• Corpus ID: 247218585

Hyperbolic site percolation

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

Figures from this paper

Percolation critical probabilities of matching lattice-pairs

. A necessary and sufficient condition is established for the strict inequality p c ( G ∗ ) < p c ( G ) between the critical probabilities of site percolation on a quasi-transitive, plane graph G and

References

SHOWING 1-10 OF 66 REFERENCES

Percolation transitive graphs as a coalescent process: relentless merging followed by simultaneous uniqueness

Consider i.i.d. percolation with retention parameter p on an infinite graph G. There is a well known critical parameter p c ∈ [0, 1] for the existence of infinite open clusters. Recently, it has been

Group-invariant Percolation on Graphs

Abstract. Let G be a closed group of automorphisms of a graph X. We relate geometric properties of G and X, such as amenability and unimodularity, to properties of G-invariant percolation processes

Percolation on Transitive Graphs as a Coalescent Process : Relentless Merging Followed by Simultaneous

Consider i.i.d. percolation with retention parameter p on an infinite graph G. There is a well known critical parameter pc ∈ [0, 1] for the existence of infinite open clusters. Recently, it has been

Existence of phase transition for percolation using the Gaussian free field

In this paper, we prove that Bernoulli percolation on bounded degree graphs with isoperimetric dimension $d>4$ undergoes a non-trivial phase transition (in the sense that $p_c<1$). As a corollary, we

Multiplicity of Phase Transitions and Mean-Field Criticality on Highly Non-Amenable Graphs

Abstract: We consider independent percolation, Ising and Potts models, and the contact process, on infinite, locally finite, connected graphs.It is shown that on graphs with edge-isoperimetric

Automorphisms of graphs

This chapter surveys automorphisms of finite graphs, concentrating on the asymmetry of typical graphs, prescribing automorphism groups (as either permutation groups or abstract groups), and special

Percolation ?

572 NOTICES OF THE AMS VOLUME 53, NUMBER 5 Percolation is a simple probabilistic model which exhibits a phase transition (as we explain below). The simplest version takes place on Z2, which we view

Stability of infinite clusters in supercritical percolation

Abstract. A recent theorem by Häggström and Peres concerning independent percolation is extended to all the quasi-transitive graphs. This theorem states that if 0<p1<p2≤1 and percolation occurs at

Cubic graphs and the golden mean

...