# Stochastic growth of radial clusters: weak convergence to the asymptotic profile and implications for morphogenesis 

Carlos Escudero Departamento de Econom´ıa Cuantitativa & Instituto de Ciencias Matem´ aticas (CSIC-UAM-UC3M-UCM), Universidad Aut´ onoma de Madrid, Ciudad Universitaria de Cantoblanco, 28049 Madrid, Spain 

## Abstract 

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. 

PACS numbers: 05.40.-a, 05.65.+b, 87.10.-e, 87.10.Mn 

