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arxiv: 2605.30299 · v1 · pith:2FQ7WFEJnew · submitted 2026-05-28 · 🧮 math.PR · math-ph· math.MP

On reversing the Simon-Lieb inequality in high-dimensional percolation

Romain Panis , Bruno Schapira This is my paper

Pith reviewed 2026-06-29 05:28 UTC · model grok-4.3

classification 🧮 math.PR math-phmath.MP
keywords percolationSimon-Lieb inequalityvan den Berg-Kesten inequalityhigh dimensionscritical probabilityone-arm probabilitytwo-point function
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The pith

In dimensions above six, percolation admits a partial reversal of the Simon-Lieb inequality that bounds φ_pc(S) uniformly in S.

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

The paper proves that a classical consequence of the van den Berg-Kesten inequality, known as the Simon-Lieb inequality, can be partially reversed for Bernoulli percolation on Z^d when d exceeds 6. This reversal is used to show that the Duminil-Copin-Tassion quantity φ at the critical point stays bounded by a constant independent of the choice of finite set S. A sympathetic reader would care because the same reversal supplies short, self-contained derivations of standard near-critical two-point estimates and sharp critical one-arm bounds.

Core claim

For Bernoulli percolation on the d-dimensional lattice with d greater than 6, the Simon-Lieb inequality admits a partial reversal. This yields the uniform upper bound on φ_pc(S) for every finite S subset of Z^d. The same reversal produces short proofs of near-critical two-point function estimates and sharp bounds on the critical one-arm probability.

What carries the argument

The partial reversal of the Simon-Lieb inequality, which converts a classical consequence of the van den Berg-Kesten inequality into an upper bound that controls the Duminil-Copin-Tassion quantity φ_pc(S).

If this is right

  • Near-critical estimates for the two-point function follow directly from the reversal.
  • Sharp bounds on the critical one-arm probability are obtained.
  • Several standard high-dimensional percolation results admit short self-contained proofs.

Where Pith is reading between the lines

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

  • The uniform bound on φ_pc(S) may simplify the analysis of other scale-invariant quantities at criticality in high dimensions.
  • Similar reversal techniques could be tested in models where mean-field behavior is known but the Simon-Lieb step has not yet been reversed.

Load-bearing premise

The reversal is proved only where mean-field decay and the van den Berg-Kesten inequality are already known to hold, which requires dimension strictly larger than six.

What would settle it

Numerical or rigorous evidence that φ_pc(S) grows unbounded as S ranges over larger and larger finite sets in some dimension d greater than 6 would falsify the main application.

Figures

Figures reproduced from arXiv: 2605.30299 by Bruno Schapira, Romain Panis.

Figure 1
Figure 1. Figure 1: An illustration of the event Euv introduced in (1.30). The cluster of u (resp. v) in ω [uv] is the light red (resp. blue) region. Our entire strategy revolves around the observation that, if u is regular and d > 6, then the blue and red regions have a positive probability to avoid each other. The points u ∈ ∂S which satisfy the above property are called regular points. They were first introduced by Kozma a… view at source ↗
Figure 2
Figure 2. Figure 2: An illustration of why Euv and Eu′v ′ cannot occur simultaneously. The bold lines represent open paths in a percolation configuration that would satisfy both events. We observe that in each case, we can close {u, v} without destroying the connection between 0 and x. (ii) one has Euv = {o S ←→ u}∩ {ωuv = 1}∩ {v ←→ x in Λ\C[uv] (u; Λ)} where C [uv] (u; Λ) denotes the cluster of u in ω [uv] ∩ Λ; (iii) the eve… view at source ↗
Figure 3
Figure 3. Figure 3: An illustration of a vertex u that is K-escapable. The blue region represents C(u). We consider the set X (K,T) S,Λ of pioneers of S that are (K, T)-regular and K-escapable in Λ, and let X K S,Λ = X (K,20) S,Λ . If u ∈ X K S,Λ , we denote by (v, w, γ) the earliest triplet (according to some arbitrary order) given by the K-escapability in Λ of u. We say that a realisation A of C(u; Λ) is u-good if2 it is su… view at source ↗
Figure 4
Figure 4. Figure 4: An illustration of the map φ. On the left: a depiction of γ, γ1, w, v, which can be constructed from a configuration ω ∈ A. On the right: a depiction of φ(ω), which is obtained from ω by (essentially) opening the edges of γ ′ and closing all the other edges in ΛK−1(u). It is clear from construction that v is always K-escapable in φ(ω). Let ω ∈ A. Let γ be any self-avoiding path from u to 0, that lies in S,… view at source ↗
Figure 5
Figure 5. Figure 5: An illustration of the proof of the bound χ(p) ≳ L(p) 2 . The idea is to reconstruct τp(0, x) for x1 = −k according to the above picture. The black line (resp. blue line) represents the two-point function τHj ,p(0, u) (resp. τHj+1\C(u;Hj+1),p(v, x)). The parameters satisfy j ≥ k and range over an interval of length ≍ L(p). By Theorem 1.2, summing this diagram over j, k, {u, v}, x yields a lower bound on χ(… view at source ↗
Figure 6
Figure 6. Figure 6: Left: An illustration of the proof of Lemma 7.2. The facet F is the bold red line. On the bottom, we illustrated the sequence (Hi)1≤i≤5 in the case F ̸= F + 1 (k). Right: An illustration of the proof of Proposition 7.1. The set Q′ is the light green region. The facet F (resp. F ′ ) is the bold red (resp. blue) line. Proof of Proposition 7.1. Let c, k0 > 0 be given by Lemma 7.2. Let ε > 0. Let α, p0 to be f… view at source ↗
read the original abstract

We study Bernoulli percolation on $\mathbb Z^d$ in dimensions ${d>6}$. We prove that a classical consequence of the van den Berg-Kesten inequality, often referred to as the Simon-Lieb inequality in the context of the Ising model, admits a partial reversal. As a main application, we show that the quantity $\varphi_{p_c}(S)$, introduced by Duminil-Copin and Tassion (Comm.\ Math.\ Phys., 2016), is uniformly bounded over all $S\subset \mathbb Z^d$. This partial reversal further yields a short and self-contained route to several key results, including near-critical estimates on the two-point function and sharp bounds on the critical one-arm probability.

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.

Referee Report

0 major / 2 minor

Summary. The paper proves a partial reversal of the Simon-Lieb inequality (a consequence of the van den Berg-Kesten inequality) for Bernoulli percolation on Z^d in dimensions d>6. The main application is to show that φ_{p_c}(S), as defined by Duminil-Copin and Tassion, is uniformly bounded over all subsets S of Z^d. The reversal is further used to derive near-critical estimates for the two-point function and sharp bounds on the critical one-arm probability, providing a short self-contained route to these results in the mean-field regime.

Significance. If the central claims hold, the partial reversal supplies a useful new tool for high-dimensional percolation that simplifies several key estimates without relying on additional fitted parameters or external uniformity assumptions beyond the known two-point decay for d>6. The uniform bound on φ_{p_c}(S) strengthens control over near-critical quantities, and the self-contained derivations constitute a clear technical contribution in the mean-field regime where the required decay is already established by prior work.

minor comments (2)
  1. [Introduction] The abstract and introduction should explicitly reference the precise statement of the Simon-Lieb inequality being reversed (e.g., the form appearing in §2) to make the partial reversal immediately comparable to the classical statement.
  2. [Section 3] Notation for the event {0 ↔ S} and the definition of φ_p(S) could be recalled in a short display equation when first used in the application section to aid readers unfamiliar with the Duminil-Copin–Tassion quantity.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary, assessment of significance, and recommendation to accept the manuscript.

Circularity Check

0 steps flagged

No significant circularity in the derivation chain

full rationale

The paper proves a partial reversal of the Simon-Lieb inequality (a consequence of the van den Berg-Kesten inequality) in d>6, where the required two-point decay is known from independent prior mean-field results. The uniform bound on φ_{p_c}(S) for all S follows directly as an application of this reversal, without any self-definitional steps, fitted inputs renamed as predictions, or load-bearing self-citations. The argument is explicitly restricted to the regime where external decay estimates hold and provides a self-contained route to near-critical estimates, making the central claims independent of the paper's own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only the abstract is supplied, so the ledger is necessarily incomplete.

axioms (1)
  • domain assumption The result holds for d>6, the regime of mean-field percolation.
    Explicitly stated in the abstract.

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