  The asymptotic shape of randomly growing radial clusters is studied. We pose
the problem in terms of the dynamics of stochastic partial differential
equations. We concentrate on the properties of the realizations of the
stochastic growth process and in particular on the interface fluctuations. Our
goal is unveiling under which conditions the developing radial cluster
asymptotically weakly converges to the concentrically propagating spherically
symmetric profile or either to a symmetry breaking shape. We demonstrate that
the long range correlations of the surface fluctuations obey a self-affine
scaling and that scale invariance is achieved by means of the introduction of
three critical exponents. These are able to characterize the large scale
dynamics and to describe those regimes dominated by system size evolution. The
connection of these results with mathematical morphogenetic problems is also
outlined.
