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

• 成果类型: Article
• 署名作者: Buckley, Jeremiah; Nishry, Alon; Peled, Ron; Sodin, Mikhail
• 署名单位: Consejo Superior de Investigaciones Cientificas (CSIC); CSIC - Instituto de Ciencia de Materiales de Madrid (ICMM); CSIC - Instituto de Ciencias Matematicas (ICMAT); University of Michigan System; University of Michigan; Tel Aviv University
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-017-0782-0
• 发表日期: 2018
• 页码: 377-430
• 关键词: random complex zeros; random polynomials

摘要

We study a family of random Taylor series F(z) = n= 0.nan zn with radius of convergence almost surely 1 and independent, identically distributed complex Gaussian coefficients ; 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 as . Our bounds take different forms according to whether the non-random coefficients grow, decay or remain of the same order. The results apply more generally to a class of Gaussian Taylor series whose coefficients display power-law behavior.