One-dimensional Multi-particle DLA -- a PDE approach

In the present note we analyze the one-dimensional multi-particle diffusion limited aggregation (MDLA) model: the initial number of particles at each positive integer site has Poisson distribution with mean $\mu$, independently of all other sites. Particles perform independent continuous-time simple symmetric random walks until they come to the site neighbouring the sticky aggregate, which initially consists only of the origin. If a particle tries to jump on the aggregate, the size of the aggregate increases by one, i.e., its rightmost point moves to the right by one unit. All particles which are present at the site neighbouring the aggregate at the moment when the aggregate advances, are immediately deleted. The $d-$dimensional MDLA model, $d \geq 1$, was introduced in 1980 by Rosenstock and Marquardt, and studied numerically by Voss (1984). The one dimensional model exhibits a phase transition for the rate of growth of the aggregate: it was proven by Kesten and Sidoravicius (2008) that if $\mu<1$ then the size $R(t)$ of the aggregate grows like $\sqrt{t}$ and Sly (2016+) proved that if $\mu>1$ then $R(t)$ grows linearly. In this note we give heuristic predictions about the constant $c(\mu)$ for which $R(t)\approx c(\mu)\sqrt{t}$ in the subcritical case $\mu<1$, $R(t)\approx c(1+\varepsilon)t$ in the barely supercritical case $\mu=1+\varepsilon$ and $R(t) \approx c(1) t^{2/3}$ in the critical case $\mu=1$. We compare our predictions with new computer simulation results of the 1-dimensional multi-particle DLA model.