  We consider the range of random analytic functions with finite radius of
convergence. We show that any unbounded random Taylor series with rotationally
invariant coefficients has dense image in the plane. We moreover show that if
in addition the coefficients are complex Gaussian with sufficiently regular
variances, then the image is the whole complex plane. We do this by exploiting
an approximate connection between the coverage problem and spatial branching
processes. This answers a long-standing open question of J.-P. Kahane, with
sufficient regularity.
