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Kozma-Nitzan Conjecture

The Kozma-Nitzan conjecture is a correlation inequality for independent bond percolation on a finite graph. Each edge e is open independently with probability p_e in [0,1]. Write s<->t if an open graph path joins the vertices s and t, and s<->A if s is joined to at least one vertex of a nonempty set A of vertices. The multiplicative formulation asserts

P(s<->A, s<->t)>=P(s<->A)min_(a in A)P(a<->t).

Gouéré (2026) gives a human exposition of a proof attributed to OpenAI. Blanc-Renaudie (2026) gives an earlier exposition of this stronger multiplicative inequality. The AI-generated Lean development for bond percolation proves a weaker correlation statement, not this multiplicative inequality.

See also

Bond Percolation

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References

Blanc-Renaudie, A. "Notes on Kozma and Nitzan's Proof of theta(p_c)=0." 7 Oct 2026. https://arxiv.org/abs/2610.10502.Gouéré, J.-B. "On the Proof of the Kozma-Nitzan Conjecture." 21 Oct 2026. https://arxiv.org/abs/2610.30063.Kozma, G. and Nitzan, S. "A Reduction of the theta(p_c)=0 Problem to a Conjectured Inequality." 22 Jan 2024. https://arxiv.org/abs/2401.12397.

Cite this as:

Weisstein, Eric W. "Kozma-Nitzan Conjecture." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Kozma-NitzanConjecture.html

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