  In this note we consider a certain class of Gaussian entire functions,
characterized by some asymptotic properties of their covariance kernels, which
we call admissible (as defined by Hayman). A notable example is the Gaussian
Entire Function, whose zero set is well-known to be invariant with respect to
the isometries of the complex plane.
  We explore the rigidity of the zero set of Gaussian Taylor series, a
phenomenon discovered not long ago by Ghosh and Peres for the Gaussian Entire
Function. In particular, we find that for a function of infinite order of
growth, and having an admissible kernel, the zero set is "fully rigid". This
means that if we know the location of the zeros in the complement of any given
compact set, then the number and location of the zeros inside that set can be
determined uniquely. As far as we are aware, this is the first explicit
construction in a natural class of random point processes with full rigidity.s
with full rigidity.
