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- W4300000022 abstract "We study the percolation time of the $r$-neighbour bootstrap percolation model on the discrete torus $(Z/nZ)^d$. For $t$ at most a polylog function of $n$ and initial infection probabilities within certain ranges depending on $t$, we prove that the percolation time of a random subset of the torus is exactly equal to $t$ with high probability as $n$ tends to infinity. Our proof rests crucially on three new extremal theorems that together establish an almost complete understanding of the geometric behaviour of the $r$-neighbour bootstrap process in the dense setting. The special case $d-r=0$ of our result was proved recently by Bollob'as, Holmgren, Smith and Uzzell." @default.
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- W4300000022 date "2012-09-19" @default.
- W4300000022 modified "2023-09-29" @default.
- W4300000022 title "The time of bootstrap percolation with dense initial sets for all thresholds" @default.
- W4300000022 doi "https://doi.org/10.48550/arxiv.1209.4339" @default.
- W4300000022 hasPublicationYear "2012" @default.
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