Work

Random zeros on complex manifolds: conditional expectations

Year: 2011

Type: article

Abstract: We study the conditional distribution of zeros of a Gaussian system of random polynomials (and more generally, holomorphic sections), given that the polynomials or sections vanish at a point p (or a fixed finite set of points). The conditional distribution is analogous to the pair correlation function of zeros but we show that it has quite a different small distance behaviour. In particular, the conditional distribution does not exhibit repulsion of zeros in dimension 1. To prove this, we give universal scaling asymptotics for around p. The key tool is the conditional Szegő kernel and its scaling asymptotics.

Source: Journal of the Institute of Mathematics of Jussieu

DOI: 10.1017/s1474748011000041

Authors: Bernard Shiffman, Steve Zelditch, Qi Zhong

Institutions: Johns Hopkins University, Northwestern University, Vanderbilt University

Language: English

Topic: Random Matrices and Applications

Subfield: Statistics and Probability

Field: Mathematics

Domain: Physical Sciences

Open Access status: green

Funder: National Science Foundation

Awards: Random Holomorphic Sections and Complex Geometry, Global Harmonic Analysis and Asymptotic Geometry