Two-stage Bootstrap Percolation
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912595809665024 |
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| author | Fang, Zihao Gravner, Janko Sivakoff, David |
| author_facet | Fang, Zihao Gravner, Janko Sivakoff, David |
| contents | We introduce and study two variants of two-stage growth dynamics in $\mathbb{Z}^2$ with state space $\{0,1,2\}^{\mathbb{Z}^2}$. In each variant, vertices in state $0$ can be changed irreversibly to state $1$, and vertices in state $1$ can be changed permanently to state $2$. In the standard variant, a vertex flips from state $i$ to $i+1$ if it has at least two nearest-neighbors in state $i+1$. In the modified variant, a $0$ changes to a $1$ if it has both a north or south neighbor and an east or west neighbor in state $1$, and a $1$ changes to a $2$ if it has at least two nearest-neighbors in state $2$. We assume that the initial configuration is given by a product measure with small probabilities $p$ and $q$ of $1$s and $2$s. For both variants, as $p$ and $q$ tend to $0$, if $q$ is large compared to $p^{2+o(1)}$, then the final density of $0$s tends to $1$. When $q$ is small compared to $p^{2+o(1)}$, for standard variant the final density of $2$s tends to $1$, while for the modified variant the final density of $1$s tends to $1$. In fact, for the modified variant, the final density of $2$s approaches $0$ regardless of the relative size of $q$ versus $p$. These results remain unchanged if, in either variant, a $1$ changes to a $2$ only if it has both a north or south neighbor and an east or west neighbor in state $2$. An essential feature of these dynamics is that they are not monotone in the initial configuration. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_16541 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Two-stage Bootstrap Percolation Fang, Zihao Gravner, Janko Sivakoff, David Probability We introduce and study two variants of two-stage growth dynamics in $\mathbb{Z}^2$ with state space $\{0,1,2\}^{\mathbb{Z}^2}$. In each variant, vertices in state $0$ can be changed irreversibly to state $1$, and vertices in state $1$ can be changed permanently to state $2$. In the standard variant, a vertex flips from state $i$ to $i+1$ if it has at least two nearest-neighbors in state $i+1$. In the modified variant, a $0$ changes to a $1$ if it has both a north or south neighbor and an east or west neighbor in state $1$, and a $1$ changes to a $2$ if it has at least two nearest-neighbors in state $2$. We assume that the initial configuration is given by a product measure with small probabilities $p$ and $q$ of $1$s and $2$s. For both variants, as $p$ and $q$ tend to $0$, if $q$ is large compared to $p^{2+o(1)}$, then the final density of $0$s tends to $1$. When $q$ is small compared to $p^{2+o(1)}$, for standard variant the final density of $2$s tends to $1$, while for the modified variant the final density of $1$s tends to $1$. In fact, for the modified variant, the final density of $2$s approaches $0$ regardless of the relative size of $q$ versus $p$. These results remain unchanged if, in either variant, a $1$ changes to a $2$ only if it has both a north or south neighbor and an east or west neighbor in state $2$. An essential feature of these dynamics is that they are not monotone in the initial configuration. |
| title | Two-stage Bootstrap Percolation |
| topic | Probability |
| url | https://arxiv.org/abs/2509.16541 |