Improved bounds for 1-independent percolation on $\mathbb{Z}^n$
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2022
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| _version_ | 1866916803768221696 |
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| author | Balister, Paul Johnston, Tom Savery, Michael Scott, Alex |
| author_facet | Balister, Paul Johnston, Tom Savery, Michael Scott, Alex |
| contents | A 1-independent bond percolation model on a graph $G$ is a probability distribution on the spanning subgraphs of $G$ in which, for all vertex-disjoint sets of edges $S_1$ and $S_2$, the states of the edges in $S_1$ are independent of the states of the edges in $S_2$. Such a model is said to percolate if the random subgraph has an infinite component with positive probability. In 2012 the first author and Bollobás defined $p_{\max}(G)$ to be the supremum of those $p$ for which there exists a 1-independent bond percolation model on $G$ in which each edge is present in the random subgraph with probability at least $p$ but which does not percolate.
A fundamental and challenging problem in this area is to determine the value of $p_{\max}(G)$ when $G$ is the lattice graph $\mathbb{Z}^2$. Since $p_{\max}(\mathbb{Z}^n)\leq p_{\max}(\mathbb{Z}^{n-1})$, it is also of interest to establish the value of $\lim_{n\to\infty} p_{\max}(\mathbb{Z}^n)$. In this paper we significantly improve the best known upper bound on this limit and obtain better upper and lower bounds on $p_{\max}(\mathbb{Z}^2)$. In proving these results, we also give an upper bound on the critical probability for a 1-independent model on the hypercube graph to contain a giant component asymptotically almost surely. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2206_12335 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Improved bounds for 1-independent percolation on $\mathbb{Z}^n$ Balister, Paul Johnston, Tom Savery, Michael Scott, Alex Probability Combinatorics A 1-independent bond percolation model on a graph $G$ is a probability distribution on the spanning subgraphs of $G$ in which, for all vertex-disjoint sets of edges $S_1$ and $S_2$, the states of the edges in $S_1$ are independent of the states of the edges in $S_2$. Such a model is said to percolate if the random subgraph has an infinite component with positive probability. In 2012 the first author and Bollobás defined $p_{\max}(G)$ to be the supremum of those $p$ for which there exists a 1-independent bond percolation model on $G$ in which each edge is present in the random subgraph with probability at least $p$ but which does not percolate. A fundamental and challenging problem in this area is to determine the value of $p_{\max}(G)$ when $G$ is the lattice graph $\mathbb{Z}^2$. Since $p_{\max}(\mathbb{Z}^n)\leq p_{\max}(\mathbb{Z}^{n-1})$, it is also of interest to establish the value of $\lim_{n\to\infty} p_{\max}(\mathbb{Z}^n)$. In this paper we significantly improve the best known upper bound on this limit and obtain better upper and lower bounds on $p_{\max}(\mathbb{Z}^2)$. In proving these results, we also give an upper bound on the critical probability for a 1-independent model on the hypercube graph to contain a giant component asymptotically almost surely. |
| title | Improved bounds for 1-independent percolation on $\mathbb{Z}^n$ |
| topic | Probability Combinatorics |
| url | https://arxiv.org/abs/2206.12335 |