Asymptotic shape in a continuum growth model

Abstract

A continuum growth model is introduced. The state at time $t$, $S_t$, is a subset of $\mathbb{R}^d$ and consists of a connected union of randomly sized Euclidean balls, which emerge from outbursts at their center points. An outburst occurs somewhere in $S_t$ after an exponentially distributed time with expected value $|S_t|^{-1}$ and the location of the outburst is uniformly distributed over $S_t$. The main result is that if the distribution of the radii of the outburst balls has bounded support, then $S_t$ grows linearly and $S_t/t$ has a non-random shape as $t\rightarrow \infty$. Due to rotation invariance the asymptotic shape must be a Euclidean ball.