Polluted Modified Bootstrap Percolation
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866917963176607744 |
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| author | Gravner, Janko Holroyd, Alexander Lee, Sangchul Sivakoff, David |
| author_facet | Gravner, Janko Holroyd, Alexander Lee, Sangchul Sivakoff, David |
| contents | In the polluted modified bootstrap percolation model, sites in the square lattice are independently initially occupied with probability $p$ or closed with probability $q$. A site becomes occupied at a subsequent step if it is not closed and has at least one occupied nearest neighbor in each of the two coordinates. We study the final density of occupied sites when $p$ and $q$ are both small. We show that this density approaches $0$ if $q\ge Cp^2/\log p^{-1}$ and $1$ if $q\le p^2/(\log p^{-1})^{1+o(1)}$. Thus we establish a logarithmic correction in the critical scaling, which is known not to be present in the standard model, settling a conjecture of Gravner and McDonald from 1997. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_15746 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Polluted Modified Bootstrap Percolation Gravner, Janko Holroyd, Alexander Lee, Sangchul Sivakoff, David Probability In the polluted modified bootstrap percolation model, sites in the square lattice are independently initially occupied with probability $p$ or closed with probability $q$. A site becomes occupied at a subsequent step if it is not closed and has at least one occupied nearest neighbor in each of the two coordinates. We study the final density of occupied sites when $p$ and $q$ are both small. We show that this density approaches $0$ if $q\ge Cp^2/\log p^{-1}$ and $1$ if $q\le p^2/(\log p^{-1})^{1+o(1)}$. Thus we establish a logarithmic correction in the critical scaling, which is known not to be present in the standard model, settling a conjecture of Gravner and McDonald from 1997. |
| title | Polluted Modified Bootstrap Percolation |
| topic | Probability |
| url | https://arxiv.org/abs/2503.15746 |