In this very interesting paper, the authors introduce a two-type internal DLA model. 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 1 water particle present, it fires by sending 1 oil particle and 1 water particle each to an independent random neighbor \(x\pm 1\). Firing continues until every site has at most one type of particles.
For \(x\in \mathbb{Z}\), let \(u(x)\) be the total number of times \(x\) fires before fixation. The random function \(u: \mathbb{Z}\to \mathbb{N}\) is called the odometer of the process. The first result of the paper concerning the order of magnitude of the odometer is as follows:
Theorem 1.1. There exist positive numbers \(\varepsilon,c,C\) such that for large \(n\):
i. \[ \mathbb{ P}\left(\sup_{x\in \mathbb{Z}}u(x)>Cn^{4/3}\right)<e^{-n^{\varepsilon}}, \]
ii. \[ \mathbb{P}\left(\inf_{x: |x|\leq cn^{1/3}}u(x)<cn^{4/3}\right)<e^{-n^{\varepsilon}}. \]
Let \(F(r)\) be the number of particles that fixate outside the interval \([-r,r]\). The following result shows that most particles do not travel very far.
Theorem 1.2. For sufficiently small \(\varepsilon>0\), there exists \(\delta>0\) such that \[ \mathbb{P}\left(F(n^{\frac{1}{3}+\varepsilon})>n^{1-\frac{\varepsilon}{2}}\right)<e^{-n^{\delta}}. \]
Let \(\tilde{u}:=\mathbb{E}u\). The authors present the following conjecture.
Conjecture 1.3.
(i) For any \(\delta>0\), \[ \mathbb{P}\left(\sup_{x\in \mathbb{Z}}\left|\frac{u(x)-\tilde{u}(x)}{n^{4/3}}\right|>\delta\right)\to 0\;\;\text{as}\;n\to\infty. \]
(ii) There is a function \(w: \mathbb{R}\to\mathbb{R}\) such that \[ \frac{\tilde{u}(\lfloor n^{1/3}\xi\rfloor)}{n^{4/3}}\to w(\xi), \] uniformly in \(\xi\).
Conditionally on Conjecture 1.3, the limit function \(w(x)\) is given in Theorem 1.4. In addition, some open questions and a conjecture about the process on higher dimensional lattices are given in Section 7.