Catalan percolation
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arXiv
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| Auteurs principaux: | , , , , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866908654969552896 |
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| author | Archer, Eleanor Hartarsky, Ivailo Kolesnik, Brett Olesker-Taylor, Sam Schapira, Bruno Valesin, Daniel |
| author_facet | Archer, Eleanor Hartarsky, Ivailo Kolesnik, Brett Olesker-Taylor, Sam Schapira, Bruno Valesin, Daniel |
| contents | In Catalan percolation, all nearest-neighbor edges $\{i,i+1\}$ along $\mathbb Z$ are initially occupied, and all other edges are open independently with probability $p$. Open edges $\{i,j\}$ are occupied if some pair of edges $\{i,k\}$ and $\{k,j\}$, with $i<k<j$, become occupied. This model was introduced by Gravner and the third author, in the context of polluted graph bootstrap percolation.
We prove that the critical $p_{\mathrm c}$ is strictly between that of oriented site percolation on $\mathbb Z^2$ and the Catalan growth rate $1/4$. Our main result shows that an enhanced oriented percolation model, with non-decaying infinite-range dependency, has a strictly smaller critical parameter than the classical model. This is reminiscent of the work of Duminil-Copin, Hilário, Kozma and Sidoravicius on brochette percolation. Our proof differs, however, in that we do not use Aizenman--Grimmett enhancements or differential inequalities. Two key ingredients are the work of Hilário, Sá, Sanchis and Teixeira on stretched lattices, and the Russo--Seymour--Welsh result for oriented percolation by Duminil-Copin, Tassion and Teixeira. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_19583 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Catalan percolation Archer, Eleanor Hartarsky, Ivailo Kolesnik, Brett Olesker-Taylor, Sam Schapira, Bruno Valesin, Daniel Probability Combinatorics 60K35 (Primary) 82B43, 05A15 (Secondary) In Catalan percolation, all nearest-neighbor edges $\{i,i+1\}$ along $\mathbb Z$ are initially occupied, and all other edges are open independently with probability $p$. Open edges $\{i,j\}$ are occupied if some pair of edges $\{i,k\}$ and $\{k,j\}$, with $i<k<j$, become occupied. This model was introduced by Gravner and the third author, in the context of polluted graph bootstrap percolation. We prove that the critical $p_{\mathrm c}$ is strictly between that of oriented site percolation on $\mathbb Z^2$ and the Catalan growth rate $1/4$. Our main result shows that an enhanced oriented percolation model, with non-decaying infinite-range dependency, has a strictly smaller critical parameter than the classical model. This is reminiscent of the work of Duminil-Copin, Hilário, Kozma and Sidoravicius on brochette percolation. Our proof differs, however, in that we do not use Aizenman--Grimmett enhancements or differential inequalities. Two key ingredients are the work of Hilário, Sá, Sanchis and Teixeira on stretched lattices, and the Russo--Seymour--Welsh result for oriented percolation by Duminil-Copin, Tassion and Teixeira. |
| title | Catalan percolation |
| topic | Probability Combinatorics 60K35 (Primary) 82B43, 05A15 (Secondary) |
| url | https://arxiv.org/abs/2404.19583 |