# Random-cluster interfaces: critical, disordered, thermal, and natural-time scaling

> Critical random-cluster interfaces on the square grid converge to SLE curves for every cluster weight up to 4. Plain-English explainer with sources.

Canonical: https://closedform.nishachay.tech/e/openai-math-2026-223/
Last updated: 2026-10-14

**Critical random-cluster interfaces on the square grid converge to SLE curves for every cluster weight up to 4.**

**Trust:** Claimed, not checked · **Papers:** 6 · **First paper:** September 23, 2026 · **Field:** Probability and statistical mechanics · **Entry:** 223

*Explained by AI.* Drafted by one AI model and checked sentence by sentence against the paper by a different one. Spot a mistake? Report an error.

The random-cluster model opens each edge of a grid at random, with an extra reward or penalty for each cluster formed. At its critical point, the boundary between clusters is expected to become a random fractal curve called SLE. These six papers claim that limit on the square grid for every cluster weight q from just above 0 up to 4. They also claim limits for all loops, a disordered case, a near-critical case, and the curve's length. No computer proof checker has checked them.

## At a glance

- **First step:** 2005 (Rohde and Schramm, the predicted SLE parameter for random-cluster interfaces)
- **Significance:** Landmark. If they hold, they settle the long-standing conformal-invariance prediction for square-lattice random-cluster interfaces across 0 < q ≤ 4.
- **Trust:** Claimed, not checked
- **Field:** Probability and statistical mechanics (6 papers)

## The explanation

Take a square grid and open each edge with some chance p. Open edges join points into clusters. The random-cluster model adds a twist. Each configuration also gets a factor q for every cluster[1]. With q = 1 this is plain bond percolation. With q = 2 it is tied to the Ising magnet. The critical edge chance is √q / (1 + √q). That is 0.5 for q = 1, and about 0.586 for q = 2.

Now force a boundary condition. Wire one arc of the border into a single cluster and leave the other arc free. A curve then runs between the two points where the arcs meet. It traces the outer edge of the wired cluster. This is the Dobrushin interface. At the critical point it wiggles at every scale.

The prediction is that this curve tends to chordal SLE_κ as the grid shrinks. SLE_κ is a random fractal curve driven by √κ times a Brownian motion[2]. The predicted parameter is κ = 4π / arccos(−√q / 2). It gives κ = 6 for q = 1, 16/3 for q = 2, 4.8 for q = 3, and 4 for q = 4. Earlier proofs covered special cases. Smirnov proved Cardy's crossing formula for site percolation on the triangular lattice[3]. Chelkak, Duminil-Copin, Hongler, Kemppainen and Smirnov proved the q = 2 interface limit.

The papers here claim to close that range. One claims SLE_κ limits for every q from 1 up to, but not including, 4. It allows every bounded Jordan domain. A Jordan domain is a region bounded by one simple closed curve. The same paper claims the full nested family of loops in the plane tends to the conformal loop ensemble CLE_κ[4]. At q = 1 this gives Cardy's formula for square-lattice bond percolation[5]. A second paper covers 0 < q < 1, where κ lies between 6 and 8, but needs a smooth boundary[6]. A further paper covers q = 4 as well[7].

Three more papers go sideways. One adds weak random bond strengths to the q = 2 case, held fixed as the grid shrinks. It claims the SLE_16/3 limit survives, in probability over the environment[8]. Another moves slightly off the critical point, with a fixed mass of either sign. Its limit is a massive SLE_16/3 curve[9]. The last counts the curve's steps. It claims the rescaled count tends to the natural length measure of the SLE limit[10].

The obstacle is that one interface limit is not the whole story. A limit for one curve does not fix how it is coupled to the others. Loop limits on compact sets can also lose multiplicities, nesting or full traversals[11]. The papers track these features explicitly, through local winding identities first studied by Smirnov, Riva and Cardy.

How sure is this? No Lean formalization is listed for these papers in the openai/math release. So no computer proof checker has checked them. No specialist review has been published.

**What it doesn't show.** This does not cover cluster weights above 4, or other lattices for general q.

## Where it came from

This result comes from an unreleased OpenAI model, published in the openai/math release on 6 October 2026.

## Hype check

Landmark only if all six papers hold. No Lean formalization is listed and no specialist review is published.

## Words used

- **Random-cluster model:** A model that opens grid edges at random, weighting each configuration by a factor q per cluster.
- **Cluster weight q:** The factor given for each connected cluster; q = 1 is percolation and q = 2 is linked to the Ising magnet.
- **Dobrushin interface:** The curve forced between two boundary points when one arc of the border is wired together and the other is free.
- **SLE_κ:** Schramm–Loewner evolution: a random fractal curve driven by √κ times a Brownian motion.
- **CLE_κ:** Conformal loop ensemble: the random family of nested loops matching SLE_κ.
- **Jordan domain:** A region in the plane bounded by one simple closed curve.
- **Cardy's formula:** An exact formula for the chance that a critical percolation cluster crosses between two arcs of a shape.

## The original papers (6)

- [Square-lattice FK interfaces and nested loops for 1 <= q < 4](https://github.com/openai/math/blob/main/preprints/Square-lattice-FK-interfaces-and-nested-loops-for-1-leq-q-lt-4-September-23-2026/paper.pdf) - September 23, 2026
- [Self-dual random-cluster interfaces below one](https://github.com/openai/math/blob/main/preprints/Self-dual-random-cluster-interfaces-below-one-September-23-2026/paper.pdf) - September 23, 2026
- [Quenched SLE Universality for Weakly Disordered FK–Ising Interfaces](https://github.com/openai/math/blob/main/preprints/Quenched-SLE-Universality-for-Weakly-Disordered-FK-Ising-Interfaces-October-5-2026/quenched-fk-ising.pdf) - October 5, 2026
- [Thermal FK–Ising interfaces and massive SLE](https://github.com/openai/math/blob/main/preprints/Thermal-FK-Ising-interfaces-and-massive-SLE-October-5-2026/paper.pdf) - October 5, 2026
- [Natural Occupation Measures for Critical Square-Lattice FK Interfaces](https://github.com/openai/math/blob/main/preprints/Natural-Occupation-Measures-for-Critical-Square-Lattice-FK-Interfaces-October-5-2026/natural-occupation-measures-critical-square-lattice-fk-interfaces.pdf) - October 5, 2026
- [Conformal Limits of Critical Square-Lattice Random-Cluster Interfaces](https://github.com/openai/math/blob/main/preprints/Conformal-Limits-of-Critical-Square-Lattice-Random-Cluster-Interfaces-October-5-2026/paper.pdf) - October 5, 2026

## Quoted lines from the paper (11)

[1] From [Conformal Limits of Critical Square-Lattice Random-Cluster Interfaces](https://github.com/openai/math/blob/main/preprints/Conformal-Limits-of-Critical-Square-Lattice-Random-Cluster-Interfaces-October-5-2026/paper.pdf):

> For $A\subseteq E$, let $k_\xi(A)$ be the number of components of $(V,A)$ after identification within each block of $\xi$, including isolated vertices.

[2] From [Conformal Limits of Critical Square-Lattice Random-Cluster Interfaces](https://github.com/openai/math/blob/main/preprints/Conformal-Limits-of-Critical-Square-Lattice-Random-Cluster-Interfaces-October-5-2026/paper.pdf):

> Chordal $\SLE_\kappa$ in $(D;a,b)$ is the conformal image of the upper-half-plane Loewner trace driven by $\sqrt\kappa B_t$ from $0$ to $\infty$, with $B$ a standard Brownian motion.

[3] From [Conformal Limits of Critical Square-Lattice Random-Cluster Interfaces](https://github.com/openai/math/blob/main/preprints/Conformal-Limits-of-Critical-Square-Lattice-Random-Cluster-Interfaces-October-5-2026/paper.pdf):

> Cardy's prediction for critical percolation crossings was proved by Smirnov for site percolation on the triangular lattice.

[4] From [Square-lattice FK interfaces and nested loops for 1 <= q < 4](https://github.com/openai/math/blob/main/preprints/Square-lattice-FK-interfaces-and-nested-loops-for-1-leq-q-lt-4-September-23-2026/paper.pdf):

> For every fixed $1\le q<4$, critical square-lattice random-cluster Dobrushin interfaces converge as ordered curves to chordal $\SLE_{\kappa(q)}$, where $\kappa(q)=4\pi/\arccos(-\sqrt q/2)$, confirming the Rohde--Schramm prediction in this parameter range.

[5] From [Square-lattice FK interfaces and nested loops for 1 <= q < 4](https://github.com/openai/math/blob/main/preprints/Square-lattice-FK-interfaces-and-nested-loops-for-1-leq-q-lt-4-September-23-2026/paper.pdf):

> At $q=1$ we obtain Cardy's formula for square-lattice bond percolation with free boundary edges.

[6] From [Self-dual random-cluster interfaces below one](https://github.com/openai/math/blob/main/preprints/Self-dual-random-cluster-interfaces-below-one-September-23-2026/paper.pdf):

> Thus $\pi/3<\lambda<\pi/2$, $1/3<\rho<1/2$, and $6<\kappa<8$.

[7] From [Conformal Limits of Critical Square-Lattice Random-Cluster Interfaces](https://github.com/openai/math/blob/main/preprints/Conformal-Limits-of-Critical-Square-Lattice-Random-Cluster-Interfaces-October-5-2026/paper.pdf):

> Fix a real number $q\in[1,4]$ and put \begin{equation} p_q=\frac{\sqrt q}{1+\sqrt q}, \qquad \kappa(q)=\frac{4\pi}{\arccos(-\sqrt q/2)}. \end{equation}

[8] From [Quenched SLE Universality for Weakly Disordered FK–Ising Interfaces](https://github.com/openai/math/blob/main/preprints/Quenched-SLE-Universality-for-Weakly-Disordered-FK-Ising-Interfaces-October-5-2026/quenched-fk-ising.pdf):

> The disorder strength is fixed as the mesh tends to zero, and convergence of the conditional curve laws holds in probability over the environment.

[9] From [Thermal FK–Ising interfaces and massive SLE](https://github.com/openai/math/blob/main/preprints/Thermal-FK-Ising-interfaces-and-massive-SLE-October-5-2026/paper.pdf):

> For positive mass it is the unique massive SLE$_{16/3}$ law with locally finite-energy drift prescribed by a massive boundary value problem; negative mass follows by duality and reversal.

[10] From [Natural Occupation Measures for Critical Square-Lattice FK Interfaces](https://github.com/openai/math/blob/main/preprints/Natural-Occupation-Measures-for-Critical-Square-Lattice-FK-Interfaces-October-5-2026/natural-occupation-measures-critical-square-lattice-fk-interfaces.pdf):

> Convergence is joint with the ordered curve and includes the total mass, giving the scaling limit of the interface's total number of steps.

[11] From [Conformal Limits of Critical Square-Lattice Random-Cluster Interfaces](https://github.com/openai/math/blob/main/preprints/Conformal-Limits-of-Critical-Square-Lattice-Random-Cluster-Interfaces-October-5-2026/paper.pdf):

> These are distinct conclusions: a limit for one interface does not by itself identify its coupling with other interfaces, and convergence of loop traces on compact sets does not by itself retain multiplicities, nesting, or complete traversals.

## Other sources

- [openai/math release notes (2026): model provenance and review status](https://github.com/openai/math/blob/main/README.md): “manuscripts and supporting proof artifacts produced by an internal OpenAI model”
- [S. Rohde and O. Schramm, “Basic properties of SLE”, Annals of Mathematics 161 (2005), 883–924](https://doi.org/10.4007/annals.2005.161.883)

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