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Site percolation and isoperimetric inequalities for plane graphs
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Haslegrave, John and Panagiotis, Christoforos (2021) Site percolation and isoperimetric inequalities for plane graphs. Random Structures and Algorithms, 58 (1). pp. 150-163. doi:10.1002/rsa.20946
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Official URL: https://doi.org/10.1002/rsa.20946
Abstract
We use isoperimetric inequalities combined with a new technique to prove upper bounds for the site percolation threshold of plane graphs with given minimum degree conditions. In the process we prove tight new isoperimetric bounds for certain classes of hyperbolic graphs. This establishes the vertex isoperimetric constant for all triangular and square hyperbolic lattices, answering a question of Lyons and Peres. We prove that plane graphs of minimum degree at least 7 have site percolation threshold bounded away from 1/2, which was conjectured by Benjamini and Schramm, and make progress on a conjecture of Angel, Benjamini, and Horesh that the critical probability is at most 1/2 for plane triangulations of minimum degree 6. We prove additional bounds for stronger minimum degree conditions, and for graphs without triangular faces.
| Item Type: | Journal Article | ||||||||
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| Subjects: | Q Science > QA Mathematics | ||||||||
| Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||||||
| Library of Congress Subject Headings (LCSH): | Graph theory, Isoperimetric inequalities, Percolation (Statistical physics) | ||||||||
| Journal or Publication Title: | Random Structures and Algorithms | ||||||||
| Publisher: | John Wiley & Sons, Inc. | ||||||||
| ISSN: | 1042-9832 | ||||||||
| Official Date: | January 2021 | ||||||||
| Dates: |
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| Volume: | 58 | ||||||||
| Number: | 1 | ||||||||
| Page Range: | pp. 150-163 | ||||||||
| DOI: | 10.1002/rsa.20946 | ||||||||
| Status: | Peer Reviewed | ||||||||
| Publication Status: | Published | ||||||||
| Access rights to Published version: | Open Access | ||||||||
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