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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 ISSN 1042-9832.

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Official URL: https://doi.org/10.1002/rsa.20946

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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
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:
DateEvent
January 2021Published
25 June 2020Available
14 May 2020Accepted
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 (Creative Commons)
Date of first compliant deposit: 22 May 2020
Date of first compliant Open Access: 20 July 2020
RIOXX Funder/Project Grant:
Project/Grant IDRIOXX Funder NameFunder ID
639046H2020 European Research Councilhttp://dx.doi.org/10.13039/100010663
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