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

1 · The essentials

These manuscripts study random paths drawn on a square grid and ask what happens as the grid becomes finer and finer. Their central claim is that, under specified boundary conditions, the paths approach a particular family of continuous random curves. A parameter called $q$ , which controls the weight given to clusters, determines which curve appears. Together, the manuscripts cover $0 < q \leq 4$ , with smoother boundaries required when $q < 1$ . For $1 \leq q \leq 4$ , they also describe the limiting collection of nested loops, rather than just one path. Companion manuscripts examine weak randomness in the bonds, a thermal change to the model, and how the number of path steps scales. These are results shown in the manuscripts; the main result is not formalized in Lean and has not been formally verified.

A picture to help (an imperfect image, not to be taken literally)

As an imperfect image, imagine drawing a winding route on finer and finer graph paper and then viewing each drawing from the same distance. The question is whether the random routes settle into a consistent overall pattern, and whether counting their tiny steps also has a consistent limit after rescaling.

Why it matters

The family connects discrete random-cluster interfaces with precisely specified continuous random curves and loop collections. It also distinguishes convergence of a path's shape from convergence of its rescaled step-counting measure. The disorder and thermal companions examine related limits under specific changes to the FK–Ising model, rather than asserting universality for arbitrary changes.

2 · Going further

The central problem concerns the scaling limit of critical square-lattice random-cluster Dobrushin interfaces. Manuscript 1 shows in the manuscript that, for each fixed $1 \leq q < 4$ , these ordered curves converge to chordal $SLE_{κ (q)}$ , with $κ (q) = 4 π / arccos (- q /2)$ , in every bounded Jordan domain under uniform marked boundary approximation. Manuscript 6 includes the endpoint $q = 4$ . Both also describe convergence of complete nested plane loop collections to whole-plane $CLE_{κ (q)}$ , retaining loop multiplicities and traversals; Manuscript 1 specifies a spherical matching topology. At $q = 1$ , they obtain Cardy's formula for square-lattice bond percolation with free boundary edges. Manuscript 1 describes its interface result as confirming the Rohde–Schramm prediction in its stated parameter range.

Manuscript 2 treats each fixed $0 < q < 1$ at the self-dual parameter, where $κ \in (6, 8)$ . Here the approximating domains must be simple closed nearest-neighbor lattice polygons with distinct marked vertices, and their marked boundary parametrizations must converge uniformly to those of a bounded smooth Jordan domain. Convergence holds along the full mesh sequence, in the uniform metric on oriented curves modulo increasing reparametrization. Its abstract describes an argument combining finite connection comparisons below one, localization in irregular tiled disks, and a boundary observable that determines the limiting Loewner driver. The abstracts of Manuscripts 1 and 6 do not give a corresponding account of their argument.

Two companions concern FK–Ising interfaces. Manuscript 3 shows convergence to chordal $SLE_{16/3}$ for sufficiently weak, symmetric, independent two-valued bond disorder. The disorder strength stays fixed as the mesh tends to zero, and the conditional curve laws converge in probability over the environment. Manuscript 4 treats thermal interfaces with any fixed nonzero mass of either sign on uniformly angle-bounded isoradial lattices in bounded simply connected domains. For positive mass, its limit is the unique massive $SLE_{16/3}$ law with locally finite-energy drift prescribed by a massive boundary value problem; negative mass is handled by duality and reversal. The stated limit is independent of the lattice and admissible approximation, including the allowed Carathéodory approximations whose diameters diverge.

Manuscript 5 adds an occupation-measure limit for critical square-lattice interfaces in the unit square, for each fixed $1 \leq q < 4$ . The rescaled counting measure converges jointly with the ordered curve to the Minkowski-content measure of its Schramm–Loewner limit. The normalization is a single deterministic constant times the predicted power of the mesh, and convergence includes total mass, hence the rescaled total number of interface steps. This supplements the curve limits with a natural measure along the limiting path.

Useful notions

Random-cluster models and cluster weight
Scaling limits as lattice mesh tends to zero
Marked domains and boundary approximation
Convergence of random curves and measures
Schramm–Loewner evolution

What was known before / what is new

Before

The source identifies a Rohde–Schramm prediction for the interface limit, which Manuscript 1 says it confirms for $1 \leq q < 4$ . It also refers to a predicted mesh power for the occupation-measure normalization. The supplied abstracts do not give a broader account of previously established parameter ranges or methods.

New

The manuscripts show critical square-lattice interface limits across $0 < q \leq 4$ , with boundary hypotheses that differ below one, and nested plane loop limits for $1 \leq q \leq 4$ . Manuscript 6 includes $q = 4$ , beyond the range stated in Manuscript 1. The companions add the specified weak-disorder and thermal FK–Ising limits, together with joint curve and occupation-measure convergence for $1 \leq q < 4$ in the unit square.

Limits and assumptions

The critical results fix q before taking the mesh limit: bounded Jordan domains with uniform marked boundary approximation are allowed for $1 \leq q \leq 4$ , while the below-one manuscript requires bounded smooth Jordan domains and its specified lattice-polygon approximations. The loop results are stated only for $1 \leq q \leq 4$ , and the occupation-measure result only for $1 \leq q < 4$ in the unit square. The disorder result requires sufficiently weak, symmetric, independent two-valued bond disorder; the thermal result requires fixed nonzero mass and uniformly angle-bounded isoradial lattices with admissible domain approximations. The source does not identify a list of remaining open problems, and no extension beyond these hypotheses is asserted here. The main result is not formalized in Lean and has not been formally verified.

Mini glossary

Scaling limit: A limiting random curve, loop collection, or measure obtained as the lattice mesh tends to zero, with rescaling where specified.
Cluster weight: The parameter q of the random-cluster model that determines the limiting SLE parameter through the stated formula.
Chordal SLE: The family of continuous random interface laws, indexed by κ, appearing as the marked-domain curve limits in these manuscripts.
Whole-plane CLE: The random loop-collection law appearing as the limit of the complete nested plane loops.
Quenched convergence: Here, convergence of curve laws conditional on the bond environment, holding in probability over that environment.
Minkowski-content measure: The measure along the Schramm–Loewner limiting curve identified in Manuscript 5 as the limit of rescaled interface step counts.

Keywords : Random-cluster interfaces · Schramm–Loewner evolution · Scaling limits

AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.

3 · Technical details and sources

Claim made in the manuscript (source text)

05/10/2026

To be verified

For every fixed $1 \leq q \leq 4$ , critical square-lattice random-cluster Dobrushin interfaces converge to chordal $SLE_{κ (q)}$ , where $κ (q) = 4 π / arccos (- q /2)$ , and the complete nested plane loop collections converge to whole-plane $CLE_{κ (q)}$ . The results hold under uniform marked Jordan boundary approximation and retain loop multiplicities and traversals. At q = 1 we obtain Cardy's formula for square-lattice bond percolation with free boundary edges.

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