Site percolation and isoperimetric inequalities for plane graphs. Issue 1 (25th June 2020)

• Record Type: Journal Article
• Authors: Haslegrave, John; Panagiotis, Christoforos
• Is Part Of: Random structures & algorithms. Volume 58:Issue 1(2021)
• Page Start: 150
• Page End: 163
• Publication Date: 2020-06-25
• Subjects: isoperimetric inequality -- percolation -- planar graphs; Random graphs -- Periodicals; Mathematical analysis -- Periodicals; 519
• DOI: 10.1002/rsa.20946
• Languages: English
• ISSNs: 1042-9832

Abstract

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.