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 $\lambda$, where the critical value equals ${\lambda }_c=-\log (1-p_c^{\mathrm{site}}({\mathbb{Z}}^2))$. Below ${\lambda }_c$, the process is nonexplosive and has a linear upper bound on its radius. For $2/3<\lambda <{\lambda }_c$, it also has a linear lower bound on the growth rate. Above ${\lambda }_c$, an infinite component is swallowed in finite time a.s. At ${\lambda }_c$, every update is finite, but the updates accumulate in finite time, causing a blowup. Excluding criticality, the results are generalized to all higher dimensions.

arXiv comment33 pages