Overcrowding and hole probabilities for random zeros on complex manifolds

Details Overcrowding and hole probabilities for random zeros on complex manifolds Overcrowding and hole probabilities for random zeros on complex manifolds

2008-05-16

arxiv.org/abs/0805.2598v2

Indiana Univ. Math. J. 57 (2008), 1977-1997

Mathematics

Complex Variables

16 pages; stylistic changes, added corollary

Scientific paper

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

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