Indeed, the original Eden problem can be greatly generalized by means of the use of stochastic partial differential equations. They allow a systematic study of the properties of the colony periphery, particularly of the interface fluctuations. At the same time, they allow us to get rid of the undesirable lattice anisotropy. In this work we will concentrate on the properties of the realizations of the stochastic growth process. Our goal is unveiling under which conditions the developing radial cluster asymptotically weakly converges to the concentrically propagating spherically symmetric profile. Let us emphasize that we are in- terested in determining the presence or absence of spherical symmetry in every realization of the stochastic growth process in the long time limit. As we are considering isotropic growth which is free from deterministic instabilities, averaging the cluster profile over many real- izations yields an immediate spherical symmetry. The properties of the realizations can be ascertained, as we will see, calculating suitable correlations adapted from classical elements of stochastic growth theory [7], and keeping in mind that we should expect weak rather than strong convergence of the asymptotic profiles. Weak convergence is a mathematically totally precise concept which physically corresponds to the convergence resulting from the self-averaging of wildly oscillating quantities or structures [8]. Apart from the overall shape, the microscopic fluctuations of growing radial clusters have been studied as well. It turns out that these fluctuations sculpture a fractal surface which is statistically self-affine [7, 9, 10]. This microscopic roughness, in those cases in which the pointwise width [11] is well defined (which actually rules out all flat interfaces due to the microscopic properties of white noise) is described by the familiar scaling of planar systems [12]. This is not so, however, for long range fluctuations [13]. Long range radial correlations cannot be in general deduced from planar scalings. They still adopt a self-affine form for macroscopic length scales, but scale invariance is this time achieved for different values of the critical exponents. The following sections will discuss the numerical values and physical meaning of these exponents. We will see that they are able to characterize which regimes asymptotically weakly converge to the radially symmetric shape and which do not. In planar situations of non-equilibrium growth the interface is parameterized by the height function h ( x, t ), which indicates the displacement of the interface with respect to some II. RADIAL CORRELATIONS 3