Oil and water: a two-type internal aggregation model

TL;DR

This paper introduces a two-type internal DLA model with oil and water particles diffusing on the integer lattice, analyzing its behavior, scaling limits, and establishing key statistical properties of the process.

Contribution

It presents the first detailed analysis of a two-type internal DLA model, including its scaling limit and statistical behavior, expanding understanding of multi-type abelian networks.

Findings

Established the order of several key statistics.

Identified the scaling limit under certain assumptions.

Demonstrated the model's behavior converges to a specific limit.

Abstract

We introduce a two-type internal DLA model which is an example of a non-unary abelian network. Starting with n "oil" and n "water" particles at the origin, the particles diffuse in Z according to the following rule: whenever some site x 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+1 or x-1. 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.