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The Rohde-Schramm conjecture: critical square-lattice FK interfaces converge to SLE for 1 <= q < 4 (and 0 < q < 1 in smooth domains)

In the random-cluster (FK) model on the square lattice with cluster weight qq at the self-dual point pq=q/(1+q)p_q=\sqrt q/(1+\sqrt q), the Dobrushin interface in a domain with two marked boundary points is expected to have a conformally invariant scaling limit. Rohde and Schramm (2005, Conjecture 9.7), restated by Schramm (ICM 2006, Problem 2.6), conjectured for every fixed 0<q<40<q<4 that it converges to chordal SLEκ\mathrm{SLE}_{\kappa} with κ=4π/arccos⁡(−q/2)\kappa=4\pi/\arccos(-\sqrt q/2). Before this work it was proved only for q=2q=2 (FK-Ising, Smirnov and Chelkak-Smirnov) and, for site percolation on the triangular lattice, the analogous q=1q=1 statement. Does the critical square-lattice FK interface converge to SLEκ(q)\mathrm{SLE}_{\kappa(q)} for every 0<q<40<q<4?

Result
Proved(see note)
Status
Partial result
AI contribution
AI-discovered
Method
Argument
Field
Statistical mechanics; random-cluster model, SLE and CLE scaling limits
Posed by
Steffen Rohde and Oded Schramm, Basic properties of SLE, Ann. of Math. 161 (2005), Conjecture 9.7; Schramm, ICM 2006 (2007), Problem 2.6
Year posed
2005
Years open
21y
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
52 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

For every fixed 1≤q<41\le q<4 (September 23), and 1≤q≤41\le q\le4 (October 5 companion), critical square-lattice FK Dobrushin interfaces converge to chordal SLEκ(q)\mathrm{SLE}_{\kappa(q)} in every bounded Jordan domain with uniformly approximated marks, and the complete nested loop collection of the free infinite-volume measure converges to whole-plane CLEκ(q)\mathrm{CLE}_{\kappa(q)}, keeping multiplicities and traversals. For 0<q<10<q<1 the interface converges to SLEκ\mathrm{SLE}_\kappa, κ∈(6,8)\kappa\in(6,8), in smooth Jordan domains only. No uniformity in qq is asserted. Not covered: 0<q<10<q<1 in nonsmooth domains such as the square, CLE for q<1q<1, and other lattices.

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. Two September 23 manuscripts cover 1≤q<41\le q<4 and 0<q<10<q<1; October 5 companions extend to q=4q=4 and add disordered, thermal (massive) and natural-parametrization results.

Verification

No independent mathematician has checked this yet. Checked here: Theorem 1.1 of the September 23 manuscript was read against Rohde-Schramm Conjecture 9.7 as the papers cite it. For 1≤q<41\le q<4 it claims convergence in every bounded Jordan domain under uniform marked-boundary approximation (including the square), plus nested whole-plane CLE convergence. The companion for 0<q<10<q<1 proves the smooth-Jordan-domain form and states that Rohde-Schramm's unit-square formulation has different boundary regularity and is not covered; that is why this entry is Partial. The proofs were not refereed. They use recent inputs including Duminil-Copin-Kozlowski-Krachun-Manolescu-Oulamara rotational invariance (cited as 2026). No Lean formalization exists for this family.

Sources

Changelog1 change

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