Overcrowding and hole probabilities for random zeros on complex manifolds

TL;DR

This paper derives large deviation estimates for the volume of zero sets of random holomorphic sections on complex manifolds, providing bounds on the probability of significant deviations and hole probabilities.

Contribution

It introduces asymptotic large deviations estimates for zero set volumes of random sections on complex manifolds, extending understanding of their probabilistic behavior.

Findings

Probability of volume deviation > δN decays as exp(-C_{δ,U}N^{m+1})

Hole probability decays as exp(-C_{U}N^{m+1})

Provides quantitative bounds on zero set fluctuations

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 holomorphic section of the N-th power of a positive line bundle on a compact Kaehler 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 $\exp(-C_{U}N^{m+1})$.