TY - JOUR
T1 - Asymptotics of the hole probability for zeros of random entire functions
AU - Nishry, Alon
N1 - Funding Information:
The research was supported by the Israel Science Foundation of the Israel Academy of Sciences and Humanities, grant 171/07.
PY - 2010
Y1 - 2010
N2 - Consider the random entire function f(z) = Σ∞ n=0 Φn zn/n (*) where the φn are independent and identically distributed (i.i.d.) standard complex Gaussian variables. The zero set of this function is distinguished by invariance of its distribution with respect to the isometries of the plane. We study the probability PH(r) that f has no zeros in the disk {|z| < r} (hole probability). Improving a result of Sodin and Tsirelson, we show that log P H(r) = -3e2/4•r4 + o(r4) as r → ∞. The proof does not use distribution invariance of the zeros, and can be extended to other Gaussian Taylor series. If φn are compactly supported random variables instead of Gaussians, we get a very different result: there exists r0 so that every random function of the form (*) must vanish in the disk {|z| < r0}.
AB - Consider the random entire function f(z) = Σ∞ n=0 Φn zn/n (*) where the φn are independent and identically distributed (i.i.d.) standard complex Gaussian variables. The zero set of this function is distinguished by invariance of its distribution with respect to the isometries of the plane. We study the probability PH(r) that f has no zeros in the disk {|z| < r} (hole probability). Improving a result of Sodin and Tsirelson, we show that log P H(r) = -3e2/4•r4 + o(r4) as r → ∞. The proof does not use distribution invariance of the zeros, and can be extended to other Gaussian Taylor series. If φn are compactly supported random variables instead of Gaussians, we get a very different result: there exists r0 so that every random function of the form (*) must vanish in the disk {|z| < r0}.
UR - http://www.scopus.com/inward/record.url?scp=77955871438&partnerID=8YFLogxK
U2 - 10.1093/imrn/rnp229
DO - 10.1093/imrn/rnp229
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AN - SCOPUS:77955871438
SN - 1073-7928
VL - 2010
SP - 2925
EP - 2946
JO - International Mathematics Research Notices
JF - International Mathematics Research Notices
IS - 15
ER -