Oil and water: A two-type internal aggregation model

Abstract

We introduce a two-type internal DLA model which is an example of a nonunary abelian network. Starting with $n$ “oil” and $n$ “water” particles at the origin, the particles diffuse in $\mathbb{Z}$ according to the following rule: whenever some site $x\in\mathbb{Z}$ has at least $1$ oil and at least $1$ water particle present, it fires by sending $1$ oil particle and $1$ water particle each to an independent random neighbor $x\pm1$. Firing continues until every site has at most one type of particles. We establish the correct order for several statistics of this model and identify the scaling limit under assumption of existence.