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 -