Monotonicity properties for Bernoulli percolation on layered graphs -- a Markov chain approach
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866909547979866112 |
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| author | König, Philipp Richthammer, Thomas |
| author_facet | König, Philipp Richthammer, Thomas |
| contents | A layered graph $G^\times$ is the Cartesian product of a graph $G = (V,E)$ with the linear graph $Z$, e.g. $Z^\times$ is the 2D square lattice $Z^2$. For Bernoulli percolation with parameter $p \in [0,1]$ on $G^\times$ one intuitively would expect that $P_p((o,0) \leftrightarrow (v,n)) \ge P_p((o,0) \leftrightarrow (v,n+1))$ for all $o,v \in V$ and $n \ge 0$. This is reminiscent of the better known bunkbed conjecture. Here we introduce an approach to the above monotonicity conjecture that makes use of a Markov chain building the percolation pattern layer by layer. In case of finite $G$ we thus can show that for some $N \ge 0$ the above holds for all $n \ge N$ $o,v \in V$ and $p \in [0,1]$. One might hope that this Markov chain approach could be useful for other problems concerning Bernoulli percolation on layered graphs. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2207_13173 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Monotonicity properties for Bernoulli percolation on layered graphs -- a Markov chain approach König, Philipp Richthammer, Thomas Probability Mathematical Physics 60K35, 60J10 (Primary) 82B43, 15B51 (Secondary) A layered graph $G^\times$ is the Cartesian product of a graph $G = (V,E)$ with the linear graph $Z$, e.g. $Z^\times$ is the 2D square lattice $Z^2$. For Bernoulli percolation with parameter $p \in [0,1]$ on $G^\times$ one intuitively would expect that $P_p((o,0) \leftrightarrow (v,n)) \ge P_p((o,0) \leftrightarrow (v,n+1))$ for all $o,v \in V$ and $n \ge 0$. This is reminiscent of the better known bunkbed conjecture. Here we introduce an approach to the above monotonicity conjecture that makes use of a Markov chain building the percolation pattern layer by layer. In case of finite $G$ we thus can show that for some $N \ge 0$ the above holds for all $n \ge N$ $o,v \in V$ and $p \in [0,1]$. One might hope that this Markov chain approach could be useful for other problems concerning Bernoulli percolation on layered graphs. |
| title | Monotonicity properties for Bernoulli percolation on layered graphs -- a Markov chain approach |
| topic | Probability Mathematical Physics 60K35, 60J10 (Primary) 82B43, 15B51 (Secondary) |
| url | https://arxiv.org/abs/2207.13173 |