Cardy's formula for critical Poisson-Voronoi percolation (annealed crossing part of Schramm's Problem 2.12)
In critical planar Voronoi percolation, the cells of a Poisson point process of intensity are coloured black or white independently with probability . Cardy predicted, and Smirnov proved for site percolation on the triangular lattice, that crossing probabilities of a conformal quadrilateral converge to Cardy's hypergeometric function of its conformal modulus. Schramm (ICM 2006, Problem 2.12) asked for the corresponding conformal-invariance theorem for Voronoi percolation. Do the crossing probabilities of critical Poisson-Voronoi percolation in a Jordan quadrilateral converge to Cardy's formula, and is the scaling limit conformally invariant?
- Result
- Proved(see note)
- Status
- Partial result
- AI contribution
- AI-discovered
- Method
- Argument
- Field
- Percolation, conformal invariance
- Posed by
- Oded Schramm
- Year posed
- 2006
- Years open
- 20y
- 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
- 30 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
Theorem 1.1: for every bounded Jordan domain with four marked boundary points, the annealed probability of a black crossing in critical Poisson-Voronoi percolation converges to Cardy's function of the conformal cross-ratio . A corollary gives a quenched limit for fixed rectangles. The companions prove that the expected number of pivotal cells for a unit-square crossing is asymptotic to and that conditional near-critical crossing-threshold laws for rational polygonal quads converge in probability over tessellations to the triangular-lattice law. Not shown: convergence of interfaces to SLE(6) or the full conformal invariance of Schramm's problem.
What the AI did
The release README says every result in it was produced by an unreleased internal OpenAI model following a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not among the README's two exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region, whose write-up was human-edited). The manuscripts are authored 'OpenAI' and name no human author. The principal manuscript (September 23, 2026) proves the annealed Cardy formula; two companions (October 5, 2026) take it as an input to prove a pivotal-count asymptotic and quenched near-critical universality of crossing thresholds.
Verification
No independent mathematician has checked this yet. Checked here: the introduction and Theorem 1.1 were read against Schramm's Problem 2.12 as the manuscript cites it. The paper itself limits the scope: the limit is of annealed crossing probabilities (averaged over points and colours); it does not claim convergence of interfaces or a quenched crossing limit in arbitrary domains. Both companions are conditional on the principal paper's theorem. No Lean formalization accompanies this family. Uses established box-crossing and quenched-annealed comparison inputs (Tassion, Ahlberg-Griffiths-Morris-Tassion, Vanneuville).