• Corpus ID: 247218585

Hyperbolic site percolation

@inproceedings{Grimmett2022HyperbolicSP,
  title={Hyperbolic site percolation},
  author={Geoffrey R. Grimmett and Zhongyan Li},
  year={2022}
}

• G. Grimmett, Zhongyan Li
• Published 2 March 2022
• Mathematics

. Several results are presented for site percolation on quasi-transitive, planar graphs G with one end, when properly embedded in either the Euclidean or hyperbolic plane. If ( G 1 , G 2 ) is a matching pair derived from some quasi-transitive mosaic M , then p u ( G 1 ) + p c ( G 2 ) = 1, where p c is the critical probability for the existence of an infinite cluster, and p u is the critical value for the existence of a unique such cluster. This fulfils and extends to the hyperbolic plane an observation of Sykes and Essam in 1964. It follows that p u ( G ) + p c ( G ∗ ) = p u ( G ∗ ) + p c ( G ) = 1, where G ∗ denotes the matching graph of G . In particular, p u ( G ) + p c ( G ) ≥ 1 and hence, when G is amenable we have p c ( G ) = p u ( G ) ≥ 12 . When combined with the main result of the companion paper by the same authors (“Percolation critical probabilities of matching lattice-pairs”, 2022), we obtain for transitive G that the strict inequality p u ( G ) + p c ( G ) > 1 holds if and only if G is not a triangulation. A key technique is a method for expressing a planar site percolation process on a matching pair in terms of a dependent bond process on the corresponding dual pair of graphs. Amongst other things, the results reported here answer positively two conjectures of Benjamini and Schramm (Conjectures 7 and 8, Electron. Comm. Probab. 1 (1996) 71–82) in the case of quasi-transitive graphs.