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

M3 - ???researchoutput.researchoutputtypes.contributiontojournal.article???

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 -