Percolation critical probabilities of matching lattice-pairs
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2022
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866914685222125568 |
|---|---|
| author | Grimmett, Geoffrey R. Li, Zhongyang |
| author_facet | Grimmett, Geoffrey R. Li, Zhongyang |
| contents | 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 on its matching graph $G_*$. It is assumed that $G$ is properly embedded in either the Euclidean or the hyperbolic plane. When $G$ is transitive, strict inequality holds if and only if $G$ is not a triangulation.
The basic approach is the standard method of enhancements, but its implemention has complexity arising from the non-Euclidean (hyperbolic) space, the study of site (rather than bond) percolation, and the generality of the assumption of quasi-transitivity.
This result is complementary to the work of the authors ("Hyperbolic site percolation", arXiv:2203.00981) on the equality $p_u(G) + p_c(G_*) = 1$, where $p_u$ is the critical probability for the existence of a unique infinite open cluster. It implies for transitive $G$ that $p_u(G) + p_c(G) \ge 1$, with equality if and only if $G$ is a triangulation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2205_02734 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Percolation critical probabilities of matching lattice-pairs Grimmett, Geoffrey R. Li, Zhongyang Probability Mathematical Physics 60K35, 82B43 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 on its matching graph $G_*$. It is assumed that $G$ is properly embedded in either the Euclidean or the hyperbolic plane. When $G$ is transitive, strict inequality holds if and only if $G$ is not a triangulation. The basic approach is the standard method of enhancements, but its implemention has complexity arising from the non-Euclidean (hyperbolic) space, the study of site (rather than bond) percolation, and the generality of the assumption of quasi-transitivity. This result is complementary to the work of the authors ("Hyperbolic site percolation", arXiv:2203.00981) on the equality $p_u(G) + p_c(G_*) = 1$, where $p_u$ is the critical probability for the existence of a unique infinite open cluster. It implies for transitive $G$ that $p_u(G) + p_c(G) \ge 1$, with equality if and only if $G$ is a triangulation. |
| title | Percolation critical probabilities of matching lattice-pairs |
| topic | Probability Mathematical Physics 60K35, 82B43 |
| url | https://arxiv.org/abs/2205.02734 |