  Our main discovery is a surprising interplay between quadrature domains on
the one hand, and the zero process of the Gaussian Entire Function (GEF) on the
other. Specifically, consider the GEF conditioned on the rare hole event that
there are no zeros in a given large Jordan domain. We show that in the natural
scaling limit, a quadrature domain enclosing the hole emerges as a forbidden
region, where the zero density vanishes. Moreover, we give a description of
those holes for which the forbidden region is a disk.
  The connecting link between random zeros and potential theory is supplied by
a constrained extremal problem for the Zeitouni-Zelditch functional. To solve
this problem, we recast it in terms of a seemingly novel obstacle problem,
where the solution is forced to be harmonic inside the hole.
