Universality and scaling of zeros on symplectic manifolds
| dc.creator | Bleher, Pavel | |
| dc.creator | Shiffman, Bernard | |
| dc.creator | Zelditch, Steve | |
| dc.date | 2000-02-16 | |
| dc.date | 2000-10-17 | |
| dc.date.accessioned | 2026-07-07T06:24:32Z | |
| dc.date.available | 2026-07-07T06:24:32Z | |
| dc.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. | |
| dc.description | 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 | |
| dc.identifier | https://arxiv.org/abs/math-ph/0002039 | |
| dc.identifier | http://arxiv.org/abs/math-ph/0002039 | |
| dc.identifier | Random Matrix Models and Their Applications, MSRI Publications 40, Cambridge Univ. Press, 2001, pp. 31-69. | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/96572 | |
| dc.subject | Mathematical Physics | |
| dc.subject | Probability | |
| dc.subject | Symplectic Geometry | |
| dc.title | Universality and scaling of zeros on symplectic manifolds | |
| dc.type | text |