VibeMathedMath problems solved with AI
Sign in

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

Cardy (1992) predicted that for critical two-dimensional percolation the probability of an open crossing between two boundary arcs of a conformal quadrilateral converges to an explicit conformally invariant function, ∫0x[t(1−t)]−2/3dt /∫01[t(1−t)]−2/3dt\int_0^x[t(1-t)]^{-2/3}dt\,/\int_0^1[t(1-t)]^{-2/3}dt in the half-plane coordinate xx. Smirnov (2001) proved it for site percolation on the triangular lattice, and Camia-Newman built the full loop limit there, but the proof uses the lattice's special symmetry. For the most standard model, bond percolation on Z2\mathbb Z^2 at p=1/2p=1/2, only partial or disputed claims existed. Does Cardy's formula, and conformal invariance of the exploration interface (convergence to SLE6\mathrm{SLE}_6), hold for critical bond percolation on the square lattice?

Result
Proved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Argument
Field
Percolation; conformal invariance
Posed by
John Cardy, Critical percolation in finite geometries, J. Phys. A 25 (1992), as a physical prediction; the square-lattice case is the q = 1 case of Rohde-Schramm Conjecture 9.7 (2005)
Year posed
1992
Years open
34y
Solved
2026-09-23
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Unreviewed
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
48 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

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.

What the AI did

The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author.

Verification

No independent mathematician has checked this yet. Checked here: Theorem 1.2 (Cardy's formula) and the q=1q=1 case of Theorem 1.1 of the September 23 manuscript were read against Cardy's prediction as the paper states it; they claim the crossing limit in every Jordan quadrilateral with uniformly approximated marks under the stated free-boundary convention, and SLE6\mathrm{SLE}_6 convergence of the interface. The proof was not refereed. The manuscript itself notes overlapping prior public claims for square-lattice bond percolation (Tsai-Yam-Zhou, arXiv 1112.2017; Zhou, arXiv 2409.03235) that were not checked here. No Lean formalization exists.

Sources

Changelog1 change

Discussion