Sensitive bootstrap percolation second term
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866909086702895104 |
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| author | Hartarsky, Ivailo |
| author_facet | Hartarsky, Ivailo |
| contents | In modified two-neighbour bootstrap percolation in two dimensions each site of $\mathbb Z^2$ is initially independently infected with probability $p$ and on each discrete time step one additionally infects sites with at least two non-opposite infected neighbours. In this note we establish that for this model the second term in the asymptotics of the infection time $τ$ unexpectedly scales differently from the classical two-neighbour model, in which arbitrary two infected neighbours are required. More precisely, we show that for modified bootstrap percolation with high probability as $p\to0$ it holds that \[τ\le \exp\left(\frac{π^2}{6p}-\frac{c\log(1/p)}{\sqrt p}\right)\] for some positive constant $c$, while the classical model is known to lack the logarithmic factor. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_14910 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Sensitive bootstrap percolation second term Hartarsky, Ivailo Probability 60C05, 60K35 In modified two-neighbour bootstrap percolation in two dimensions each site of $\mathbb Z^2$ is initially independently infected with probability $p$ and on each discrete time step one additionally infects sites with at least two non-opposite infected neighbours. In this note we establish that for this model the second term in the asymptotics of the infection time $τ$ unexpectedly scales differently from the classical two-neighbour model, in which arbitrary two infected neighbours are required. More precisely, we show that for modified bootstrap percolation with high probability as $p\to0$ it holds that \[τ\le \exp\left(\frac{π^2}{6p}-\frac{c\log(1/p)}{\sqrt p}\right)\] for some positive constant $c$, while the classical model is known to lack the logarithmic factor. |
| title | Sensitive bootstrap percolation second term |
| topic | Probability 60C05, 60K35 |
| url | https://arxiv.org/abs/2303.14910 |