article

Title: Overcrowding and hole probabilities for random zeros on complex manifolds

Authors: Bernard Shiffman, Steve Zelditch and Scott Zrebiec

Issue: Volume 57 (2008), Issue 5, 1977-1998

Abstract:

$We give asymptotic large deviations estimates for the volume inside a domain $U$ of the zero set of a random polynomial of degree $N$, or more generally, of a random holomorphic section of the $N$-th power of a positive line bundle on a compact K\"ahler manifold. In particular, we show that for all $\delta > 0$, the probability that this volume differs by more than $\delta N$ from its average value is less than $\exp (-C_{\delta, U}N^{m + 1})$, for some constant $C_{\delta,U} > 0$. As a consequence, the "hole probability" that a random section does not vanish in $U$ has an upper bound of the form $\exp (-C_{U} N^{m + 1})$.$