Criticality and conformally invariant scaling limits of the planar random-cluster model for $0<q<1$ and for $1<q\le4$, $q\ne2$

The Fortuin–Kasteleyn random-cluster model on the square lattice — the dependent bond-percolation representation of the Potts model — has, for cluster weight $q\ge1$, a completely settled critical diagram: the critical point is the self-dual value $p_{\rm sd}(q)=\sqrt q/(1+\sqrt q)$ (Beffara and Duminil-Copin, 2012), the transition is continuous for $1\le q\le4$ (Duminil-Copin, Sidoravicius, Tassion) and first order for $q>4$ (Duminil-Copin, Gagnebin, Harel, Manolescu, Tassion), and crossing estimates uniform in boundary conditions hold for $q\in[1,4)$ (Duminil-Copin, Manolescu, Tassion). The unresolved core of the critical-behavior question posed in the 2012 paper is the following. Two components remain open. First, for cluster weight $0<q<1$, where the FKG lattice condition (positive association) fails and with it every monotonicity-based tool, it is not even known where — or whether sharply — the percolative transition occurs: only strict bounds around the self-dual point are known, with the best current results (Beffara, Faipeur, Oke, 2025) merely enlarging the known sub- and supercritical windows. Second, for $1<q\le4$ with $q\ne2$, conformal invariance of the critical scaling limit and its identification with the predicted Schramm–Loewner Evolution processes remain unproved — established only for $q=1$ (percolation) and $q=2$ (Ising).

Background

The Fortuin–Kasteleyn (FK) random-cluster model is a dependent bond-percolation model on a graph $G=(V,E)$: each edge is declared open or closed, and a configuration $\omega$ receives probability proportional to $p^{o(\omega)}(1-p)^{c(\omega)}q^{k(\omega)}$, where $o(\omega)$ and $c(\omega)$ count open and closed edges, $k(\omega)$ counts connected components (clusters), $p\in[0,1]$ is the edge parameter, and $q>0$ is the cluster weight. For $q=1$ this is Bernoulli bond percolation; for integer $q\ge2$ the model is coupled to the $q$-state Potts spin model (Potts spin correlations are FK connection probabilities), so its phase diagram is the Potts phase diagram.

On the square lattice $\mathbb Z^2$, planar duality maps the model with parameters $(p,q)$ to the random-cluster model on the dual lattice with the same $q$ and dual parameter $p^*=q(1-p)/(p+q(1-p))$. The unique self-dual value is $p_{\rm sd}(q)=\sqrt q/(1+\sqrt q)$. A decades-old conjecture, proved by Beffara and Duminil-Copin (2012) for every $q\ge1$, is that this self-dual point is the critical point $p_c$ separating a phase with exponentially decaying connection probabilities from a percolating phase with an infinite cluster.

For $q\ge1$ the model satisfies the FKG lattice condition (positive association: increasing events are non-negatively correlated), which yields stochastic monotonicity in $p$ and underlies essentially all planar technology: Russo–Seymour–Welsh (RSW) type crossing estimates — scale-uniform lower bounds on crossing probabilities of rectangles at criticality, the workhorse behind critical exponents and scaling limits — sharp-threshold arguments, and non-coexistence arguments. Beffara and Duminil-Copin ended their paper by posing the general critical-behavior question: they conjectured a second-order (continuous) transition for $q\in(0,4)$, with the scaling limit believed conformally invariant, and a first-order (discontinuous) transition for $q>4$, calling a proof for every $q$ a major open problem; they noted their method relies on positive association and does not extend to $q<1$; and they flagged the absence of crossing estimates uniform in boundary conditions, expected true for $q\in[1,4)$ and false for $q\ge4$.

Most of this program was completed in the following decade. A first-order transition means multiple infinite-volume Gibbs states coexist at criticality (an order-parameter jump); second-order/continuous means a unique critical Gibbs state and no infinite cluster. Duminil-Copin, Sidoravicius and Tassion proved continuity for $1\le q\le4$ (unique critical Gibbs state, no infinite cluster under wired boundary conditions, subexponential decay of the free two-point function, together with RSW-type crossing estimates valid under arbitrary boundary conditions), while Duminil-Copin, Gagnebin, Harel, Manolescu and Tassion proved the transition is first order for $q>4$ (multiple critical Gibbs states, ordering under wired or monochromatic boundary conditions, exponential decay under free boundary conditions, correlation lengths behaving like $\exp(\pi^2/\sqrt{q-4})$ as $q\downarrow4$ — confirming that uniform boundary-condition estimates fail for $q>4$, as anticipated). Sharpness for $q\ge1$, boundary-condition-uniform quad crossings (crossing estimates for quads, i.e. topological quadrilaterals with four marked boundary arcs, whose value depends only on the quad's conformal modulus) for $q\in[1,4)$, and an independent proof of continuity via six-vertex delocalisation round out the settled $q\ge1$ picture.

Conformal invariance of the critical scaling limit is a separate, mostly open chapter. The Schramm–Loewner Evolution ${\rm SLE}_\kappa$ is the canonical one-parameter family of random conformally invariant planar curves; a conformal loop ensemble (CLE) is the corresponding loop version. Lattice interface convergence to ${\rm SLE}_\kappa$ is proved at $q=1$ (percolation interfaces, ${\rm SLE}_6$, via Cardy's formula, the closed-form conformally invariant crossing probability predicted by conformal field theory) and $q=2$ (FK-Ising interfaces, ${\rm SLE}_{16/3}$; full scaling limit a branching SLE / ${\rm CLE}_3$). For $q=3$, $q=4$, and non-integer $q\in(1,4)$ no conformal-invariance result exists, although the predicted values are explicit: FK interfaces should converge to ${\rm SLE}_\kappa$ with $\sqrt q=-2\cos(4\pi/\kappa)$, giving $\kappa=6,16/3,24/5,4$ at $q=1,2,3,4$ (with logarithmic corrections expected at the borderline $q=4$).

For $0<q<1$ the situation is qualitatively harder: FKG fails (the model is conjectured to be negatively dependent, but even pairwise negative correlation between edges is open on general graphs), monotonicity in $p$ is unavailable, and duality preserves $q$ so the self-dual point is still defined but nothing pins the transition to it. Stochastic comparison with Bernoulli percolation classically gives subcriticality for $p\le q/(1+q)$ and supercriticality for $p\ge1/2$; recent work extends both windows but leaves open the interval containing $p_{\rm sd}(q)$, which lies strictly between $q/(1+q)$ and $1/2$ for every $0<q<1$.

Problem Statement

Complete the critical theory of the random-cluster model on the square lattice $\mathbb Z^2$ with cluster weight $q>0$, beyond the settled $q\ge1$ phase diagram, by resolving the two remaining components of the critical-behavior question posed by Beffara and Duminil-Copin (2012).

Component A (the non-FKG regime $0<q<1$). For the random-cluster measure on $\mathbb Z^2$ with cluster weight $0<q<1$ — where the FKG lattice condition and monotonicity in $p$ fail — determine the percolative phase structure: prove or disprove that the transition occurs at the self-dual point $p_{\rm sd}(q)=\sqrt q/(1+\sqrt q)$, in the sense that (i) connection probabilities decay (exponentially, as for $q\ge1$) for $p<p_{\rm sd}(q)$, and (ii) an infinite cluster exists for $p>p_{\rm sd}(q)$ in one or in every infinite-volume random-cluster measure (a DLR measure: a probability measure on edge configurations whose finite-volume conditional distributions are the random-cluster specifications); determine the order of the transition at $p_{\rm sd}(q)$, conjectured to be second order (continuous: unique infinite-volume measure and no infinite cluster at criticality), consistent with the conjectured negative-dependence structure of the model; and, in this conjectured second-order regime, establish conformal invariance of the critical scaling limit, which the source conjectures for every $q\in(0,4)$ and hence also for $0<q<1$. Currently only the windows $p\le q/(1+q)$ (subcritical) and $p\ge1/2$ (supercritical) are classified, with recent extensions, and the interval containing $p_{\rm sd}(q)$ is open. A refutation — e.g. a proof that for some $0<q<1$ the transition is not at the self-dual point, or is first order, or does not exist — equally answers the component (in which case the conjectured conformally invariant scaling limit does not arise).

Component B (conformal invariance of scaling limits for $1<q\le4$, $q\ne2$). For cluster weight $q\in(1,4)$ with $q\ne2$, and at the borderline $q=4$ where logarithmic corrections are expected, prove conformal invariance of the critical model at $p=p_{\rm sd}(q)$ and identify the scaling limit, which comprises: that critical exploration-process interfaces (the random curves separating open from closed clusters, revealed progressively from the domain boundary) in Jordan domains converge in law, under conformal maps, to chordal ${\rm SLE}_\kappa$ with the predicted parameter $\sqrt q=-2\cos(4\pi/\kappa)$ (so $q=3$ corresponds to ${\rm SLE}_{24/5}$, and $q=4$ to ${\rm SLE}_4$); that quad crossing probabilities (for a quad, i.e. a topological quadrilateral with four marked boundary arcs, the probability that opposite arcs are connected through open clusters) converge to their predicted conformally invariant limits; and that the full interface ensemble is described by a conformal loop ensemble — as has been established for $q=1$ (percolation) and $q=2$ (Ising).

The common objective is the same one stated by the source — the behavior at criticality in the general case, at the level of completeness achieved for $q\ge1$ and $q\le2$ — and the two components are exactly its unresolved parts: component A extends the order-of-transition classification and the conjectured conformally invariant scaling limit to the cluster weights where positive association fails, and component B extends the conformal-invariance picture from $q\in\{1,2\}$ to the remaining critical weights. The problem is fully solved when both components are resolved (in the proving or refuting direction) for their stated ranges of $q$.

References

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