Local Bootstrap Percolation

Janko Gravner (University of California Davis)
Alexander E. Holroyd (University of British Columbia, Microsoft Research)

Abstract


We study a variant of bootstrap percolation in which growth is restricted to a single active cluster. Initially there is a single active site at the origin, while other sites of $\mathbb{Z}^2$ are independently occupied with small probability $p$, otherwise empty. Subsequently, an empty site becomes active by contact with two or more active neighbors, and an occupied site becomes active if it has an active site within distance 2. We prove that the entire lattice becomes active with probability $\exp [\alpha(p)/p]$, where $\alpha(p)$ is between $-\pi^2/9+c\sqrt p$ and $-\pi^2/9+C\sqrt p(\log p^{-1})^3$. This corrects previous numerical predictions for the scaling of the correction term.

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Pages: 385-399

Publication Date: February 9, 2009

DOI: 10.1214/EJP.v14-607

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