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Cardy's formula and conformal invariance for critical bond percolation on the square lattice

Theorem 1.2: for critical bond percolation on δZ2\delta\mathbb Z^2 in a Jordan domain with four uniformly approximated marks, all boundary edges sampled and no external connections, the probability of an open crossing between the arcs [a,b][a,b] and [c,d][c,d] converges to Cardy's hypergeometric function of the cross-ratio. The q=1q=1 case of Theorem 1.1 gives chordal SLE6\mathrm{SLE}_6 convergence of the exploration interface and CLE6_6 convergence of all nested loops. The crossing result is proved directly, not from the curve limit. It does not treat other lattices or general boundary conditions beyond those stated.

Exact FrontierDelta

Prior state unknown→proved

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Occurred: Sep 23, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: unreviewed. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: Cardy's formula and conformal invariance for critical bond percolation on the square lattice · Cardy's formula on the square lattice

Confidence: Not scored

Registry verification: unreviewed · announcement · candidate

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Unreleased internal OpenAI model
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Lineage and corrections

Companion: Natural Occupation Measures for Critical Square-Lattice FK Interfaces evidence for this event

Companion: Conformal Limits of Critical Square-Lattice Random-Cluster Interfaces evidence for this event

This event attributed to Unreleased internal OpenAI model

VibeMathed record: Cardy's formula and conformal invariance for critical bond percolation on the square lattice evidence for this event

Cardy's formula and conformal invariance for critical bond percolation on the square lattice evidence for this event

Theorem 1.2: for critical bond percolation on δZ2\delta\mathbb Z^2 in a Jordan domain with four uniformly approximated marks, all boundary edges sampled and no external connections, the probability of an open crossing between the arcs [a,b][a,b] and [c,d][c,d] converges to Cardy's hypergeometric function of the cross-ratio. The q=1q=1 case of Theorem 1.1 gives chordal SLE6\mathrm{SLE}_6 convergence of the exploration interface and CLE6_6 convergence of all nested loops. The crossing result is proved directly, not from the curve limit. It does not treat other lattices or general boundary conditions beyond those stated. parent of this event

Cardy's formula and conformal invariance for critical bond percolation on the square lattice parent of this event

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