caa a22 4500 467891311 CHVBK 20180406152750.0 cr unu---uuuuu 170328e20060901xx s 000 0 eng 10.1007/s10955-006-9159-y doi (NATIONALLICENCE)springer-10.1007/s10955-006-9159-y Krishnapur Manjunath Department of Statistics, U.C., 94720, Berkeley, CA, USA aut Overcrowding estimates for zeroes of Planar and Hyperbolic Gaussian analytic functions [Elektronische Daten] [Manjunath Krishnapur] 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. Springer Science + Business Media, Inc., 2006 random analytic functions nationallicence chaotic analytic zero points nationallicence CAZP nationallicence large deviations nationallicence fluctuations nationallicence coulomb gas nationallicence one component plasma nationallicence Journal of Statistical Physics Kluwer Academic Publishers-Plenum Publishers; http://www.springer-ny.com 124/6(2006-09-01), 1399-1423 0022-4715 124:6<1399 2006 124 10955 https://doi.org/10.1007/s10955-006-9159-y text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/s10955-006-9159-y text/html Onlinezugriff via DOI NATIONALLICENCE 100 1- Krishnapur Manjunath Department of Statistics, U.C., 94720, Berkeley, CA, USA aut NATIONALLICENCE 773 0- Journal of Statistical Physics Kluwer Academic Publishers-Plenum Publishers; http://www.springer-ny.com 124/6(2006-09-01), 1399-1423 0022-4715 124:6<1399 2006 124 10955 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer