Random zeros on complex manifolds: Conditional expectations

We study the conditional distribution K_κ^Nof 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 K_κ^N around p. The key tool is the conditional Szeg kernel and its scaling asymptotics.