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Planar percolation and the loop O(n) model

Alexander Glazman, Matan Harel, Nathan Zelesko

TL;DR

This work proves a general dichotomy for planar site percolation under tail triviality, positive association, and domination by the complement, showing that the number of infinite clusters is either $0$ or $\infty$ and obtaining a sharp bound $p_c(G)\ge \tfrac{1}{2}$ for invariantly amenable unimodular planar graphs. The authors then apply this framework to the loop $O(n)$ model on the hexagonal lattice, establishing that for $n\in[1,2]$ and $x\in[1/\sqrt{2},1]$ every face is surrounded by infinitely many loops under translation-invariant Gibbs measures, and they derive RS-W-type crossing bounds in the subcritical region. The approach leverages Edwards–Sokal-type graphical representations and divide-and-color constructions to obtain positive association on quenched measures, enabling percolation arguments even when monotonicity is absent. Together, these results advance understanding of macroscopic structures in planar, nonmonotone models and provide robust tools to analyze percolation phenomena on general planar graphs and random planar maps.

Abstract

We show that a large class of site percolation processes on any planar graph contains either zero or infinitely many infinite connected components. The assumptions that we require are: tail triviality, positive association (FKG) and that the set of open vertices is stochastically dominated by the set of closed ones. This covers the case of Bernoulli site percolation at parameter $p\leq 1/2$ and resolves Conjecture 8 from the work of Benjamini and Schramm from 1996. Our result also implies that $p_c\geq 1/2$ for any invariantly amenable unimodular random rooted planar graph. Furthermore, we apply our statement to the loop O(n) model on the hexagonal lattice and confirm a part of the phase diagram conjectured by Nienhuis in 1982: the existence of infinitely many loops around every face whenever $n\in [1,2]$ and $x\in [1/\sqrt{2},1]$. The point $n=2,x=1/\sqrt{2}$ is conjectured to be critical. This is the first instance that this behavior has been proven in such a large region of parameters. In a big portion of this region, the loop O(n) model has no known FKG representation. We apply our percolation result to quenched distributions that can be described as divide and color models.

Planar percolation and the loop O(n) model

TL;DR

This work proves a general dichotomy for planar site percolation under tail triviality, positive association, and domination by the complement, showing that the number of infinite clusters is either or and obtaining a sharp bound for invariantly amenable unimodular planar graphs. The authors then apply this framework to the loop model on the hexagonal lattice, establishing that for and every face is surrounded by infinitely many loops under translation-invariant Gibbs measures, and they derive RS-W-type crossing bounds in the subcritical region. The approach leverages Edwards–Sokal-type graphical representations and divide-and-color constructions to obtain positive association on quenched measures, enabling percolation arguments even when monotonicity is absent. Together, these results advance understanding of macroscopic structures in planar, nonmonotone models and provide robust tools to analyze percolation phenomena on general planar graphs and random planar maps.

Abstract

We show that a large class of site percolation processes on any planar graph contains either zero or infinitely many infinite connected components. The assumptions that we require are: tail triviality, positive association (FKG) and that the set of open vertices is stochastically dominated by the set of closed ones. This covers the case of Bernoulli site percolation at parameter and resolves Conjecture 8 from the work of Benjamini and Schramm from 1996. Our result also implies that for any invariantly amenable unimodular random rooted planar graph. Furthermore, we apply our statement to the loop O(n) model on the hexagonal lattice and confirm a part of the phase diagram conjectured by Nienhuis in 1982: the existence of infinitely many loops around every face whenever and . The point is conjectured to be critical. This is the first instance that this behavior has been proven in such a large region of parameters. In a big portion of this region, the loop O(n) model has no known FKG representation. We apply our percolation result to quenched distributions that can be described as divide and color models.

Paper Structure

This paper contains 8 sections, 11 theorems, 46 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Site percolation models on planar graphs
  3. Corollaries for percolation on unimodular invariantly amenable graphs
  4. Loop $\mathrm{O}(n)$ model
  5. Sketch of the proof of Theorem \ref{['thm:perco']}
  6. Planar percolation models: proof of Theorem \ref{['thm:perco']}
  7. Proof of Corollaries \ref{['cor:p-c']} and \ref{['cor:divide-color']}
  8. Loop $\mathrm{O}(n)$ model: proof of Theorem \ref{['thm:LoopOn']}

Key Result

Theorem 1

Let $G=(V,E)$ be an infinite locally finite planar graph. Consider a random variable $\sigma\colon V\to \{0,1\}$ that is tail trivial, positively associated and is stochastically dominated by $1-\sigma$. Then, $\{\sigma=1\}$ contains either no infinite connected component a.s. or infinitely many of

Figures (4)

  • Figure 1: Left: two copies of $\mathbb{Z} \times \mathbb{Z}^+$ connected by one edge; at any $p > p_c(\mathbb{Z}^2)$, the probability that the number of infinite components is one or two is positive. Right: a half plane connected to a graph of branching number $\sqrt{2}$. For $p_c(\mathbb{Z}^2) < p \leq 1/\sqrt{2}$, Bernoulli percolation contains a unique infinite component, almost surely; for $1/\sqrt{2}< p < 1$, there are infinitely many connected components in Bernoulli percolation.
  • Figure 2: The phase diagram of the loop $\mathrm{O}(n)$ model: above $n=2$ and to the left of $x_c(n)$, loop lengths should have exponential tails; below and on the curve $x_c(n)$, the loops should be macroscopic and converge to $\mathrm{CLE}_\kappa$. Established regions are in orange and red respectively. The current work treats the opaque red rectangle.
  • Figure 3: Left: A portion $\phi(R)$ bounding the face $F_1$ in black, the graph $G_1$ in cyan, and the edges of $E_1$ in dashed gray. The half edges $\gamma_{1,k}^1$, with $y_{1,k}$ as their endpoints, in blue. Middle: The new embedding of $G_1$ in a ball contained in the interior in green. The image of the half edges $\gamma_{1,k}^2$ with their endpoints $z_{1,k}$ in burgundy. Right: A choice for the paths connecting the first three sets of endpoints, and the domain $T_3$ in purple.
  • Figure 4: Left: a configuration in $\mathrm{Arm}_n^{2k}(i_1, \dots, i_{2k})$ for $k=3$ with open (blue) and closed (red) infinite paths. Right: a coloring of the interior of the domain. As shown by the green circles, planar topology restricts the possible connectivity of the infinite paths for any coloring. In the illustrated example, there are five open and closed infinite components in total, which is greater than $k+1=4$.

Theorems & Definitions (23)

  • Theorem 1
  • Corollary 1.1
  • Corollary 1.2
  • Theorem 2
  • Lemma 2.1
  • proof
  • Proposition 2.2
  • proof
  • Lemma 2.3
  • proof
  • ...and 13 more