Percolative MDLA: phase transition and critical behaviour

Abstract

In this paper we introduce the model of Percolative MDLA (PMDLA) on $\mathbb{Z}^2$ which engulfs the entire occupied percolation component of each particle that reaches the aggregate. We prove a phase transition of finite time explosion, as a function of the particle density $λ$, where the critical value equals $λ_c=-\log(1-p_c^{\rm site}(\mathbb{Z}^2))$. Below $λ_c$, the process is nonexplosive and has a linear upper bound on its radius. For $2/3<λ<λ_c$, it also has a linear lower bound on the growth rate. Above $λ_c$, an infinite component is swallowed in finite time a.s. At $λ_c$, every update is finite, but the updates accumulate in finite time, causing a blowup. Excluding criticality, the results are generalized to all higher dimensions.