Comparison of limit shapes for Bernoulli first-passage percolation
We consider Bernoulli first-passage percolation on the [Formula: see text]-dimensional hypercubic lattice with [Formula: see text]. The passage time of edge [Formula: see text] is 0 with probability [Formula: see text] and 1 with probability [Formula: see text], independently of each other. Let...
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| Format: | Article |
| Language: | English |
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World Scientific Publishing
2022-12-01
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| Series: | International Journal of Mathematics for Industry |
| Subjects: | |
| Online Access: | https://www.worldscientific.com/doi/10.1142/S2661335222500058 |
| _version_ | 1797854851229548544 |
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| author | Naoki Kubota Masato Takei |
| author_facet | Naoki Kubota Masato Takei |
| author_sort | Naoki Kubota |
| collection | DOAJ |
| description | We consider Bernoulli first-passage percolation on the [Formula: see text]-dimensional hypercubic lattice with [Formula: see text]. The passage time of edge [Formula: see text] is 0 with probability [Formula: see text] and 1 with probability [Formula: see text], independently of each other. Let [Formula: see text] be the critical probability for percolation of edges with passage time 0. When [Formula: see text], there exists a nonrandom, nonempty compact convex set [Formula: see text] such that the set of vertices to which the first-passage time from the origin is within [Formula: see text] is well approximated by [Formula: see text] for all large [Formula: see text], with probability one. The aim of this paper is to prove that for [Formula: see text], the Hausdorff distance between [Formula: see text] and [Formula: see text] grows linearly in [Formula: see text]. Moreover, we mention that the approach taken in the paper provides a lower bound for the expected size of the intersection of geodesics, that gives a nontrivial consequence for the critical case. |
| first_indexed | 2024-04-09T20:14:01Z |
| format | Article |
| id | doaj.art-2ee2551c988a4b7797a751d49405b2d2 |
| institution | Directory Open Access Journal |
| issn | 2661-3352 2661-3344 |
| language | English |
| last_indexed | 2024-04-09T20:14:01Z |
| publishDate | 2022-12-01 |
| publisher | World Scientific Publishing |
| record_format | Article |
| series | International Journal of Mathematics for Industry |
| spelling | doaj.art-2ee2551c988a4b7797a751d49405b2d22023-03-31T12:10:15ZengWorld Scientific PublishingInternational Journal of Mathematics for Industry2661-33522661-33442022-12-01140110.1142/S2661335222500058Comparison of limit shapes for Bernoulli first-passage percolationNaoki Kubota0Masato Takei1College of Science and Technology, Nihon University, Funabashi Campus, 24-1, Narashinodai 7-Chome, Funabashi-Shi, Chiba 274–8501, JapanDepartment of Applied Mathematics, Faculty of Engineering, Yokohama National University, 79-5 Tokiwadai, Hodogaya-Ku, Yokohama 240–8501, JapanWe consider Bernoulli first-passage percolation on the [Formula: see text]-dimensional hypercubic lattice with [Formula: see text]. The passage time of edge [Formula: see text] is 0 with probability [Formula: see text] and 1 with probability [Formula: see text], independently of each other. Let [Formula: see text] be the critical probability for percolation of edges with passage time 0. When [Formula: see text], there exists a nonrandom, nonempty compact convex set [Formula: see text] such that the set of vertices to which the first-passage time from the origin is within [Formula: see text] is well approximated by [Formula: see text] for all large [Formula: see text], with probability one. The aim of this paper is to prove that for [Formula: see text], the Hausdorff distance between [Formula: see text] and [Formula: see text] grows linearly in [Formula: see text]. Moreover, we mention that the approach taken in the paper provides a lower bound for the expected size of the intersection of geodesics, that gives a nontrivial consequence for the critical case.https://www.worldscientific.com/doi/10.1142/S2661335222500058First-passage percolationtime constantslimit shapes |
| spellingShingle | Naoki Kubota Masato Takei Comparison of limit shapes for Bernoulli first-passage percolation International Journal of Mathematics for Industry First-passage percolation time constants limit shapes |
| title | Comparison of limit shapes for Bernoulli first-passage percolation |
| title_full | Comparison of limit shapes for Bernoulli first-passage percolation |
| title_fullStr | Comparison of limit shapes for Bernoulli first-passage percolation |
| title_full_unstemmed | Comparison of limit shapes for Bernoulli first-passage percolation |
| title_short | Comparison of limit shapes for Bernoulli first-passage percolation |
| title_sort | comparison of limit shapes for bernoulli first passage percolation |
| topic | First-passage percolation time constants limit shapes |
| url | https://www.worldscientific.com/doi/10.1142/S2661335222500058 |
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