(M,L^N)$ by \begin{equation} \label{zhat} \wh Z^N_s(z):=Z_{s}\left(\frac{z}{\sqrtn}\right)\,. \end{equation} The scaled $n$-point correlation currents are then defined by: \begin{equation}\label{def-var} \E\left( \wh{Z}^N_s(z^1) \otimes \wh{Z}^N_s(z^2) \otimes \cdots \otimes \wh{Z}^N_s(z^n)\right) =\vec K_n^N\left(\frac{z^1}{\sqrtn},\dots,\frac{z^n}{\sqrtn}\right)\in \dcal'{}^{n,n}(M^n). \end{equation} Following the approach of \cite{SZ}, we fix an orthonormal basis $\{S^N_j\}$ of $H^0(M, L^{N})$ and write $S^N_j = f^N_j e_L^{\otimes N}$ over $U$. Any section in $H^0(M, L^{N})$ may then be written as $s = \sum_{j=1}^{d_N} c_j f^N_j e_L^N$. To simplify the notation we let $f^N=(f^N_1,\ldots,f^N_{d_N}):U\to \C^{d_N}$ and we put $$ \sum_{j=1}^{d_N} c_j f_j = c\cdot f^N\;.$$ Hence \begin{equation}\label{PL2} Z_s = \frac{\sqrt{-1}}{\pi} \ddbar \log| c\cdot f^N|\;,\quad \wh Z^N_s =\frac{\sqrt{-1}}{ \pi}\d_z\dbar_z\log\big|c\cdot f^N\big(\frac{z}{\sqrtn}\big) \big| \end{equation} and therefore \begin{equation} \wh Z_s(z^1) \otimes\cdots\otimes \wh Z_s(z^n)= \left(\frac{i }{ \pi}\right)^n\d_{z^1}\dbar_{z^1}\cdots\d_{z^n}\dbar_{z^n} \left[\log|c\cdot f^N(\frac{z^1}{\sqrtn})|\cdots\log| c\cdot f^N(\frac{z^n}{\sqrtn})|\right]\;. \end{equation} We then can write the rescaled correlation currents in the form \begin{equation}\label{PLcorscaled}\begin{array}{l}\displaystyle \vec K^N_n\left(\frac{z^1}{\sqrtn},\dots,\frac{z^n}{\sqrtn}\right)= \E\big(\wh Z_s(z^1) \otimes\cdots\otimes \wh Z_s(z^n)\big)\\[12pt] \displaystyle\quad = \left(\frac{i }{ \pi}\right)^n \d_{z^1}\dbar_{z^1}\cdots\d_{z^n}\dbar_{z^n} \int_{\C^{d_N}} \log\left|c\cdot f^N\left(\frac{z^1}{\sqrtn}\right)\right| \cdots\log\left|c\cdot f^N\left(\frac{z^n}{\sqrtn}\right) \right|\frac{e^{-|c|^2}}{\pi^{d_N}} dc\;.\end{array} \end{equation} \subsection{Scaling limit of the Szeg\"o kernel}\label{scalingszego} The asymptotics of the Szeg\"o kernel along the diagonal were given by \cite{Ti} and \cite{Z}: \begin{equation}\label{diag}\frac{\pi^m}{N^m}\Pi_N(x,x)= 1 +O(N^{-1})\,.\end{equation} For our proof of the Main Theorem, we need the following lemma from \cite{BSZ}, which gives the `near-diagonal' asymptotics of the Szeg\"o kernel. \begin{lem} \label{neardiag} Let $z^0\in M$ and choose local coordinates $\{z^j\}$ in a neighborhood of $z_0$ so that $z^0=0$ and $\Theta_h(z_0)=\sum dz^j\wedge d\bar z^j$. Then \begin{eqnarray*}\frac{\pi^m}{N^m}\Pi_N(\frac{z}{\sqrtn},\frac{\theta}{N }; \frac{w}{\sqrtn},\frac{\phi}{N})&=&e^{i2\pi (\theta-\phi)+i\Im (z\cdot \bar w)-\half |z-w|^2} + O(N^{-1/2})\;.\end{eqnarray*} \end{lem} Here, $(z,\theta)$ denotes the point $e^{i\theta}\|e_L(z)\|_he^*_L(z)\in X$, and similarly for $(w,\phi)$. In (\ref{diag}) and Lemma \ref{neardiag}, the expression $O(N^\al)$ means a term with $k^{\rm th}$ order derivatives $\le C_k N^\al$, for all $k\ge 0$. Lemma \ref{neardiag} says that the Szeg\"o kernel has a universal scaling limit. In fact, its scaling limit is the first Szeg\"o kernel of the reduced Heisenberg group; see \cite{BSZ}. \section{Universality}\label{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 (\ref{PLcorscaled}) with $n=2$) \begin{equation}\label{PLcor2}\begin{array}{l}\displaystyle \vec K^N_2\left(\frac{z}{\sqrtn},\frac{w}{\sqrtn}\right)= \E\left(\wh Z^N_s(z) \otimes \wh Z^N_s(w)\right)\\[10pt] \displaystyle\quad\quad\quad=\frac{-1 }{ \pi^2} \d_{z}\dbar_{z}\d_w\dbar_w \int_{\C^{d_N}}\log\left|c\cdot f^N(\frac{z}{\sqrtn})\right|\,\log\left| c\cdot f^N(\frac{w}{\sqrtn})\right| \frac{e^{-|c|^2}}{\pi^{d_N}} dc \end{array} \end{equation} is universal. As in \cite{SZ}, we write $f^N=|f^N|u^N$ and expand the integrand in (\ref{PLcor2}): \begin{eqnarray}\log|c\cdot f^N(\frac{z}{\sqrtn})|\log|c\cdot f^N(\frac{w}{\sqrtn})|&=& \log |f^N( \frac{z}{\sqrt{N}})| \log |f^N( \frac{w}{\sqrt{N}})|\nonumber \\ &&+\log|f^N(\frac{z}{\sqrt{N}})| \log |c\cdot u^N(\frac{w}{\sqrt{N}}) |\nonumber \\&&+ \log |f^N(\frac{w}{\sqrt{N}})| \log |c\cdot u^N( \frac{z}{\sqrt{N}}) |\nonumber \\ && +\log |c\cdot u^N(\frac{z}{\sqrt{N}})| \log |c\cdot u^N(\frac{w}{\sqrt{N}}) |\;.\label{expand}\end{eqnarray} Let us denote the terms resulting from this expansion by $E_1,\ E_2,\ E_3,\ E_4$, respectively. In particular, \begin{equation}\label{E1}E_1=\frac{-1 }{ \pi^2}\d_z\dbar_z\d_w\dbar_w \left[\log \big|f^N( \frac{z}{\sqrt{N}}, \frac{z}{\sqrt{N}})\big| \log \big|f^N( \frac{w}{\sqrt{N}}, \frac{w}{\sqrt{N}})\big| \right]\;.\end{equation} By (\ref{snhat}), $\wh S^N_j(z,\theta)=e^{iN\theta} \|e_L(z)\|_h^{N} f^N_j (z)$, where $(z,\theta)$ are the coordinates in $X$ given in \S \ref{scalingszego}. By (\ref{Szego}), \begin{equation}\label{szego1} \Pi_N(z,w)=\|e_L(z)\|_h^{N} \|e_L(w)\|_h^{N}\langle f^N(z), f^N(w)\rangle\,, \end{equation} where we write $\Pi_N(z,w)=\Pi_N(z,0;w,0)$. Since $\Pi_N(z,z)^{1/2} = \|e_L(z)\|^N_h |f^N(z)|$, each factor in (\ref{E1}) has the form $\half\log \Pi_N(\frac{z}{\sqrt{N}}, \frac{z}{\sqrt{N}}) - N \log \|e_L(\frac{z}{\sqrt{N}})\|_h$. By (\ref{diag}), $\log \Pi_N(\frac{z}{\sqrt{N}}, \frac{z}{\sqrt{N}})\to 0$ as $N\to\infty$. On the other hand $$-iN\d_z\dbar_z \log \|e_L(\frac{z}{\sqrt{N}})\|_h = \omega (\frac{z}{\sqrt{N}})\;.$$ Hence the first term converges to the normalized Euclidean (double) \kahler form: \begin{equation}\label{E1limit} E_1=\frac{i}{ 2\pi}\ddbar|z|^2 \wedge \frac{i}{ 2\pi}\ddbar|w|^2 +O(\frac{1}{N})\;.\end{equation} The middle two terms vanish since the integrals in $E_2$ and $E_3$ are independent of $w$ and $z$ respectively (see \cite[\S 3.2]{SZ}). The ``interesting term'' is therefore \begin{equation}\label{interesting} E_4= \frac{-1 }{\pi^2}\d_z\dbar_z\d_w\dbar_w \int_{\C^{d_N}}\log| c\cdot u^N(\frac{z}{\sqrtn})|\log|c\cdot u^N(\frac{w}{\sqrtn})| \frac{e^{-|c|^2}}{\pi^{d_N}} dc \; .\end{equation} To evaluate $E_4$, we consider the integral \begin{equation}\label{GN} G_2^N(x^1,x^2) := \int_{\C^{d_N}} \log |c\cdot x^1| \log | c\cdot x^2| \frac{e^{-|c|^2}}{\pi^{d_N}} dc\quad (x^1, x^2\in \C^{d_N}) \end{equation} with $x^1 = u^N(\frac{z}{\sqrtn}),\ x^2 = u^N(\frac{w}{\sqrtn})$. To simplify it, we construct a Hermitian orthonormal basis $\{e_1, \dots, e_{d_N}\}$ for $\C^{d_N}$ such that $x^1=e_1$ and \begin{equation}\label{2D} x^2 = \xi_{1} e_1 + \xi_{2} e_2,\quad\xi_{1} = \langle x^2, x^1\rangle,\;\; \xi_{2} = \sqrt{1 - |\xi_{1}|^2}. \end{equation} This is possible because we can always multiply $e_2$ by a phase $e^{i \theta}$ so that $\xi_{2}$ 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, (\ref{GN}) simplifies to \begin{equation} \label{2dim} G_2^N(x^1,x^2) = G_2(\xi_1,\xi_2)= \frac{1}{\pi^2} \int_{\C^2 }e^{-(|c_1|^2 + |c_2|^2)} \log |\xi_{1}|\log |c_1\xi_{1} + c_2 \xi_{2} | dc_1 dc_2\end{equation} (where we used the fact that the Gaussian integral in each $c_j, j \geq 3$ equals one by construction). By performing a rotation of the $c_1$ variable, we may replace $\xi_1$ with $|\xi_1|$ and replace $G_2(\xi_1,\xi_2)$ with \begin{equation}\label{Gcos}G(\cos\theta):=G_2(\cos \theta, \sin\theta) \,,\end{equation} where $\cos\theta = |\xi_1|=|\langle x^1,x^2\rangle|$, $0\leq\theta\leq\pi/2$. Hence (\ref{interesting}) becomes \begin{equation}\label{interesting-simp} E_4= \frac{-1 }{\pi^2}\d_z\dbar_z\d_w\dbar_w G(\cos\theta_N)\;,\quad \cos\theta_N= \big|\big\langle u^N(\frac{z}{\sqrt{N}}), u^N(\frac{w}{\sqrt{N}})\big\rangle\big|\;.\end{equation} By the universal scaling formula for the Szeg\"o kernel (Lemma~\ref{neardiag}) and (\ref{szego1}), we have \begin{equation}\label{thetalimit}\cos\theta_N =\frac{|\Pi_N(z,w)|}{\Pi_N(z,z)^{1/2}\Pi_N(w,w)^{1/2}} = e^{-\half |z-w|^2} + O(N^{-\half})\;.\end{equation} Thus we get the universal formula: \begin{equation}\label{UPCC} \vec K_2^{\infty}(z,w) = \frac{i}{ 2\pi}\ddbar|z|^2 \wedge \frac{i}{ 2\pi}\ddbar|w|^2 + \frac{-1 }{\pi^2}\d_z\dbar_z\d_w\dbar_w G(e^{-\half |z - w|^2}). \end{equation} 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 (\ref{PLcorscaled}): $$ \begin{array}{l} \log|c\cdot f^N(\frac{z^1}{\sqrtn})|\log|c\cdot f^N(\frac{z^2}{\sqrtn})|\cdots \log|c\cdot f^N(\frac{z^n}{\sqrtn})| \\ \\ = \log |f^N( \frac{z^1}{\sqrt{N}})| \log |f^N( \frac{z^2}{\sqrt{N}})| \cdots \log|f^N(\frac{z^n}{\sqrtn})| \nonumber \\ \\ +\log|f^N(\frac{z^1}{\sqrt{N}})|\log |f^N( \frac{z^2}{\sqrt{N}})| \cdots \log|f^N(\frac{z^{n-1}}{\sqrtn})| \log |c\cdot u^N(\frac{z^n}{\sqrt{N}}) |\nonumber \\ \\ + \cdots \\ \\+\log |c\cdot u^N(\frac{z^1}{\sqrt{N}})| \log |c\cdot u^N(\frac{z^2}{\sqrt{N}})| \cdots \log |c\cdot u^N(\frac{z^n}{\sqrt{N}}) |\;.\end{array}$$ We denote the terms resulting from this expansion by $E_1,\dots, E_{2^n}$, respectively. As before, the first term converges to the normalized Euclidean ``$n$-fold" \kahler form: $$E_1=\frac{i}{2\pi}\ddbar|z^1|^2\wedge \cdots \wedge \frac{i}{2\pi}\ddbar|z^n|^2 +O(\frac{1}{N})\,.$$ The $E_{2^n}$ term is obtained from the function \begin{equation}\label{GNn} G_n^N(x^1, x^2, \dots, x^n) := \int_{\C^{d_N}} \log |c\cdot x^1| \log |c\cdot x^2|\cdots \log |c\cdot x^n| \frac{e^{-|c|^2}}{\pi^{d_N}} dc\,, \end{equation} $x^1,x^2, \dots, x^n\in \C^{d_N}$. Precisely, we substitute \begin{equation}\label{xj} x^j = u^N(\frac{z^j}{\sqrt{N}})\end{equation} in (\ref{GNn}) and apply the operator $\left(\frac{i }{ \pi}\right)^n \d_{z^1}\dbar_{z^1}\cdots\d_{z^n}\dbar_{z^n}$. As above, we define a special Hermitian orthonormal basis $\{e_1, \dots, e_n\}$ for the n-dimensional complex subspace spanned by $\{x_1, \dots, x_n\}.$ We put: $$\begin{array}{ll} x^1 = e_1 \\ x^2 = \xi_{21} e_1 + \xi_{22}e_2 & \xi_{22} = \sqrt{1 - | \xi_{21}|^2} \\ \vdots \\ x^n = \xi_{n1} e_1 + \dots + \xi_{nn} e_n\qquad & \xi_{nn} = \sqrt{1 - \sum_{j \leq n-1} |\xi_{nj}|^2 }. \end{array}$$ Such a basis exists because we can always multiply $e_j$ by a phase $e^{i \theta}$ so that the last component $\xi_{jj}$ 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 $\{c_{n+1}, \dots, c_{d_N}\}$, (\ref{GNn}) simplifies to the $n$-dimensional complex Gaussian integral \begin{equation}\label{ndim}\begin{array}{lll} G^N_n(x^1,\dots,x^n)&=& G_n(\xi_{21}, \xi_{22}, \dots, \xi_{nn})\\[10pt] &=& \frac{1}{\pi^n} \int_{\C^n} e^{-|c|^2} \log|c_1| \log |c_1 \xi_{21} + c_2 \xi_{22}|\cdots \log |c_1 \xi_{n1} + \dots c_n \xi_{nn}| dc\,.\end{array}\end{equation} Note that the variables $\xi_{jk}$ depend on $N$; we write $\xi_{jk} = \xi_{jk}^N$ when we need to indicate this dependence. To prove universality, we observe that the $\xi_{jk}$ are universal algebraic functions of the inner products $\langle x^a, x^b \rangle$. Indeed, \begin{equation}\label{lex} \xi_{j1}\bar\xi_{k1}+\cdots+\xi_{jk}\bar\xi_{kk}=\langle x^j,x^k\rangle\,,\quad 1\le k\le j\le n\,,\end{equation} where we set $\xi_{11}=1$. These algebraic functions are obtained by induction (lexicographically) using (\ref{lex}). (The triangular matrix $(\xi_{jk})$ is just the inverse of the matrix describing the Gram-Schmidt process.) By (\ref{xj}), it follows that the $\xi_{jk}^N$ are universal algebraic functions of the variables $$\left\langle u^N({\textstyle \frac{z^j}{\sqrt{N}}}), u^N({\textstyle\frac{z^k}{\sqrt{N}}}) \right\rangle=\frac{\Pi_N(\frac{z^j}{\sqrt{N}}, \frac{z^k}{\sqrt{N}})}{ \Pi_N(\frac{z^j}{\sqrt{N}},\frac{z^j}{\sqrt{N}})^{1/2} \Pi_N(\frac{z^k}{\sqrt{N}},\frac{z^k}{\sqrt{N}})^{1/2}} =e^{i\Im (z^j\cdot \bar z^k)-\half |z^j-z^k|^2} + O({\textstyle\frac{1}{\sqrtn}})\,.$$ We note here that \begin{equation}\label{det} |x^1\wedge\cdots\wedge x^n|^2=\det( \langle x^j,x^k\rangle) \to \det \left(e^{i\Im (z^j\cdot \bar z^k)-\half |z^j-z^k|^2}\right) =e^{-\sum|z^j|^2}\det\left(e^{z_j\cdot \bar z_k}\right)\,.\end{equation} When the $z_j$ are distinct (i.e., $(z^1,\dots,z^n)\in \gcal^m_n$), the limit determinant in (\ref{det}) is nonzero (see \cite{BSZ}) and thus $\xi_{jk}^N= \xi_{jk}^\infty+O(\frac{1}{\sqrtn})$, where the $\xi_{jk}^\infty$ are universal real-analytic functions of $z\in\gcal^m_n$. We conclude that the $E_{2^n}$ term converges to a universal current: $$E_{2^n}=\left(\frac{i}{\pi}\right)^n \d_{z^1}\dbar_{z^1}\cdots \d_{z^n}\dbar_{z^n}G_n(\xi_{21}^\infty,\dots,\xi_{kk}^\infty) + O({\textstyle\frac{1}{\sqrtn}})\,.$$ Consider now a general term $E_a$. Suppose without loss of generality that $E_a$ comes from $$ \log |c\cdot u^N(\frac{z^1}{\sqrt{N}})|\cdots \log |c\cdot u^N(\frac{z^k}{\sqrt{N}})|\log|f^N(\frac{z^{k+1}}{\sqrt{N}})|\cdots \log|f^N(\frac{z^n}{\sqrtn})|\,.$$ As above we obtain $$E_a=\left(\frac{i}{\pi}\right)^k \d_{z^1}\dbar_{z^1}\cdots \d_{z^k}\dbar_{z^k}G_k(\xi_{21}^\infty,\dots,\xi_{kk}^\infty) \wedge \frac{i}{2\pi} \ddbar |z^{k+1}|^2\wedge\cdots \wedge \frac{i}{2\pi} \ddbar |z^{n}|^2 +O({\textstyle\frac{1}{\sqrtn}})\,.$$ Hence this term also approaches a universal current. (As in the pair correlation case, terms with only one $u^N$ vanish.) \qed \section{Explicit formulae} We now calculate explicitly the limit pair correlation measures $\wt K^{\infty}_2(z,w)$. \subsection{Preliminaries} The first step is to compute $\Delta G(e^{-\half r^2})$, where $\Delta$ is the Euclidean Laplacian on $\C^m$ and $r=|\zeta|$ ($\zeta\in\C^m$). To begin this computation, we write $a_j = r_j e^{i \phi_j}$ and then rewrite (\ref{2dim})--(\ref{Gcos}) as \begin{equation} G(\cos\theta)=\frac{2}{\pi}\int_0^{\infty} \int_0^{\infty}\int_0^{2\pi} r_1 r_2 e^{-(r_1^2 + r_2^2)} \log r_1 \log |r_1 \cos\theta+ r_2 e^{i \phi}\sin\theta| d\phi dr_1 dr_2\;.\end{equation} We now evaluate the inner integral by Jensen's formula, which gives \begin{equation} \int_0^{2\pi} \log|r_1 \cos \theta + r_2 \sin \theta e^{i \phi}| d\phi = \left\{ \begin{array}{ll}2\pi \log (r_1 \cos \theta) & \mbox{for}\;\;r_2 \sin \theta \leq r_1 \cos \theta \\ & \\ 2\pi\log (r_2 \sin \theta) & \mbox{for} \;\;r_2 \sin \theta\geq r_1 \cos \theta \end{array} \right. \end{equation} Hence \begin{equation} G(\cos\theta) = 4 \int_0^{\infty} \int_0^{\infty} r_1 r_2 e^{-(r_1^2 + r_2^2)} \log r_1 \log \max ( r_1 \cos \theta , r_2 \sin \theta)dr_1 dr_2. \end{equation} Now change variables again with $r_1 = \rho \cos \phi, r_2 = \rho \sin \phi$ to get \begin{equation} G(\cos\theta) = 4 \int_0^{\infty} \int_0^{\pi/2} \rho^3 e^{-\rho^2} \log (\rho \cos \phi) \log\max (\rho \cos \phi \cos \theta , \rho \sin \phi \sin \theta) \cos\phi\sin \phi d\phi d\rho\;. \end{equation} Since $$ \log \max( \rho \cos \phi \cos \theta , \rho \sin \phi \sin \theta) = \log (\rho \cos \phi \cos \theta) + \log^+ (\tan \phi \tan \theta)\;,$$ we can write $G=G_1+G_2$, where \begin{eqnarray} \label{G1} G_1(\cos\theta)&=&4 \int_0^{\infty} \int_0^{\pi/2} \rho^3 e^{-\rho^2} \log (\rho \cos \phi) \log ( \rho \cos \phi \cos \theta) \cos\phi \sin \phi d\phi d\rho\\ \label{G2} G_2(\cos\theta) &=&4\int_0^{\infty} \int_{\pi/2 - \theta}^{\pi/2}\rho^3 e^{-\rho^2} \log (\rho \cos \phi) \log ( \tan \phi \tan \theta) \cos\phi\sin \phi d\phi d\rho\;. \end{eqnarray} {From} (\ref{G1}), $G_1(\cos\theta)=C_1+C_2\log\cos\theta$ and thus $$G_1(e^{-\half r^2})= C_1-\half C_2 r^2\;,$$ so that \begin{equation}\label{forgetG1}\Delta G_1(e^{-\half r^2})=\left(\frac{d^2 }{ dr^2} +\frac{2m-1}{ r}\frac{d }{ dr}\right)(C_1-\half C_2 r^2)=-2mC_2\;.\end{equation} We now evaluate $\Delta G_2(e^{-\half r^2})$. Since the integrand in (\ref{G2}) vanishes when $\phi=\pi/2 -\theta$, we have $$\frac{d }{ dr}G_2(\cos\theta)= 4 \left(\frac{d }{ dr}\log\tan\theta \right) \int_0^{\infty} \int_{\pi/2 - \theta}^{\pi/2}\rho^3 e^{-\rho^2} \log (\rho \cos \phi) \cos\phi\sin \phi d\phi d\rho\;.$$ Substituting $\tan^2 \theta = e^{r^2}-1$, we have $$\frac{d }{ dr}\log\tan\theta =\frac{r}{ 1-e^{-r^2}}\;.$$ Thus $$\frac{d }{ dr}G_2(e^{-\half r^2})= \frac{4r}{ 1-e^{-r^2}}(I_1+I_2)\;,$$ where $$I_1=\int_0^{\infty} \int_{\pi/2 - \theta}^{\pi/2}\rho^3 e^{-\rho^2} (\log \rho) \cos\phi\sin \phi d\phi d\rho= C \sin^2 \theta = C(1-e^{-r^2})\;,$$ $$I_2=\int_0^{\infty} \int_{\pi/2 - \theta}^{\pi/2}\rho^3 e^{-\rho^2} (\log \cos\phi) \cos\phi\sin \phi d\phi d\rho\;.$$ We compute \begin{eqnarray*}I_2 &=& \half\int_{\pi/2 - \theta}^{\pi/2} (\log \cos\phi) \cos\phi\sin \phi d\phi \ =\half\int_0^{\sin\theta}t\log tdt\\ &=&\frac{1}{ 8}(\sin^2\theta\log\sin^2\theta-\sin^2\theta)=\frac{1}{ 8}(1-e^{-r^2})\left[\log (1-e^{-r^2})-1\right] \end {eqnarray*} Thus \begin{equation} \label{d1} \frac{d }{ dr}G_2(e^{-\half r^2})= \frac{r}{ 2}\log (1-e^{-r^2}) +C'r\;.\end{equation} Hence by (\ref{forgetG1}) and (\ref{d1}), \begin{eqnarray}\Delta G(e^{-\half r^2})&=& -2mC_2+ \left(\frac{d }{ dr}+\frac{2m-1}{ r}\right)\left(\frac{r}{ 2}\log(1-e^{-r^2})+C'r\right)\nonumber \\&=& m\log(1-e^{-r^2})+\frac{r^2}{ e^{r^2}-1} +C''\;.\label{d2}\end{eqnarray} \subsection{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 one-dimensional case. Our formula agrees with that of Bogomolny-Bohigas-Leboeuf \cite{BBL} and Hannay \cite{H} for $\SU(2)$ polynomials. \begin{theo} \label{Hformula} Suppose $\dim M=1$. Then $$\vec K^N_2(\frac{z}{\sqrtn},\frac{w}{\sqrtn})\to \vec K^\infty_2(z,w)=\left[\pi\delta_0(z-w) +H({\textstyle{\half}}|z-w|^2)\right]\frac{ i}{ 2\pi}\ddbar |z|^2 \wedge \frac{i}{ 2\pi}\ddbar|w|^2\;,$$ where $$H(t)= \frac{(\sinh^2 t + t^2) \cosh t -2t \sinh t }{ \sinh^3 t}=t-\frac{2}{ 9} t^3+\frac{2}{ 45}t^5+O(t^7)\;.$$ \end{theo} \begin{proof} Making the change of variables $\zeta=z-w$, we have by (\ref{UPCC}), $$\begin{array}{lll}\displaystyle \E\left(\wh Z^N(z) \otimes \wh Z^N(w)\right) &\to &\displaystyle\frac{i}{ 2\pi}\ddbar|z|^2 \wedge \frac{i}{ 2\pi}\ddbar|w|^2-\frac {1}{ \pi^2} \d_z\dbar_z\d_w\dbar_w G(e^{-\half |z-w|^2})\\[14pt] &&=\ \displaystyle \left[1+{4}\frac{\d^2}{\d z \dbar z}\frac{\d^2}{\d w \dbar w} G(e^{-\half |z-w|^2}) \right]\frac{i}{ 2\pi}\ddbar|z|^2 \wedge\frac {i}{ 2\pi}\ddbar|w|^2\\[14pt] &&=\ \displaystyle \left[1+{4}\left(\frac{\d^2}{\d\zeta \dbar\zeta}\right)^2 G(e^{-\half |\zeta|^2}) \right]\frac{i}{ 2\pi}\ddbar|z|^2 \wedge \frac{i}{ 2\pi}\ddbar|w|^2\\[14pt] &&=\ \displaystyle \left[1+\frac{1}{ 4}\Delta^2 G(e^{-\half r^2}) \right]\frac{i}{ 2\pi}\ddbar|z|^2 \wedge \frac{i}{ 2\pi}\ddbar|w|^2 \end{array}$$ By (\ref{d2}) with $m=1$, we have \begin{eqnarray*} \Delta^2 G(e^{-\half r^2}) &=&\left(\frac{d^2 }{ dr^2} +\frac{1}{ r}\frac{d }{ dr}\right) \left[\log(1-e^{-r^2})+\frac{r^2}{ e^{r^2}-1}\right]\\ &=&4\pi\delta_0 +\frac{8(e^{r^2}-1)^2 -16r^2e^{r^2}(e^{r^2}-1)+4r^4e^{r^2}(e^{r^2}+1)}{ (e^{r^2}-1)^3}\;. \end{eqnarray*} Finally, \begin{eqnarray*} \left[1+\frac{1}{ 4}\Delta^2 G(e^{-\half r^2}) \right] &=& \pi\delta_0 +\frac{(e^{r^2}+1)(e^{r^2}-1)^2 -4r^2e^{r^2}(e^{r^2}-1)+r^4e^{r^2}(e^{r^2}+1)}{ (e^{r^2}-1)^3}\\ &=&\pi\delta_0 +\frac{(\sinh^2\half r^2 +\frac{1}{ 4}r^4)\cosh \half r^2 - r^2 \sinh \half r^2 }{ \sinh^3 \half r^2}\;. \end{eqnarray*} \end{proof} \subsection{Pair correlation in higher dimensions} The limit pair correlation measure is given by \begin{eqnarray*} \wt K^{\infty}_2(z,w) &=& \lim_{N\to\infty} N^{2(m-1)}\wt K^N_2(\frac{z}{\sqrtn},\frac{w}{\sqrtn})\\ & = & \vec K^{\infty}_2(z,w)\wedge \frac{1}{(m-1)!}\left(\frac{i}{2}\ddbar |z|^2\right)^{m-1}\wedge \frac{1}{(m-1)!}\left(\frac{i}{2}\ddbar |w|^2\right)^{m-1}\,.\end{eqnarray*} (The scaling $N^{2(m-1)}$ comes from the fact that $N\om(\frac{z}{\sqrtn}) =N(\tau_{\sqrt{N}})_*\om\to \frac{i}{2}\ddbar |z|^2$.) We now compute $\wt K^{\infty}_2$ 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 \begin{equation}\label{expden} \wt K^\infty_1(z)= \lim_{N\to\infty}N^{m-1}\wt K^N_1(\frac{z}{\sqrtn})= \frac{m}{\pi}dV_{\C^m} =\frac{1}{\pi(m-1)!}\left(\frac{i}{2}\ddbar|z|^2\right)^m\,. \end{equation} 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. \begin{theo} \label{mformula} Suppose $\dim M=m>1$. Then $$\wt K^\infty_2(z,w)= \left[\gamma_m({\textstyle{\half}}|z-w|^2)\right] \wt K^\infty_1(z)\wedge \wt K^\infty_1(w) \;,$$ where \begin{eqnarray*}\gamma_m(t)&=& \frac{\left[{\textstyle\half}(m^2+m)\sinh^2t + t^2\right]\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\\ &&\quad\quad -\frac{(m+4)(m+3)}{ 90m^2}t^3+ {\frac {\left (m+6\right )\left (m+5\right )} {945{m}^{2}}}{t}^{5} +O(t^7) \;.\end{eqnarray*}\end{theo} \begin{proof} By (\ref{UPCC}) and (\ref{d2}), again writing $\zeta=z-w$ (except this time $\zeta\in\C^m$), \begin{eqnarray}&&\wt K^\infty_2(z,w)\ = \ \left[1+\frac{4}{m^2}\sum_{j,k=1}^m \frac{\partial^2}{\partial z_j \partial \bar{z}_j } \frac{\partial^2}{\partial w_k \partial \bar{w}_k } G(e^{-\half|z-w|^2})\right] \wt K^\infty_1(z)\wedge \wt K^\infty_1(w)\nonumber \\ &&\qquad =\ \left[1+ \frac{1}{4 m^2} \Delta^2_\zeta G(e^{-\half|\zeta|^2})\right] \wt K^\infty_1(z)\wedge \wt K^\infty_1(w)\nonumber\\ &&\qquad =\ \left[ 1 + \frac{1}{4m^2} \left(\frac{d^2}{dr^2} + \frac{2m -1}{r} \frac{d}{dr}\right)\left( m \log(1 - e^{-r^2}) + \frac{r^2}{e^{r^2} - 1}\right)\right]\wt K^\infty_1(z)\wedge \wt K^\infty_1(w)\;.\label{pc-m} \end{eqnarray} Computing the Laplacian in (\ref{pc-m}) leads to the stated formula. \end{proof} Note that if we substitute $m=1$ in the expression for $\gamma_m(t)$, we obtain Hannay's function $H(t)$. 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