Overcrowding and hole probabilities for random zeros on complex manifolds

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 $\exp (-C_U{N}^{m+1})$.

16 pages; stylistic changes, added corollary

Cited by in corpus (1)

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