Overcrowding estimates for zeroes of Planar and Hyperbolic Gaussian analytic functions期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

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Overcrowding estimates for zeroes of Planar and Hyperbolic Gaussian analytic functions

Authors:	Manjunath Krishnapur

Institution:	(1) Department of Statistics, U.C., Berkeley, CA 94720, USA

Abstract:	We consider the point process of zeroes of certain Gaussian analytic functions and find the asymptotics for the probability that there are more than m points of the process in a fixed disk of radius r, as $r\rightarrow \infty$ . For the planar Gaussian analytic function, $\sum_{n \geq 0}\frac{a_n z^n}{\sqrt{n!}}$ , we show that this probability is asymptotic to ${e^{-\frac{1}{2}m^2\log(m)}}$ . For the hyperbolic Gaussian analytic functions, ${\sum_{n\geq 0}{-\rho \choose n}^{1/2} a_n z^n}, \rho > 0$ , we show that this probability decays like $e^{-cm^2}$ .In the planar case, we also consider the problem posed by Mikhail Sodin² on moderate and very large deviations in a disk of radius r, as $r\rightarrow \infty$ . We partially solve the problem by showing that there is a qualitative change in the asymptotics of the probability as we move from the large deviation regime to the moderate.Research supported by NSF grant #DMS-0104073 and NSF-FRG grant #DMS-0244479.

Keywords:	random analytic functions chaotic analytic zero points CAZP large deviations fluctuations coulomb gas one component plasma
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