Universality and scaling of zeros on symplectic manifolds

Bleher, Pavel; Shiffman, Bernard; Zelditch, Steve

Universality and scaling of zeros on symplectic manifolds

Authors

Bleher, Pavel

Shiffman, Bernard

Zelditch, Steve

Description

This article is concerned with random holomorphic polynomials and their generalizations to algebraic and symplectic geometry. A natural algebro-geometric generalization studied in our prior work involves random holomorphic sections $H^0(M,L^N)$ of the powers of any positive line bundle $L \to M$ over any complex manifold. Our main interest is in the statistics of zeros of $k$ independent sections (generalized polynomials) of degree $N$ as $N\to\infty$. We fix a point $P$ and focus on the ball of radius $1/\sqrt{N}$ about $P$. Under a microscope magnifying the ball by the factor $\sqrt{N}$, the statistics of the configurations of simultaneous zeros of random $k$-tuples of sections tends to a universal limit independent of $P,M,L$. We review this result and generalize it further to the case of pre-quantum line bundles over almost-complex symplectic manifolds $(M,J,ω)$. Following [SZ2], we replace $H^0(M,L^N)$ in the complex case with the `asymptotically holomorphic' sections defined by Boutet de Monvel-Guillemin and (from another point of view) by Donaldson and Auroux. Using a generalization to an $m$-dimensional setting of the Kac-Rice formula for zero correlations together with the results of [SZ2], we prove that the scaling limits of the correlation functions for zeros of random $k$-tuples of asymptotically holomorphic sections belong to the same universality class as in the complex case.
Added results on the decay of connected correlations; corrected typographical errors . To appear in the Proceedings of the 1999 MSRI Workshop on Random Matrices and Their Applications