Universality and Scaling of Correlations between Zeros on
Complex Manifolds
Pavel Bleher
Abstract:
We study the limit as $N\to\infty$ 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.
(Joint work with Bernard Shiffman and Steve Zelditch.)