科研之友
微信
新浪微博
Facebook
分享链接
摘要
Consider the Gaussian entire function f(z) = Sigma(infinity)(n=0) xi(n)a(n)z(n), where {xi(n)} is a sequence of independent and identically distributed standard complex Gaussians and {a(n)} is some sequence of non-negative coefficients, with a(0) > 0. We study the asymptotics (for large values of r) of the hole probability for f(z), that is, the probability P-H(r) that f(z) has no zeros in the disk {vertical bar z vertical bar < r}. We prove that log P-H (r) = -S(r) + o(S(r)), where S(r) = 2 . Sigma(n >= 0) log(+)(a(n)r(n)) as r tends to infinity outside a deterministic exceptional set of finite logarithmic measure.
- 出版日期2012-11
