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