Asymptotic shape in a continuum growth model

TL;DR

This paper introduces a continuum growth model where a set of Euclidean balls expands over time through random outbursts, and proves that the shape of the growing set converges to a Euclidean ball under certain conditions.

Contribution

The paper establishes that in a continuum growth model with bounded radii, the shape of the expanding set converges to a Euclidean ball as time approaches infinity.

Findings

Growth is linear over time.

The asymptotic shape is a Euclidean ball.

The shape convergence is non-random.

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.