Geometry and statistics of simultaneous-arrival clouds in the monotone coupling of the planar FK-Ising model

The planar FK-Ising model (two-dimensional random-cluster model with cluster-weight $q=2$) has a near-critical regime qualitatively different from Bernoulli percolation: as the edge-parameter $p$ is raised through the critical point $p_c(2)$, new edges do not open one by one but in simultaneous groups called clouds, a self-organized mechanism discovered by Duminil-Copin, Garban and Pete (2014), who proved that non-trivial clouds occur with probability tending to one in large boxes but left their structure quantitatively unexplained. This problem asks for a quantitative geometric and statistical theory of these clouds by resolving the four questions posed in that paper: (1) whether, in large boxes with wired boundary conditions, macroscopic clouds (diameter a positive fraction of the box size) occur with uniformly positive probability and have labels concentrating near $p_c$; (2) whether clouds whose labels lie away from $p_c$ are only of logarithmic size; (3) whether non-trivial clouds exist almost surely in the infinite-volume coupling; and (4) whether, for cluster-weights $q\le 4$, all clouds are almost surely finite. The companion analytic questions of the same paper — sharp bounds on edge influences for torus crossing events, and a near-critical scaling theory relating specific heat, influences and hyperscaling — were largely resolved by Duminil-Copin and Manolescu (2022) and by later computation of the mixing-rate exponent, work which explicitly leaves the cloud structure open.

Background

The model. The random-cluster (Fortuin–Kasteleyn, FK) model on a finite subgraph $G$ of the square lattice $\mathbb{Z}^2$ assigns to each configuration $\omega$ of open edges a probability proportional to $(p/(1-p))^{|\omega|}q^{k(\omega)}$, where $|\omega|$ is the number of open edges and $k(\omega)$ the number of connected components (clusters); boundary conditions specify how boundary vertices are wired together outside $G$ ('wired' means all boundary vertices form one component). For cluster-weight $q=2$ the model is coupled to the Ising model (whence the name FK-Ising) and is exactly solvable at the critical point $p_c(2)=\sqrt{2}/(1+\sqrt{2})$, which is self-dual: the square-lattice model is mapped to itself by the Kramers–Wannier duality exchanging high and low edge-parameters, and this fixes the critical point at the self-dual value.

Near-criticality for Bernoulli percolation. For Bernoulli percolation ($q=1$), Kesten (1987) showed that the near-critical regime is governed by pivotal edges — edges whose flip changes a large-scale crossing event; being pivotal is a four-arm event, meaning four alternating open and closed paths connect the vicinity of the edge to distance $n$, with probability decaying like $n^{-\xi_4}$. The characteristic length $L(p)$, defined as the scale at which crossing probabilities of a square stop being bounded away from 0 and 1, then satisfies $L(p)^2\pi_4(L(p))|p-p_c|\asymp 1$, and Kesten derived the scaling relations among the critical exponents of the model. Garban, Pete and Schramm (2018) later gave this near-critical stability a geometric, dynamical form.

What changes at $q=2$. For $q>1$ the states of distinct edges are dependent, and in Russo's formula for the derivative of the probability of an increasing event $A$, the pivotality probability is replaced by the conditional influence $I_A^p(e)=\varphi(A\mid e\text{ open})-\varphi(A\mid e\text{ closed})$, where $\varphi$ is the random-cluster measure. Duminil-Copin, Garban and Pete (2014) determined the crossing-based correlation length of the near-critical FK-Ising model: $c|p-p_c|^{-1}\le L(p)\le c^{-1}|p-p_c|^{-1}\log(1/|p-p_c|)$, with the logarithmic factor removable for the natural definition with $\rho>1$. The pivotal-based prediction, using the predicted four-arm exponent $\xi_4(2)=35/24$, would instead give $L(p)\approx|p-p_c|^{-24/13}$, which contradicts this: the transition happens faster than critical pivotals allow. Two effects are responsible. First, the derivative of the edge-intensity (the expected fraction of open edges) blows up near $p_c$ at a rate governed by the specific heat of the model (the second derivative of the free energy; for $q=2$ it diverges only logarithmically, i.e. the specific-heat exponent is $\alpha=0$). Second, and dominant: in any monotone coupling of the FK($p,2$) measures the arriving edges are placed in a strongly configuration-dependent, 'self-organized' way.

Grimmett's monotone coupling and clouds. Grimmett (1995) constructed a monotone coupling: a random label field $Z\in[0,1]^{E(G)}$ such that for every $p$ the sub-level configuration $\omega_p=\{e:Z(e)\le p\}$ has the random-cluster law $\varphi^\xi_{G,p,q}$; in infinite volume the coupling has a unique limiting law $\mu_{\mathbb{Z}^2}$. Since the label field has atoms — label values carried by many edges at once — edges can open simultaneously: for a sample $Z$, the cloud of an edge $e$ is $\mathrm{cloud}(e)=\{f:Z(f)=Z(e)\}$. Duminil-Copin, Garban and Pete proved that for $q>1$, as the box $\Lambda$ grows to $\mathbb{Z}^2$, the probability that some cloud of at least $N$ edges appears at some $p$ tends to 1 for every fixed $N$ — but the proof is soft and gives no quantitative information on the number, size, geometry, or label location of clouds. The clouds are the geometric signature of the long-range dependence of the model and the expected key input for constructing the near-critical scaling limit of FK-Ising, the analogue of what was done for Bernoulli percolation by Garban–Pete–Schramm.

What has since been resolved. The same paper conjectured sharp bounds on the conditional influences for the event $A_n$ that the torus $\mathbb{T}_n=\mathbb{Z}^2/n\mathbb{Z}^2$ contains an open circuit of non-trivial homotopy: $I_{A_n}^p(e)\ge c/(n\log n)$ for $p$ in the window $[p_c-\lambda\log n/n,\,p_c+\lambda\log n/n]$, and, conjecturally without the logarithmic losses, $c/n<I_{A_n}^p(e)<c^{-1}/n$ for $p\in[p_c-\lambda/n,\,p_c+\lambda/n]$, noting that the influence exponent $\iota$ should differ from the pivotal exponent $\xi_4$ for $q>1$. Duminil-Copin and Manolescu (2022) developed a substitute for pivotality — the mixing rate $\Delta_p(R)$, the effect of wired-versus-free boundary conditions at distance $R$ on the status of a central edge — proved that it governs the covariances in Russo's formula, established near-critical stability below the characteristic length and the system of scaling relations R1–R7 for planar random-cluster models with $q\in[1,4]$ (relation R6 only when $\alpha\ge 0$), including the generalized Kesten relation $\nu(2-\iota)=1$ and the hyperscaling relation $\alpha=2-2\nu$ (when $\alpha\ge 0$), with the specific heat tied to influences by $f''(p)\asymp\sum_{\ell\le L(p)}\ell\,\Delta_{p_c}(\ell)^2$, which for $q=2$ diverges logarithmically. They state that this answers the two influence conjectures of the source paper (using that the correlation length $\xi(p)$ of the subcritical FK-Ising model is explicitly known by Beffara–Duminil-Copin (2012)), and equally explicitly that it 'does not provide direct insight on the structure of these clouds', re-posing the cloud question as their own open Question 4. The mixing exponent itself was then computed for the Conformal Loop Ensemble (CLE, a conformally invariant random family of nested loops, indexed by a parameter $\kappa$, expected to describe the scaling limit of the critical FK loop ensembles) by Liu–Wu–Zhuang (2026): $\iota=3\kappa/8-1$, giving $\iota=1$ for FK-Ising ($\kappa=16/3$) as against the pivotal value $35/24$, conditionally on the convergence of critical FK loops to CLE. What remains untouched is precisely the quantitative structure of the clouds in Grimmett's coupling, in the four concrete forms posed by the source paper.

Problem Statement

Let $q=2$ and let $Z$ be a sample of Grimmett's monotone coupling of the FK-Ising model on the square lattice: on the box $\Lambda_n=[-n,n]^2$ with wired boundary conditions, with law $\mu_{\Lambda_n}$, the sub-level configuration $\omega_p=\{e:Z(e)\le p\}$ has the wired random-cluster law $\varphi^{\mathrm{wired}}_{\Lambda_n,p,2}$ for every $p\in[0,1]$; as $n\to\infty$ the couplings converge to a unique infinite-volume coupling law $\mu_{\mathbb{Z}^2}$. For an edge $e$ define $\mathrm{cloud}(e)=\{f:Z(f)=Z(e)\}$, the set of edges that become open simultaneously with $e$ when $p$ rises past the label $Z(e)$. Known (Proposition 3.7 of the source paper): for $q>1$, for every fixed $N$, the probability that some cloud of at least $N$ edges appears in $\Lambda_n$ at some label tends to 1 as $n\to\infty$; the proof is non-quantitative.

Give a quantitative theory of these clouds by resolving, each rigorously in the affirmative or negative, the four questions posed by Duminil-Copin-Garban-Pete (2014):

(Q1) Macroscopic clouds near $p_c$. Prove or disprove: there exists a universal constant $c>0$ such that for all $n\ge 1$, with $\mu_{\Lambda_n}$-probability at least $c$, the box $\Lambda_n$ contains at least one macroscopic cloud, i.e. a cloud of diameter at least $cn$ (the diameter of a cloud being the greatest lattice distance between endpoints of two of its edges); and, with probability tending to 1 as $n\to\infty$, the labels of the macroscopic clouds all lie in $(p_c-\delta,p_c+\delta)$ for every fixed $\delta>0$, where $p_c=p_c(2)=\sqrt{2}/(1+\sqrt{2})$ (label concentration around criticality).

(Q2) Local clouds away from $p_c$. Prove or disprove: for every $\delta>0$, the largest cloud in $\Lambda_n$ whose label lies outside $(p_c-\delta,p_c+\delta)$ has diameter at most $C\log n$ with probability tending to 1 (the diameter of a cloud being, as in (Q1), the greatest lattice distance between endpoints of two of its edges).

(Q3) Non-trivial clouds in infinite volume. Prove or disprove: for $Z$ with law $\mu_{\mathbb{Z}^2}$, almost surely there exists at least one edge $e$ with $|\mathrm{cloud}(e)|\ge 2$.

(Q4) Finiteness of clouds. Prove or disprove: for cluster-weights $q\le 4$, all clouds are almost surely finite under the infinite-volume coupling law (the monotone coupling and clouds being defined for general cluster-weight $q\ge 1$ exactly as above; for large $q$, infinitely many edges appear at the single label $p_c(q)$).

These four questions, posed jointly by the source paper as the open problems on the structure of emerging clouds, form the subject of this problem.

References

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