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

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dc.creatorZrebiec, Scott
dc.date2006-03-30
dc.date2006-06-18
dc.date.accessioned2026-07-07T07:07:21Z
dc.date.available2026-07-07T07:07:21Z
dc.descriptionWe 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.
dc.description26 pages, typos removed
dc.identifierhttps://arxiv.org/abs/math/0603696
dc.identifierhttp://arxiv.org/abs/math/0603696
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/110363
dc.subjectComplex Variables
dc.subjectProbability
dc.subject30B20; 30C15
dc.titleThe zeros of Gaussian random holomorphic functions on $\C^n$, and hole probability
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