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# Maximum percolation time in two-dimensional bootstrap percolation

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### Abstract

We consider a classic model known as bootstrap percolation on the $$n \times n$$ square grid. To each vertex of the grid we assign an initial state, infected or healthy, and then in consecutive rounds we infect every healthy vertex that has at least $$2$$ already infected neighbours. We say that percolation occurs if the whole grid is eventually infected. In this paper, contributing to a recent series of extremal results in this field, we prove that the maximum time a bootstrap percolation process can take to eventually infect the entire vertex set of the grid is $$13n^2/18+O(n)$$.

### Most cited references10

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### Bootstrap percolation on a Bethe lattice

(1979)
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### Metastability effects in bootstrap percolation

(1988)
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### Sharp metastability threshold for two-dimensional bootstrap percolation

(2003)
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### Author and article information

###### Journal
16 October 2013
2014-11-05
###### Article
1310.4457