Triple collisions on a comb graph
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866929530143244288 |
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| author | Croydon, David A. De Ambroggio, Umberto |
| author_facet | Croydon, David A. De Ambroggio, Umberto |
| contents | In this article, we consider the number of collisions of three independent simple random walks on a subgraph of the two-dimensional square lattice obtained by removing all horizontal edges with vertical coordinate not equal to 0 and then, for $n\in \mathbb{Z}$, restricting the vertical segment of the graph located at horizontal coordinate $n$ to the interval $\{0,1,\dots,\log^α(|n|\vee 1)\}$. Specifically, we show the following phase transition: when $α\leq 1$, the three random walks collide infinitely many times almost-surely, whereas when $α>1$, they collide only finitely many times almost-surely. This is a variation of a result of Barlow, Peres and Sousi, who showed a similar phase transition for two random walks when the vertical segments are truncated at height $|n|^α$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_04882 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Triple collisions on a comb graph Croydon, David A. De Ambroggio, Umberto Probability 60J10 (primary), 05C81, 60J35 In this article, we consider the number of collisions of three independent simple random walks on a subgraph of the two-dimensional square lattice obtained by removing all horizontal edges with vertical coordinate not equal to 0 and then, for $n\in \mathbb{Z}$, restricting the vertical segment of the graph located at horizontal coordinate $n$ to the interval $\{0,1,\dots,\log^α(|n|\vee 1)\}$. Specifically, we show the following phase transition: when $α\leq 1$, the three random walks collide infinitely many times almost-surely, whereas when $α>1$, they collide only finitely many times almost-surely. This is a variation of a result of Barlow, Peres and Sousi, who showed a similar phase transition for two random walks when the vertical segments are truncated at height $|n|^α$. |
| title | Triple collisions on a comb graph |
| topic | Probability 60J10 (primary), 05C81, 60J35 |
| url | https://arxiv.org/abs/2410.04882 |