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 Kahler manifold. In particular, we show that for all δ > 0, the probability that this volume differs by more than δ N from its average value is less than exp(-Cδ,UNm+1), for some constant Cδ,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(-CUNm+1).
Original language | English (US) |
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Pages (from-to) | 1977-1997 |
Number of pages | 21 |
Journal | Indiana University Mathematics Journal |
Volume | 57 |
Issue number | 5 |
DOIs | |
State | Published - 2008 |
Keywords
- Bergman kernel
- Large deviations
- Positive line bundle
- Random zeros
- Szego kernel
ASJC Scopus subject areas
- General Mathematics