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 ℝ d and consists of a connected union of randomly sized Euclidean balls, which emerge from outbursts at their centre 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 nonrandom shape as t → ∞. Due to rotational invariance the asymptotic shape must be a Euclidean ball.