The Planar Lattice Two-Neighbor Graph Percolates
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866916545866760192 |
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| author | Coupier, David Henry, Benoît Jahnel, Benedikt Köppl, Jonas |
| author_facet | Coupier, David Henry, Benoît Jahnel, Benedikt Köppl, Jonas |
| contents | The k-neighbor graph is a directed percolation model on the hypercubic lattice Z d in which each vertex independently picks exactly k of its 2d nearest neighbors at random, and we open directed edges towards those. We prove that the 2-neighbor graph percolates on Z 2 , i.e., that the origin is connected to infinity with positive probability. The proof rests on duality, an exploration algorithm, a comparison to i.i.d. bond percolation under constraints as well as enhancement arguments. As a byproduct, we show that i.i.d. bond percolation with forbidden local patterns has a strictly larger percolation threshold than 1/2. Additionally, our main result provides further evidence that, in low dimensions, less variability is beneficial for percolation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_20781 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Planar Lattice Two-Neighbor Graph Percolates Coupier, David Henry, Benoît Jahnel, Benedikt Köppl, Jonas Probability The k-neighbor graph is a directed percolation model on the hypercubic lattice Z d in which each vertex independently picks exactly k of its 2d nearest neighbors at random, and we open directed edges towards those. We prove that the 2-neighbor graph percolates on Z 2 , i.e., that the origin is connected to infinity with positive probability. The proof rests on duality, an exploration algorithm, a comparison to i.i.d. bond percolation under constraints as well as enhancement arguments. As a byproduct, we show that i.i.d. bond percolation with forbidden local patterns has a strictly larger percolation threshold than 1/2. Additionally, our main result provides further evidence that, in low dimensions, less variability is beneficial for percolation. |
| title | The Planar Lattice Two-Neighbor Graph Percolates |
| topic | Probability |
| url | https://arxiv.org/abs/2412.20781 |