Planar site percolation via tree embeddings

TL;DR

The paper resolves Benjamini–Schramm conjecture 7 for infinite planar graphs properly embedded in the plane with minimum degree at least 7 by constructing explicit embedded trees to generate multiple infinite clusters near criticality and by establishing exponential decay of connection probabilities via embedded forests and duality. A key component is the introduction of the matching graph $G_*$ to relate site percolation on $G$ to dual structures, alongside a generalized vertex-cut characterization of $p_c^{site}$ that does not rely on bounded degree or transitivity. The authors also develop a uniform percolation framework based on embedded binary trees to prove stability and nonuniqueness of infinite clusters for $p$ in $(p_c^{site}, 1-p_c^{site})$, and they extend the methods to broader applications, including vertex-cut characterizations and counterexamples in planar graphs with many ends. The results advance understanding of phase transitions in planar, non-amenable-like settings and provide tools for analyzing percolation without symmetry assumptions.

Abstract

We prove that if $G$ is an infinite, connected, planar graph properly embedded in $\mathbb{R}^2$ with minimum degree at least $7$, then i.i.d.\ Bernoulli$(p)$ site percolation on $G$ almost surely has infinitely many infinite open (1-)clusters for every \[ p \in \bigl(p_c^{\mathrm{site}},\, 1-p_c^{\mathrm{site}}\bigr). \] Moreover, we show that $p_c^{\mathrm{site}}<\tfrac12$, so this non-uniqueness interval is nonempty. This verifies Conjecture~7 of Benjamini and Schramm~\cite{bs96} for this class of properly embedded planar graphs. Our proof introduces a new construction of embedded trees in $G$. These trees yield infinitely many infinite clusters for percolation parameters near $\tfrac12$, and they also enable exponential decay of two-point connection probabilities by partitioning $G$ using infinitely many disjoint trees. Variants of this approach were later used in~\cite{ZL26} to construct a counterexample to Conjecture~7 of~\cite{bs96} for planar graphs with uncountably many ends. Finally, the methods developed here have further applications: in~\cite{perc24} they are used to prove a vertex-cut characterization of $p_c^{\mathrm{site}}$ (conjectured by Kahn in~\cite{JK03}) and to refute an edge-cut characterization proposed by Lyons and Peres~\cite{LP16} and Tang (\cite{Tang2023}).

Figures (4)

• Figure 2.1: Tree embedding in a degree-7 triangular tiling of the hyperbolic plane: the degree-7 triangular tiling is represented by black lines, and the embedded tree is represented by red lines.
• Figure 2.2: Infinite open clusters in the tree rooted at $v_{r01}$ is separated from the infinite open clusters in the tree rooted at $v_{r11}$ by the infinite closed cluster occupying $v_r$, $v_{r0},v_{r1}$, $v_{r00},v_{r10}$
• Figure 4.1: In the left graph, the black lines represent a chandelier, the blue lines represent incident edges on the left of a chandelier. In the right graph, the black lines represent an anti-chandelier, and blue lines represent incident edges on the right of an anti-chandelier.
• Figure 4.2: Embedded forest

Theorems & Definitions (75)

• Definition 1.1
• Conjecture 1.2
• Conjecture 1.3
• Theorem 1.4
• Definition 1.5
• Definition 1.6
• Definition 2.1
• Definition 2.2
• Lemma 2.3
• proof