The zeros of Gaussian random holomorphic functions on $\C^n$, and hole probability

Details The zeros of Gaussian random holomorphic functions on $\C^n$, and hole probability The zeros of Gaussian random holomorphic functions on $\C^n$, and hole probability

2006-03-30

arxiv.org/abs/math/0603696v2

Mathematics

Complex Variables

26 pages, typos removed

Scientific paper

We consider a class of Gaussian random holomorphic functions, whose expected zero set is uniformly distributed over $\C^n $. This class is unique (up to multiplication by a non zero holomorphic function), and is closely related to a Gaussian field over a Hilbert space of holomorphic functions on the reduced Heisenberg group. For a fixed random function of this class, we show that the probability that there are no zeros in a ball of large radius, is less than $e^{-c_1 r^{2n+2}}$, and is also greater than $e^{-c_2 r^{2n+2}}$. Enroute to this result we also compute probability estimates for the event that a random function's unintegrated counting function deviates significantly from its mean.

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