Overcrowding and hole probabilities for random zeros on complex manifolds

Shiffman, Bernard; Zelditch, Steve; Zrebiec, Scott

Overcrowding and hole probabilities for random zeros on complex manifolds

Authors

Shiffman, Bernard

Zelditch, Steve

Zrebiec, Scott

Description

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 $δ>0$, the probability that this volume differs by more than $δN$ from its average value is less than $\exp(-C_{δ,U}N^{m+1})$, for some constant $C_{δ,U}>0$. As a consequence, the "hole probability" that a random section does not vanish in U has an upper bound of the form $\exp(-C_{U}N^{m+1})$.
16 pages; stylistic changes, added corollary