We consider the zeroes of a random Gaussian Entire Function f and show that
their basins under the gradient flow of the random potential U partition the
complex plane into domains of equal area.
We find three characteristic exponents 1, 8/5, and 4 of this random
partition: the probability that the diameter of a particular basin is greater
than R is exponentially small in R; the probability that a given point z lies
at a distance larger than R from the zero it is attracted to decays as
exp(-R^{8/5}); and the probability that, after throwing away 1% of the area of
the basin, its diameter is still larger than R decays as exp(-R^4).
We also introduce a combinatorial procedure that modifies a small portion of
each basin in such a way that the probability that the diameter of a particular
modified basin is greater than R decays only slightly slower than exp(-cR^4).