Infinite collision property for the three-dimensional uniform spanning tree
Let [Formula: see text] be the three-dimensional uniform spanning tree, whose probability law is denoted by [Formula: see text]. For [Formula: see text]-a.s. realization of [Formula: see text], the recurrence of the simple random walk on [Formula: see text] is proved in Benjamini et al. (2001) [I. B...
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| Format: | Article |
| Language: | English |
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World Scientific Publishing
2023-12-01
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| Series: | International Journal of Mathematics for Industry |
| Subjects: | |
| Online Access: | https://www.worldscientific.com/doi/10.1142/S2661335223500053 |
| _version_ | 1797306793069641728 |
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| author | Satomi Watanabe |
| author_facet | Satomi Watanabe |
| author_sort | Satomi Watanabe |
| collection | DOAJ |
| description | Let [Formula: see text] be the three-dimensional uniform spanning tree, whose probability law is denoted by [Formula: see text]. For [Formula: see text]-a.s. realization of [Formula: see text], the recurrence of the simple random walk on [Formula: see text] is proved in Benjamini et al. (2001) [I. Benjamini, R. Lyons, Y. Peres and O. Schramm, Uniform spanning forests, Ann. Probab. 29(1) (2001) 1–65] and it is also demonstrated in Hutchcroft and Peres (2015) [T. Hutchcroft and Y. Peres, Collisions of random walks in reversible random graphs, Electron. Commun. Probab. 20(63) (2015) 1–6] that two independent simple random walks on [Formula: see text] collide infinitely often. In this paper, we will give a quantitative estimate on the number of collisions of two independent simple random walks on [Formula: see text], which provides another proof of the infinite collision property of [Formula: see text]. |
| first_indexed | 2024-03-08T00:46:57Z |
| format | Article |
| id | doaj.art-4ce9fc2d6fd7481ca377cba32bab5de8 |
| institution | Directory Open Access Journal |
| issn | 2661-3352 2661-3344 |
| language | English |
| last_indexed | 2024-03-08T00:46:57Z |
| publishDate | 2023-12-01 |
| publisher | World Scientific Publishing |
| record_format | Article |
| series | International Journal of Mathematics for Industry |
| spelling | doaj.art-4ce9fc2d6fd7481ca377cba32bab5de82024-02-15T05:50:11ZengWorld Scientific PublishingInternational Journal of Mathematics for Industry2661-33522661-33442023-12-01150110.1142/S2661335223500053Infinite collision property for the three-dimensional uniform spanning treeSatomi Watanabe0Department of Advanced Mathematical Sciences, Graduate School of Informatics, Kyoto University, Kyoto 606-8501, JapanLet [Formula: see text] be the three-dimensional uniform spanning tree, whose probability law is denoted by [Formula: see text]. For [Formula: see text]-a.s. realization of [Formula: see text], the recurrence of the simple random walk on [Formula: see text] is proved in Benjamini et al. (2001) [I. Benjamini, R. Lyons, Y. Peres and O. Schramm, Uniform spanning forests, Ann. Probab. 29(1) (2001) 1–65] and it is also demonstrated in Hutchcroft and Peres (2015) [T. Hutchcroft and Y. Peres, Collisions of random walks in reversible random graphs, Electron. Commun. Probab. 20(63) (2015) 1–6] that two independent simple random walks on [Formula: see text] collide infinitely often. In this paper, we will give a quantitative estimate on the number of collisions of two independent simple random walks on [Formula: see text], which provides another proof of the infinite collision property of [Formula: see text].https://www.worldscientific.com/doi/10.1142/S2661335223500053Uniform spanning treerandom walkinfinite collision property |
| spellingShingle | Satomi Watanabe Infinite collision property for the three-dimensional uniform spanning tree International Journal of Mathematics for Industry Uniform spanning tree random walk infinite collision property |
| title | Infinite collision property for the three-dimensional uniform spanning tree |
| title_full | Infinite collision property for the three-dimensional uniform spanning tree |
| title_fullStr | Infinite collision property for the three-dimensional uniform spanning tree |
| title_full_unstemmed | Infinite collision property for the three-dimensional uniform spanning tree |
| title_short | Infinite collision property for the three-dimensional uniform spanning tree |
| title_sort | infinite collision property for the three dimensional uniform spanning tree |
| topic | Uniform spanning tree random walk infinite collision property |
| url | https://www.worldscientific.com/doi/10.1142/S2661335223500053 |
| work_keys_str_mv | AT satomiwatanabe infinitecollisionpropertyforthethreedimensionaluniformspanningtree |