Random-cluster curves: conformal limits and natural time

Source and verification

This article explains family 223 of the October 6, 2026 OpenAI collection. Research conclusions are attributed to its manuscripts. The review covers statements, introductions, and selected proof mechanisms; it is not an independent verification of the complete proofs.

Overview

Random-cluster interfaces are boundaries between primal and dual connections. Family 223 claims conformal limits across cluster weights, then studies disorder, thermal perturbations, and natural parametrization. A curve’s trace, its ordered traversal, its surrounding loop ensemble, and its occupation measure carry different information.

🏷️ Preliminaries: random clusters and Loewner curves

On a finite graph, an open-edge set $\omega$ has FK weight proportional to

$$p^{|\omega |}(1-p{)}^{|E|-|\omega |}{q}^{k(\omega )},$$

where $k(\omega )$ counts clusters with the prescribed boundary identifications. The square-lattice self-dual parameter is $p=\sqrt{q}/(1+\sqrt{q})$. For $q=1$ this is independent bond percolation; for $q=2$ it is FK–Ising.

Wiring one boundary arc together and leaving the complementary arc free produces a distinguished medial-lattice exploration curve. The curve separates an open primal region from an open dual region. It is different from the boundary of an Ising spin cluster.

Chordal ${\mathrm{SLE}}_{\kappa }$ is generated by

$${\partial }_t{g}_t(z)=\frac{2}{g_t(z)-\sqrt{\kappa }{B}_t}.$$

A conformal loop ensemble, CLE, describes a collection of conformally invariant loops; its nested version retains successive generations inside other loops. Convergence as ordered curves modulo increasing reparametrization is stronger than convergence of the visited sets.

🏷️ The parameter dictionary and the main companions

The critical relation used in the family is

$$\begin{array}{c}\sqrt{q}=-2\cos (4\pi /\kappa ),\\ \kappa =\frac{4\pi }{\arccos (-\sqrt{q}/2)}.\end{array}$$

Thus $q=1$ gives $\kappa =6$, $q=2$ gives $16/3$, and $q=4$ gives $4$. As $q\downarrow 0$, $\kappa \uparrow 8$.

The main square-lattice companions claim interface and nested-loop convergence for $1\le q<4$, with a further companion including $q=4$ under its stated marked-domain approximations. (OpenAI, 2026) (OpenAI, 2026a) The below-one paper treats $0<q<1$ in its smooth-domain setting. Its proof cannot assume the positive association available for $q\ge 1$. (OpenAI, 2026b)

A disorder companion claims quenched ${\mathrm{SLE}}_{16/3}$ universality for weak symmetric two-valued random FK–Ising bonds. The conditional curve law converges in probability over the environment. (OpenAI, 2026b) A thermal companion treats every fixed nonzero mass of either sign on uniformly angle-bounded isoradial lattices. Positive mass leads to a massive ${\mathrm{SLE}}_{16/3}$ with drift specified by a massive boundary value problem; negative mass is handled by duality and reversal. This is a deformation of critical SLE, not ordinary drift-free SLE. (OpenAI, 2026e)

The occupation companion retains the time spent along the curve for $1\le q<4$. It claims convergence of the renormalized counting measure to a deterministic multiple of natural SLE occupation measure. Its stated range does not include the endpoint $q=4$. (OpenAI, 2026b)

🏷️ A worked distinction: curve length versus capacity time

For $\kappa <8$, the SLE dimension is $d=1+\kappa /8$. A lattice path of mesh $\delta$ with approximately ${\delta }^{-d}$ edges has divergent ordinary Euclidean length $\delta \cdot {\delta }^{-d}$. The proposed natural measure therefore assigns mass of order ${\delta }^d$ to each visited step.

In the continuum, a $d$-dimensional Minkowski-content normalization examines

$${\epsilon }^{d-2}\mathrm{Area}\{z:\mathrm{dist}(z,\gamma )<\epsilon \}$$

as $\epsilon \downarrow 0$, locally along the curve. A limiting content requires much more than knowing the dimension: dimension only determines a power exponent and permits oscillating multiplicative factors. Likewise, convergence of unparametrized paths gives no automatic convergence of edge counts. The occupation theorem needs separate one- and two-point control and uniform integrability.

🏷️ Crossing observables identify the driver

The critical proof starts with a four-boundary-change problem. It computes an alternating-arc connection probability as a function of the conformal cross-ratio. A finite normalization matters: an auxiliary law with four unmatched interface ends is different from the law obtained by joining designated ends with exterior wires. If the former connection probability is $f$ and loop weight is $d_0=\sqrt{q}$, the latter is

$$\frac{f}{f+d_0(1-f)}.$$

The extra cluster weight explains the denominator. Forgetting it would change the martingale used to identify the continuum curve.

During exploration, the conditional crossing probability is a martingale. Once the source proves its conformal limit in the remaining slit domain, Itô’s formula for the Loewner map identifies the driver’s quadratic variation and drift. Uniform crossing and multiple-passage bounds provide compactness and rule out hidden retracing. Iterating explorations identifies nested loops, while retaining their multiplicities and traversal order.

Below one, the manuscript develops different finite conditional comparisons before transporting the observable; it does not reuse ferromagnetic monotonicity outside its range. In the disorder extension, boundary cells require separate transfer estimates because bulk symmetry no longer cancels their errors.

For massive SLE, the observable limit alone does not establish a unique curve law. The additional task is to solve and identify the stochastic driving equation with its boundary-problem drift. For natural occupation, yet another task is to upgrade the already identified curve law to joint convergence with a measure. These dependencies explain why the six companions remain separate within one family article.

🏷️ Key steps and proof outlines

Two martingale tests determine both drift and variance

The critical proof turns a small wired interval near a real point $x$ into a bounded test of the Loewner driver. Fix a capacity horizon and a driver bound, and choose $x$ beyond that bound. After taking the mesh limit with the interval size fixed, the small-interval asymptotic of the four-change observable gives

$$M_t(x)={\left(\frac{x{g}_t^{\prime }(x)}{g_t(x)-W_t}\right)}^h,h=\frac{8}{\kappa }-1>0.$$

The process is bounded above and away from zero under this localization. One first replaces the normalized crossing probability by this bounded expression, then transfers between the auxiliary and original exploration laws. This order avoids dividing a coupling error by a crossing probability that tends to zero.

Why use two points $x_1<x_2$? Freeze the conformal-map variables at a stopping time and write $F_j(w)=A_j(X_j-w{)}^{-h}$. The ratio $F_1^{\prime }(w)/F_2^{\prime }(w)$ is strictly increasing, uniformly on the stopped range. Choosing $a=F_1^{\prime }(W)/F_2^{\prime }(W)$ cancels the linear term and gives

$$F_1(w)-F_1(W)-a(F_2(w)-F_2(W))\ge c(w-W{)}^2.$$

Approximate martingale identities, valid at the same stopping rules, then bound driver increments by elapsed capacity time. Together with localization this provides the tightness needed to pass to limiting martingales.

In the limit put $X=g_t(x)$ and $Z=X-W$. The martingale identity also expresses $W$ as a smooth function of a positive martingale and finite-variation variables, making it a continuous semimartingale. Write $B^{\mathrm{fv}}$ for its finite-variation part. Since

$$dX=2Z^{-1}dt,d\log (x^h{g}_t^{\prime }(x{)}^h)=-2h{Z}^{-2}dt,$$

Itô’s formula and the vanishing drift of $M$ give

$$(X-W)d{B}^{\mathrm{fv}}-4dt+\frac{1+h}{2}d[W]=0.$$

Subtract this identity for the two separated points. Their gap is positive, so $d{B}^{\mathrm{fv}}=0$. Substitution yields $d[W]=8(1+h{)}^{-1}dt=\kappa dt$. Lévy’s characterization identifies the Brownian driver. Passage estimates and small-gate bounds then exclude macroscopic excursions hidden in zero-capacity intervals, completing convergence as ordered curves. (OpenAI, 2026e)

What changes below one and at the endpoint

For $0<q<1$, the required finite comparisons must be rebuilt without positive association. One technical device is a finite current extraction with common external states. A local difference between two insertion positions is rewritten as

$$E_{\mathrm{out}}^{x}N(x)E_{\mathrm{in}}^x-e^{i\tau (x-y)}{E}_{\mathrm{out}}^{y}N(y)E_{\mathrm{in}}^y.$$

The surrounding tensors are common to the two terms. To achieve this, backward increments are expressed as negatives of forward increments with their common primitive prefix canceled. In a diagonal-time frame with forward steps $a,b$, append two unmarked steps to the two alternatives so both end at $z+2a+2b$; the mixed-direction case uses the common path with steps $b,b,a,a$. Background flux is rerouted with the same phase for every source position. This exact finite identity preserves the signed cancellation through the transfer estimates. A bound on the two terms separately would lose that cancellation. The source uses these finite tools in its smooth-domain proof; the $q\ge 1$ monotone coupling is not imported into this range. (OpenAI, 2026d)

At $q=4$, the endpoint companion uses a conditional GFF identification. It first transfers the field law into every complementary pocket, with protected cuts controlling rough boundaries. After the stated normalization, the first-level loop set has conditional harmonic values $\pm 2{\lambda }_0$; the marked chordal experiment has total harmonic values $\pm {\lambda }_0$.

To apply the imported local-set uniqueness theorem, the proof checks more than these values. Conditional on the set and its harmonic function, the residual field is a Dirichlet GFF in all complementary components. The harmonic function is bounded, and the distributional decomposition has no extra term supported on the set—this is the required thinness. The geometric construction verifies the component and boundary conditions. Uniqueness then identifies the first-level set with the CLE${}_4$ construction or the marked SLE${}_4$ level line. Iterating inside pockets retains labels and nesting. This endpoint argument is separate from simply substituting $q=4$ in a subcritical estimate. (OpenAI, 2026a)

Disorder must preserve tests after revealing the slit

The random FK–Ising companion uses the same two-test driver mechanism, but the tests must remain valid under a filtration revealing both the exploration and its accessible environment. Boundary cells require their own transfer estimates, and stopping rules must be transported consistently between the original and inserted-interval experiments.

The source does this on a common tree of prefix histories: apply the same stopping rule while the histories agree, and stop at the first exit from that common tree otherwise. Extend the history test boundedly outside it. This preserves adaptedness and the order of the two stopping times. Real Loewner trajectories from the inserted interval stay separated from the localized driver, providing the fixed buffer needed for the conditional comparison. The resulting martingale identities identify the driver under typical fixed environments, while additional passage control gives the full oriented curve. Averaging the environment first would discard the quenched conclusion. (OpenAI, 2026c)

Finite drift energy gives uniqueness for massive SLE

The thermal paper obtains a progressively measurable drift $b(t,W)$ from the massive boundary problem. To identify a unique law it proves finite drift energy on finite horizons and applies a stopped change of measure to

$$d{W}_t=\sqrt{\kappa }d{B}_t+b(t,W)dt.$$

For a candidate solution, stop when $A_t={\int }_0^{t}b(s,W{)}^{2}ds$ first reaches $n$, or at the horizon $T$. The exponential with stochastic integral $-{\kappa }^{-1/2}\int bdB$ and compensator $-A/(2\kappa )$ is a true density by the bounded-energy criterion. Under the changed measure the drift-corrected path is Wiener and agrees with the candidate up to the stop.

Nonanticipation makes the first attained energy level the same path functional under both descriptions. The inverse density is therefore one canonical Wiener functional, independent of which candidate solution was chosen. All stopped candidate laws coincide. Since each candidate has $A_T<\infty$, its stops eventually equal $T$, and bounded convergence identifies the full laws. The argument does not assume global drift bounds or finite energy on every arbitrary Wiener path. (OpenAI, 2026f)

The occupation measure requires an $L^2$ replacement

Fix $f\in C_c(D)$ and let $\epsilon \downarrow 0$ through the calibrated radius sequence. Write $d=1+\kappa /8$ and $x=2-d$. A microscopic segment indicator $I_e$ is replaced by the indicator $J_e^{\epsilon }$ that the distinguished strand enters a small disk about it. An unrelated loop is not counted. With the midpoint density constant $A_0$, compare

$$\begin{array}{rl}{X}_n(f)& =A_0{n}^{-d}\sum _{e}f(z_e)I_e,\\ X_{n,\epsilon }(f)& =A_0{n}^{-2}\sum _{e}f(z_e)L_{\epsilon }{J}_e^{\epsilon },L_{\epsilon }\asymp {\epsilon }^{-x}.\end{array}$$

At two probes separated by a fixed distance $g$, annular coupling compares segment and disk starts under every induced exterior partition. Choose an intermediate radius of order ${\epsilon }^b$ in physical units. The error is bounded by a sum of positive powers,

$$\eta (\epsilon )={\epsilon }^{(1-b)c_{\mathrm{out}}-bx}+{\epsilon }^{c_{\mathrm{cal}}}+{\epsilon }^{b{c}_{\mathrm{mix}}}.$$

Choose $b>0$ small enough that the first exponent is positive. Near the diagonal, two-probe arm bounds give a kernel $C|z-z^{\prime }{|}^{-2x}$. Its integral over distances below $g$ is $O(g^{2-2x})$; overlapping disk probes contribute $O({\epsilon }^{2-x})$. Both exponents are positive in the stated range. Expanding the square therefore yields

$$\underset{n}{\mathrm{limsup}}\mathbb{E}|X_n(f)-X_{n,\epsilon }(f){|}^2\le C(g^{2-2x}+{\epsilon }^{2-x})+C_g\eta (\epsilon ).$$

Take $\epsilon \downarrow 0$ first, then $g\downarrow 0$. Fixed-radius disk probes are controlled by curve convergence; their continuum limit is calibrated to Minkowski content. Boundary-strip estimates extend the compactly supported argument to the whole square. This second-moment comparison supplies the missing uniform integrability and identifies the measure jointly with the curve. (OpenAI, 2026b)

See Also

• Family 218 treats Ising spin interfaces with parameter three and explains quenched convergence.
• Family 224 derives planar limits through crossings and pivotal events.
• Family 226 emphasizes why topological loop data do not imply convergence as curves.

🏷️ Manuscripts and verification scope

• Square-lattice FK interfaces and nested loops for 1 ≤ q < 4 — pinned manuscript and companion source.
• Self-dual random-cluster interfaces below one — pinned manuscript and companion source.
• Quenched SLE Universality for Weakly Disordered FK–Ising Interfaces — pinned manuscript and companion source.
• Thermal FK–Ising interfaces and massive SLE — pinned manuscript and companion source.
• Natural Occupation Measures for Critical Square-Lattice FK Interfaces — pinned manuscript and companion source.
• Conformal Limits of Critical Square-Lattice Random-Cluster Interfaces — pinned manuscript and companion source.

No family scope page was present at lean/docs/223.md in the pinned snapshot. This does not establish that all related formalization code is absent. No Lean build, axiom audit, or manuscript-to-formal-statement equivalence check was performed for these notes.

OpenAI collection provides prerequisites and the reading sequence; coverage register records completed and pending families.

🐻  OpenAI 2026a. Conformal Limits of Critical Square-Lattice Random-Cluster Interfaces.

🐻  OpenAI 2026b. Natural Occupation Measures for Critical Square-Lattice FK Interfaces.

🐻  OpenAI 2026c. Quenched SLE Universality for Weakly Disordered FK–Ising Interfaces.

🐻  OpenAI 2026d. Self-dual random-cluster interfaces below one.

🐻  OpenAI 2026e. Square-lattice FK interfaces and nested loops for 1 ≤ q < 4.

🐻  OpenAI 2026f. Thermal FK–Ising interfaces and massive SLE.