The hole probability for Gaussian entire functions

Alon Nishry

doi:10.1007/s11856-011-0136-z

The hole probability for Gaussian entire functions

Alon Nishry

School of Mathematical Sciences

Tel Aviv University

Research output: Contribution to journal › Article › peer-review

Abstract

Consider the random entire function, where the ø_n are independent standard complex Gaussian coefficients, and the a_n are positive constants, which satisfy. We study the probability P_H(r) that f has no zeroes in the disk { {pipe}z{pipe} < r} (hole probability). Assuming that the sequence a_n is logarithmically concave, we prove that log P_H(r) = -S(r)+o(S(r)), where, and r tends to ∞ outside a (deterministic) exceptional set of finite logarithmic measure.

Original language	English
Pages (from-to)	197-220
Number of pages	24
Journal	Israel Journal of Mathematics
Volume	186
Issue number	1
DOIs	https://doi.org/10.1007/s11856-011-0136-z
State	Published - Nov 2011

All Science Journal Classification (ASJC) codes

General Mathematics

Access to Document

10.1007/s11856-011-0136-z

Cite this

@article{b2d6428853ac49a5ae54cfe2059a5e9d,

title = "The hole probability for Gaussian entire functions",

abstract = "Consider the random entire function, where the {\o}n are independent standard complex Gaussian coefficients, and the an are positive constants, which satisfy. We study the probability PH(r) that f has no zeroes in the disk \{ \{pipe\}z\{pipe\} < r\} (hole probability). Assuming that the sequence an is logarithmically concave, we prove that log PH(r) = -S(r)+o(S(r)), where, and r tends to ∞ outside a (deterministic) exceptional set of finite logarithmic measure.",

author = "Alon Nishry",

note = "Funding Information: ∗Research supported by the Israel Science Foundation of the Israel Academy of Sciences and Humanities, grant 171/07. Received November 5, 2009 and in revised form March 11, 2010",

year = "2011",

month = nov,

doi = "10.1007/s11856-011-0136-z",

language = "الإنجليزيّة",

volume = "186",

pages = "197--220",

journal = "Israel Journal of Mathematics",

issn = "0021-2172",

publisher = "Springer New York",

number = "1",

}

TY - JOUR

T1 - The hole probability for Gaussian entire functions

AU - Nishry, Alon

N1 - Funding Information: ∗Research supported by the Israel Science Foundation of the Israel Academy of Sciences and Humanities, grant 171/07. Received November 5, 2009 and in revised form March 11, 2010

PY - 2011/11

Y1 - 2011/11

N2 - Consider the random entire function, where the øn are independent standard complex Gaussian coefficients, and the an are positive constants, which satisfy. We study the probability PH(r) that f has no zeroes in the disk { {pipe}z{pipe} < r} (hole probability). Assuming that the sequence an is logarithmically concave, we prove that log PH(r) = -S(r)+o(S(r)), where, and r tends to ∞ outside a (deterministic) exceptional set of finite logarithmic measure.

AB - Consider the random entire function, where the øn are independent standard complex Gaussian coefficients, and the an are positive constants, which satisfy. We study the probability PH(r) that f has no zeroes in the disk { {pipe}z{pipe} < r} (hole probability). Assuming that the sequence an is logarithmically concave, we prove that log PH(r) = -S(r)+o(S(r)), where, and r tends to ∞ outside a (deterministic) exceptional set of finite logarithmic measure.

UR - http://www.scopus.com/inward/record.url?scp=80155171889&partnerID=8YFLogxK

U2 - 10.1007/s11856-011-0136-z

DO - 10.1007/s11856-011-0136-z

M3 - مقالة

SN - 0021-2172

VL - 186

SP - 197

EP - 220

JO - Israel Journal of Mathematics

JF - Israel Journal of Mathematics

IS - 1

ER -

The hole probability for Gaussian entire functions

Abstract

All Science Journal Classification (ASJC) codes

Access to Document

Other files and links

Fingerprint

Cite this