Hole probability for zeroes of Gaussian Taylor series with finite radii of convergence

TY - JOUR

T1 - Hole probability for zeroes of Gaussian Taylor series with finite radii of convergence

AU - Buckley, Jeremiah Anthony

AU - Nishry, Alon

AU - Peled, Ron

AU - Sodin, Mikhail

PY - 2018/6

Y1 - 2018/6

N2 - We study a family of random Taylor seriesF(z)=∑n≥0ζnanznwith radius of convergence almost surely 1 and independent, identically distributed complex Gaussian coefficients (ζn) ; these Taylor series are distinguished by the invariance of their zero sets with respect to isometries of the unit disk. We find reasonably tight upper and lower bounds on the probability that F does not vanish in the disk {|z|⩽r} as r↑1 . Our bounds take different forms according to whether the non-random coefficients (an) grow, decay or remain of the same order. The results apply more generally to a class of Gaussian Taylor series whose coefficients (an) display power-law behavior.

AB - We study a family of random Taylor seriesF(z)=∑n≥0ζnanznwith radius of convergence almost surely 1 and independent, identically distributed complex Gaussian coefficients (ζn) ; these Taylor series are distinguished by the invariance of their zero sets with respect to isometries of the unit disk. We find reasonably tight upper and lower bounds on the probability that F does not vanish in the disk {|z|⩽r} as r↑1 . Our bounds take different forms according to whether the non-random coefficients (an) grow, decay or remain of the same order. The results apply more generally to a class of Gaussian Taylor series whose coefficients (an) display power-law behavior.

U2 - 10.1007/s00440-017-0782-0

DO - 10.1007/s00440-017-0782-0

M3 - Article

SN - 0178-8051

VL - 171

SP - 377

EP - 430

JO - PROBABILITY THEORY AND RELATED FIELDS

JF - PROBABILITY THEORY AND RELATED FIELDS

IS - 1-2

ER -