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

摘要

The asymptotic shape of randomly growing radial clusters is studied. We pose the problemin terms of the dynamics of stochastic partial differential equations. We concentrate on theproperties of the realizations of the stochastic growth process and in particular on theinterface fluctuations. Our goal is unveiling under which conditions the developing radialcluster asymptotically weakly converges to the concentrically propagating spherically symmetricprofile or either to a symmetry breaking shape. We demonstrate that the long rangecorrelations of the surface fluctuations obey a self-affine scaling and that scale invariance isachieved by means of the introduction of three critical exponents. These are able to characterizethe large scale dynamics and to describe those regimes dominated by system sizeevolution. The connection of these results with mathematical morphogenetic problems isalso outlined.