Planar percolation and the loop O(n) model

Alexander Glazman; Matan Harel; Nathan Zelesko

arxiv: 2508.20917 · v3 · submitted 2025-08-28 · 🧮 math.PR · math-ph· math.CO· math.MP

Planar percolation and the loop O(n) model

Alexander Glazman , Matan Harel , Nathan Zelesko This is my paper

Pith reviewed 2026-05-18 20:34 UTC · model grok-4.3

classification 🧮 math.PR math-phmath.COmath.MP

keywords planar percolationsite percolationloop O(n) modelinfinite clustersphase diagramhexagonal latticeFKG inequalitystochastic domination

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The pith

Site percolation on planar graphs has either zero or infinitely many infinite connected components when open vertices are stochastically dominated by closed ones.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that site percolation processes on any planar graph satisfy a zero-one law for the number of infinite clusters. The proof requires only tail triviality of the measure, the FKG inequality for positive associations, and the assumption that the open set is stochastically dominated by the closed set. These conditions cover Bernoulli site percolation at every p at most 1/2 and thereby resolve the 1996 Benjamini-Schramm conjecture. The same argument applies to divide-and-color representations of the loop O(n) model and yields the existence of infinitely many loops around every face in a large region of the conjectured phase diagram.

Core claim

Under tail triviality, the FKG inequality, and stochastic domination of the open vertices by the closed vertices, any site percolation process on a planar graph has either zero or infinitely many infinite connected components. This statement is proved for general planar graphs and then specialized to Bernoulli percolation at p ≤ 1/2 and to quenched divide-and-color representations of the loop O(n) model on the hexagonal lattice.

What carries the argument

Stochastic domination of the open set by the closed set, which together with planarity and the FKG property rules out the existence of exactly one infinite component.

If this is right

Bernoulli site percolation at p ≤ 1/2 on any planar graph has either zero or infinitely many infinite clusters.
The critical probability satisfies pc ≥ 1/2 for every invariantly amenable unimodular random rooted planar graph.
The loop O(n) model on the hexagonal lattice has infinitely many loops surrounding every face for all n ∈ [1,2] and x ∈ [1/√2, 1].
The same zero-one law holds for quenched divide-and-color representations of the loop O(n) model even in regions without a known direct FKG representation.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The divide-and-color technique may extend the argument to other planar loop or interface models that lack obvious monotonicity.
A similar zero-one law could constrain the possible number of infinite clusters in percolation on random planar maps or other planar random media.
If the domination assumption can be verified for additional models, the result would give immediate information about their critical regimes without computing crossings directly.

Load-bearing premise

The open vertices are stochastically dominated by the closed vertices.

What would settle it

An explicit construction of a planar graph together with a probability measure satisfying tail triviality and FKG but having exactly one infinite cluster and violating the domination condition only on a set of measure zero would falsify the claim.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Proves zero-or-infinity for planar site percolation under tail triviality, FKG, and open-closed domination, settling the 1996 conjecture and extending to a large chunk of the loop O(n) phase diagram.

read the letter

The core result is a general theorem: any site percolation on a planar graph that satisfies tail triviality, FKG positive association, and stochastic domination of the open vertices by the closed ones has either zero or infinitely many infinite clusters. This directly resolves Conjecture 8 of Benjamini and Schramm from 1996 for Bernoulli site percolation at p ≤ 1/2 and yields p_c ≥ 1/2 for invariantly amenable unimodular random rooted planar graphs. The authors then apply the same statement to quenched divide-and-color representations of the loop O(n) model on the hexagonal lattice, obtaining infinitely many loops around every face for n in [1,2] and x in [1/√2, 1]. That interval includes regions without known FKG lattice representations, so the reach is genuinely new there.

Referee Report

1 major / 2 minor

Summary. The manuscript proves that site percolation on any planar graph has either zero or infinitely many infinite clusters, assuming tail triviality, FKG positive association, and stochastic domination of the open vertex set by the closed vertex set. This is applied to Bernoulli site percolation for p ≤ 1/2 (resolving Conjecture 8 of Benjamini-Schramm 1996) and to divide-and-color representations of the loop O(n) model on the hexagonal lattice, establishing infinitely many loops around every face for n ∈ [1,2] and x ∈ [1/√2, 1] and thereby confirming part of Nienhuis's 1982 phase diagram.

Significance. If the central theorem holds, the result supplies a general planar-percolation criterion that unifies several models and resolves a 28-year-old conjecture while extending to parameter regions of the loop O(n) model lacking direct FKG lattice representations. The combination of ergodicity from tail triviality with a planar separation argument is a clean technical contribution; the applications to quenched divide-and-color measures are a notable strength.

major comments (1)

[Theorem 1.1 / §2] Theorem 1.1 (or the main statement in §2): the proof that the domination assumption precludes a unique infinite cluster relies on a planar separation argument; the precise manner in which the stochastic ordering is transferred to the conditional law given the existence of at least one infinite cluster should be spelled out explicitly, as this step appears load-bearing for the zero-versus-infinite dichotomy.

minor comments (2)

[Introduction / §1] The statement of the three standing assumptions could be collected in a single displayed box or numbered list for quick reference when the applications are discussed later.
[§5] In the loop O(n) application, the precise range of the divide-and-color parameter that inherits the required stochastic domination should be stated as an explicit interval rather than left implicit in the quenched construction.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading, positive assessment of the significance, and recommendation for minor revision. The comment identifies a point where additional explicitness will strengthen the exposition of the central argument. We address it below and will revise accordingly.

read point-by-point responses

Referee: [Theorem 1.1 / §2] Theorem 1.1 (or the main statement in §2): the proof that the domination assumption precludes a unique infinite cluster relies on a planar separation argument; the precise manner in which the stochastic ordering is transferred to the conditional law given the existence of at least one infinite cluster should be spelled out explicitly, as this step appears load-bearing for the zero-versus-infinite dichotomy.

Authors: We agree that the transfer of stochastic domination under conditioning merits a more explicit treatment, as it underpins the contradiction to uniqueness. In the current proof, tail triviality ensures that the conditioning event (existence of at least one infinite cluster) is measurable with respect to the tail sigma-algebra, while FKG positive association and the global domination of the open set by the closed set allow the conditional law to inherit the same stochastic ordering. The planar separation argument then applies directly to this conditional measure to rule out a unique infinite cluster. In the revised manuscript we will insert a short dedicated paragraph (or lemma) immediately preceding the separation step that spells out this preservation of the ordering, including the relevant monotonicity and tail-triviality justifications. This addition clarifies the load-bearing step without altering the logic or results. revision: yes

Circularity Check

0 steps flagged

No significant circularity in central derivation

full rationale

The paper proves a general theorem establishing that site percolation on any planar graph has either zero or infinitely many infinite clusters, under the assumptions of tail triviality, FKG positive association, and stochastic domination of open vertices by closed ones. This is combined with planarity for the separation argument. Applications to Bernoulli site percolation (p ≤ 1/2) and divide-and-color representations of the loop O(n) model verify the assumptions directly in the relevant ranges, including regions without known FKG lattice representations. The result resolves an external 1996 conjecture and addresses part of a 1982 phase diagram conjecture. No self-definitional steps, fitted inputs renamed as predictions, or load-bearing self-citation chains appear in the derivation; any self-citations are contextual and non-central.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The result rests on three domain assumptions standard in percolation theory plus planarity of the underlying graph; no new free parameters or invented entities are introduced.

axioms (3)

domain assumption Tail triviality of the percolation measure
Invoked to rule out exactly one infinite cluster; stated as a required assumption in the abstract.
domain assumption Positive association (FKG inequality)
Required for the percolation result; explicitly listed among the assumptions.
domain assumption Stochastic domination of open vertices by closed vertices
Central assumption that covers Bernoulli site percolation at p ≤ 1/2 and the divide-and-color representations.

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discussion (0)

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Relation between the paper passage and the cited Recognition theorem.

Theorem 1. Let G=(V,E) be an infinite locally finite planar graph. Consider a random variable σ:V→{0,1} that is tail trivial, positively associated and is stochastically dominated by 1−σ. Then {σ=1} contains either no infinite connected component a.s. or infinitely many of them a.s.

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