Percolation critical probabilities of matching lattice-pairs

Abstract

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.

Figures (12)

• Figure 1.1: The square lattice ${\mathbb Z}^2$ and its matching graph.
• Figure 3.1: The graph $G$ is the tiling of the plane with copies of this square. Taking into account the symmetries of the square, this tiling is canonical after a suitable rescaling of the interior square. The diagonals are indicated by dashed lines.
• Figure 4.1: An illustration of Lemma \ref{['prop:10']}. The jagged (red) path crosses $L_\delta$ in the long direction.
• Figure 4.2: A square of the square lattice, its matching graph, and with its facial site added.
• Figure 4.3: An illustration of the property $\Pi_A$: a non-self-touching path of $G_*$ containing a diagonal near its middle.

Theorems & Definitions (43)

• Remark 1.1
• Theorem 1.2
• Corollary 1.3
• proof : Proof of Corollary \ref{['cor']}
• Theorem 1.4
• Remark 1.5
• Theorem 2.1
• Remark 2.2
• Definition 2.3
• Definition 2.4