Fortuin–Kasteleyn SLE scaling-limit conjecture

About 19 years old · traced to 2007

Let $q\in[0,4]$ , and consider the Fortuin–Kasteleyn random-cluster model on the square lattice at the self-dual value $p_{\mathrm{sd}}=\sqrt q/(\sqrt q+1)$ , with Dobrushin boundary conditions producing an interface from $a$ to $b$ . Fortuin–Kasteleyn SLE scaling-limit conjecture. As the lattice step goes to zero, the law of the interface converges to Schramm–Loewner Evolution with

\kappa=\frac{4\pi}{\arccos(-\sqrt q/2)}.

This predicts conformally invariant scaling limits for the FK interfaces. The source presents the claim as conjectural, and its general validity across $q\in[0,4]$ remains open.

References

Primary source

Stanislav Smirnov, “Towards conformal invariance of 2D lattice models”, arXiv:0708.0032 (2007).

conformal-invariance random-cluster-models schramm-loewner-evolution

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RemarkAI-assistedClaimed by OpenAI. Claims chordal SLE interfaces and nested whole-plane CLE loop limits for critical square-lattice random-cluster models at every fixed 1<=q<4, with kappa=4pi/arccos(-sqrt(q)/2). Interface convergence holds in bounded marked Jordan domains with uniform boundary approximation; loop convergence uses the specified free plane law. The q=4 endpoint and q<1 are outside this manuscript’s theorem.See full solutionHide full solution

Claimed by OpenAI. Claims chordal SLE interfaces and nested whole-plane CLE loop limits for critical square-lattice random-cluster models at every fixed 1<=q<4, with kappa=4pi/arccos(-sqrt(q)/2). Interface convergence holds in bounded marked Jordan domains with uniform boundary approximation; loop convergence uses the specified free plane law. The q=4 endpoint and q<1 are outside this manuscript’s theorem.

The source covers only its stated fixed q range and domain approximation hypotheses; it does not address q=0. Companion CLE loop conclusions go beyond the interface assertion.

GitHub repository: https://github.com/openai/math

Manuscript: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Square-lattice-FK-interfaces-and-nested-loops-for-1-leq-q-lt-4-September-23-2026/paper.pdf

OpenAI-223-01-Square-lattice-FK-interfaces-and-nested-loops-for-1-q-4.pdf1,002,139 bytesOpen

RemarkAI-assistedClaimed by OpenAI. Claims full-sequence chordal SLE convergence of square-lattice random-cluster Dobrushin interfaces at the self-dual parameter for every fixed 0<q<1, with kappa=4pi/arccos(-sqrt(q)/2) in (6,8). Lattice polygons and their distinct marks must uniformly approximate the marked boundary of a bounded smooth Jordan domain.See full solutionHide full solution

Claimed by OpenAI. Claims full-sequence chordal SLE convergence of square-lattice random-cluster Dobrushin interfaces at the self-dual parameter for every fixed 0<q<1, with kappa=4pi/arccos(-sqrt(q)/2) in (6,8). Lattice polygons and their distinct marks must uniformly approximate the marked boundary of a bounded smooth Jordan domain.

The source covers only its stated fixed q range and domain approximation hypotheses; it does not address q=0. Companion CLE loop conclusions go beyond the interface assertion.

GitHub repository: https://github.com/openai/math

Manuscript: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Self-dual-random-cluster-interfaces-below-one-September-23-2026/paper.pdf

OpenAI-223-02-Self-dual-random-cluster-interfaces-below-one.pdf1,569,976 bytesOpen

RemarkAI-assistedClaimed by OpenAI. Related quenched-disorder result: claims to prove that critical FK–Ising interfaces with sufficiently weak, symmetric, independent two-valued bond disorder converge to chordal SLE16/3 . The disorder strength is fixed as the mesh tends to zero, and convergence of the conditional curve laws holds in probability over the environment.See full solutionHide full solution

Claimed by OpenAI. Related quenched-disorder result: claims to prove that critical FK–Ising interfaces with sufficiently weak, symmetric, independent two-valued bond disorder converge to chordal SLE16/3 . The disorder strength is fixed as the mesh tends to zero, and convergence of the conditional curve laws holds in probability over the environment.

This is a weak quenched random-bond FK-Ising variant, related progress rather than the homogeneous model of the target.

GitHub repository: https://github.com/openai/math

Manuscript: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Quenched-SLE-Universality-for-Weakly-Disordered-FK-Ising-Interfaces-October-5-2026/quenched-fk-ising.pdf

OpenAI-223-03-Quenched-SLE-Universality-for-Weakly-Disordered-FK-Ising-Interfaces.pdf583,188 bytesOpen

RemarkAI-assistedClaimed by OpenAI. Related massive-limit result: claims to prove convergence of thermal FK–Ising interfaces, for every fixed nonzero mass of either sign, on uniformly angle-bounded isoradial lattices in bounded simply connected domains. The limit is independent of the lattice and the admissible domain approximation.See full solutionHide full solution

Claimed by OpenAI. Related massive-limit result: claims to prove convergence of thermal FK–Ising interfaces, for every fixed nonzero mass of either sign, on uniformly angle-bounded isoradial lattices in bounded simply connected domains. The limit is independent of the lattice and the admissible domain approximation.

This is a massive thermal FK-Ising limit on isoradial lattices, related progress rather than the zero-mass critical square-lattice SLE law in the target.

GitHub repository: https://github.com/openai/math

Manuscript: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Thermal-FK-Ising-interfaces-and-massive-SLE-October-5-2026/paper.pdf

OpenAI-223-04-Thermal-FK-Ising-interfaces-and-massive-SLE.pdf655,222 bytesOpen

RemarkAI-assistedClaimed by OpenAI. Related natural-parametrization result: claims to prove that the rescaled counting measure of a critical square-lattice Fortuin–Kasteleyn Dobrushin interface in the unit square converges to the Minkowski-content measure of its Schramm–Loewner limit, for every fixed cluster weight 1 ≤ q < 4. A single deterministic constant times the predicted power of the mesh gives the normalization.See full solutionHide full solution

Claimed by OpenAI. Related natural-parametrization result: claims to prove that the rescaled counting measure of a critical square-lattice Fortuin–Kasteleyn Dobrushin interface in the unit square converges to the Minkowski-content measure of its Schramm–Loewner limit, for every fixed cluster weight 1 ≤ q < 4. A single deterministic constant times the predicted power of the mesh gives the normalization.

This claims natural interface measure convergence for fixed 1<=q<4 in the unit square; the target asks the unparametrized critical interface law on the wider q range.

GitHub repository: https://github.com/openai/math

Manuscript: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Natural-Occupation-Measures-for-Critical-Square-Lattice-FK-Interfaces-October-5-2026/natural-occupation-measures-critical-square-lattice-fk-interfaces.pdf

OpenAI-223-05-Natural-Occupation-Measures-for-Critical-Square-Lattice-FK-Interfaces.pdf703,577 bytesOpen

RemarkAI-assistedClaimed by OpenAI. Claims full-sequence chordal SLE interfaces and nested whole-plane CLE loops for critical square-lattice random-cluster models at every fixed 1<=q<=4, with kappa=4pi/arccos(-sqrt(q)/2). Interfaces use uniformly approximated marked Jordan domains; loops use the specified free plane law and retain multiplicities and traversals. At q=4 the limits are SLE4 and nested CLE4.See full solutionHide full solution

Claimed by OpenAI. Claims full-sequence chordal SLE interfaces and nested whole-plane CLE loops for critical square-lattice random-cluster models at every fixed 1<=q<=4, with kappa=4pi/arccos(-sqrt(q)/2). Interfaces use uniformly approximated marked Jordan domains; loops use the specified free plane law and retain multiplicities and traversals. At q=4 the limits are SLE4 and nested CLE4.

The source covers only its stated fixed q range and domain approximation hypotheses; it does not address q=0. Companion CLE loop conclusions go beyond the interface assertion.

GitHub repository: https://github.com/openai/math

Manuscript: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Conformal-Limits-of-Critical-Square-Lattice-Random-Cluster-Interfaces-October-5-2026/paper.pdf

OpenAI-223-06-Conformal-Limits-of-Critical-Square-Lattice-Random-Cluster-Interfaces.pdf1,092,659 bytesOpen