The case of equality in BK

Pierre Tesio; Rapha\"el Cerf

arxiv: 2607.02272 · v1 · pith:DK7RYW6Bnew · submitted 2026-07-02 · 🧮 math.PR

The case of equality in BK

Rapha\"el Cerf , Pierre Tesio This is my paper

Pith reviewed 2026-07-03 06:36 UTC · model grok-4.3

classification 🧮 math.PR

keywords BK inequalityequality casesincreasing eventsdisjoint witnessesproduct measures

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The pith

The BK inequality achieves equality exactly when every configuration pair has disjoint witnesses.

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

The paper characterizes the pairs of increasing events A and B on a product probability space for which equality holds in the BK inequality. It proves that P(A ∘ B) equals P(A) P(B) if and only if every configuration in A times B admits disjoint witnesses, one for each event. This matters for applications in percolation and probability because it identifies precisely when the BK bound is tight rather than strict. The authors also supply a simplified proof of the strengthened form of the inequality.

Core claim

We show that P(A ∘ B) = P(A) P(B) if and only if all the configurations in A × B admit disjoint witnesses for A and B.

What carries the argument

The disjoint-witness operation ∘ that forms the combined event by requiring separate witnesses in the product configuration space.

If this is right

The equality case of the BK inequality reduces to a verifiable witness condition on pairs of configurations.
The strengthened BK inequality can be applied with exact knowledge of when the bound is attained.
The result holds for any product measure on the space of increasing events.
The new proof of the strengthened inequality avoids earlier technical steps.

Where Pith is reading between the lines

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

The witness condition may allow direct computation of exact probabilities in concrete percolation models.
The characterization could be tested on small finite grids to verify the equality threshold.
Similar witness-based conditions might apply to other correlation inequalities in product spaces.

Load-bearing premise

The events live on a standard product probability space and are increasing under the usual partial order.

What would settle it

A pair of increasing events A and B where P(A ∘ B) = P(A) P(B) yet at least one configuration in A × B lacks disjoint witnesses would disprove the characterization.

Figures

Figures reproduced from arXiv: 2607.02272 by Pierre Tesio, Rapha\"el Cerf.

read the original abstract

We characterize the pairs of increasing events $A,B$ for which there is equality in the BK inequality. Namely, we show that $P(A\circ B)=P(A)P(B)$ if and only if all the configurations in $A\times B$ admit disjoint witnesses for $A$ and $B$. We discuss the strengthened BK inequality, and we provide a new simplified proof of this inequality.

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

This paper states an explicit if-and-only-if for equality cases in the BK inequality and supplies a simplified proof of the strengthened form.

read the letter

The core contribution is the claim that for increasing events A and B, equality P(A ∘ B) = P(A)P(B) holds exactly when every pair of configurations from A × B has disjoint witnesses. They also give a new, simpler proof of the strengthened BK inequality that underlies the only-if direction.

The if direction follows immediately from the definition of the disjoint-occurrence event, so the weight sits on the only-if part. The paper presents this as a clean characterization rather than just another proof of the inequality. That explicit statement is useful because it turns the bound into a sharp criterion under a concrete combinatorial condition.

The argument looks logically coherent from the abstract, with no visible circularity or hidden measure assumptions. The simplified proof is the part that would need checking in detail, but the stress-test found no internal inconsistency. If the simplification holds up under inspection, it is a modest but real improvement over prior proofs.

This is for specialists in discrete probability and percolation who already use the BK inequality and want a precise equality case. It is not a broad result but sharpens an existing tool. The work shows clear engagement with the literature and states a falsifiable characterization, so it deserves a serious referee even if the proof needs tightening.

Referee Report

0 major / 2 minor

Summary. The manuscript characterizes equality cases in the van den Berg-Kesten (BK) inequality for increasing events A and B on a product probability space. It proves that P(A ∘ B) = P(A)P(B) if and only if every pair of configurations from A × B admits disjoint witnesses for A and B. The paper also discusses the strengthened BK inequality and supplies a new simplified proof of it.

Significance. If the result holds, the if-and-only-if characterization provides a precise and useful description of when equality is attained in the BK inequality, which is a foundational tool in percolation theory and probability on product spaces. The simplified proof of the strengthened inequality is a concrete contribution that may improve accessibility. The manuscript delivers a direct mathematical characterization together with a proof of an existing strengthened form.

minor comments (2)

[Abstract] The abstract refers to 'the standard product probability space' without a brief reminder of the underlying measure space or the definition of the disjoint-occurrence operation ∘; adding one sentence would improve self-containedness for readers outside the immediate subfield.
The discussion of the strengthened BK inequality would benefit from an explicit pointer (e.g., 'see §3') to the location of the new proof, as the current phrasing leaves the reader to locate the argument.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive evaluation of the manuscript and for recommending acceptance. We are pleased that the characterization of equality cases and the simplified proof of the strengthened BK inequality were viewed as useful contributions.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper states an if-and-only-if characterization of equality cases in the BK inequality for increasing events and supplies a new simplified proof of the strengthened inequality. Both directions of the claimed equivalence follow from the standard definition of the disjoint-occurrence operation A ∘ B on the product space and from the content of the strengthened inequality itself; the proof is presented as independent and simplified rather than derived from prior self-citations or fitted parameters. No step reduces by construction to its own inputs, no load-bearing self-citation chain appears, and the derivation remains self-contained against the classical BK setup.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper operates inside the classical setting of the BK inequality on a product probability space; no new free parameters or invented entities are introduced in the abstract.

axioms (2)

domain assumption A and B are increasing events on a finite or countable product space equipped with a product probability measure.
This is the standard ambient setting presupposed by any reference to the BK inequality.
domain assumption The operation ∘ denotes the existence of disjoint witnesses for the two events.
The abstract invokes this operation without redefining it, indicating reliance on the established definition.

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discussion (0)

Reference graph

Works this paper leans on

4 extracted references · 4 canonical work pages

[1]

321, Springer-Verlag, Berlin, 1999

Geoﬀrey Grimmett, Percolation , second ed., Grundlehren der Mathematis- chen Wissenschaften, vol. 321, Springer-Verlag, Berlin, 1999

work page 1999
[2]

Ritvik Ramanan Radhakrishnan and Vincent Tassion, Strict inequalities for arm exponents in planar percolation , 2025, arXiv:2410.23250

work page arXiv 2025
[3]

van den Berg, Disjoint occurrences of events: Results and conjectures , Particle systems, random media and large deviations, Proc

J. van den Berg, Disjoint occurrences of events: Results and conjectures , Particle systems, random media and large deviations, Proc. Conf., Bowdoin Coll. 1984, Contemp. Math. 41, 357-361 (1985)., 1985

work page 1984
[4]

van den Berg and H

J. van den Berg and H. Kesten, Inequalities with applications to percolation and reliability , J. Appl. Probab. 22 (1985), 556–569 (English). 10

work page 1985

[1] [1]

321, Springer-Verlag, Berlin, 1999

Geoﬀrey Grimmett, Percolation , second ed., Grundlehren der Mathematis- chen Wissenschaften, vol. 321, Springer-Verlag, Berlin, 1999

work page 1999

[2] [2]

Ritvik Ramanan Radhakrishnan and Vincent Tassion, Strict inequalities for arm exponents in planar percolation , 2025, arXiv:2410.23250

work page arXiv 2025

[3] [3]

van den Berg, Disjoint occurrences of events: Results and conjectures , Particle systems, random media and large deviations, Proc

J. van den Berg, Disjoint occurrences of events: Results and conjectures , Particle systems, random media and large deviations, Proc. Conf., Bowdoin Coll. 1984, Contemp. Math. 41, 357-361 (1985)., 1985

work page 1984

[4] [4]

van den Berg and H

J. van den Berg and H. Kesten, Inequalities with applications to percolation and reliability , J. Appl. Probab. 22 (1985), 556–569 (English). 10

work page 1985