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

TL;DR

This paper investigates the stochastic growth of radial clusters, analyzing interface fluctuations and conditions for symmetry or symmetry breaking, with implications for understanding morphogenesis.

Contribution

It introduces a framework using stochastic PDEs to characterize surface fluctuations and identifies critical exponents governing large-scale dynamics.

Findings

Surface fluctuations obey self-affine scaling.

Scale invariance characterized by three critical exponents.

Conditions for asymptotic symmetry or symmetry breaking identified.

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.