  We consider the Gaussian Entire Function (GEF) whose Taylor coefficients are
independent complex-valued Gaussian variables, and the variance of the kth
coefficient is 1/k!. This random Taylor series is distinguished by the
invariance of its zero set with respect to the isometries of the complex plane.
  We show that the law of the zero set, conditioned on the GEF having no zeros
in a disk of radius r, and properly normalized, converges to an explicit
limiting Radon measure in the plane, as r goes to infinity. A remarkable
feature of this limiting measure is the existence of a large 'forbidden region'
between a singular part supported on the boundary of the (scaled) hole and the
equilibrium measure far from the hole.
