Bollobás and Riordan proved that for a wide class of independent planar percolation models on self-dual polygon (hyperlattice) configurations — the rigorous setting of the Scullard–Ziff and Chayes–Lei duality criteria — the self-dual point is critical: connectivity decays exponentially strictly below it and percolation occurs strictly above. They explicitly left open the analogue for the random-cluster (Fortuin–Kasteleyn) model and the Potts model, for which the physics literature (Chayes–Lei, Wu) predicts exact self-dual critical frontiers without proof. The problem remains open: every rigorous criticality result for these dependent models holds only on the square lattice or on isoradial graphs with Z-invariant weights, both of which use symmetries or integrability that general self-dual polygon configurations lack. The problem is to prove, for cluster weight q≥1q\ge 1, that the self-dual parameter manifold of these configurations is critical — the exact dependent-model analogue of Bollobás–Riordan's Theorem 2.1.

Background

Percolation theory studies the connectivity of random subgraphs of a lattice: in bond percolation each edge is independently open with probability pp, and the central quantity is the critical probability pcp_c, the threshold below which the probability of an open path from a fixed vertex to graph distance rr decays exponentially in rr, and above which an infinite open cluster exists almost surely. Every planar graph GG has a planar dual G∗G^\ast whose vertices are the faces of GG; a primal open edge corresponds to a closed dual edge, so the event of a horizontal open crossing of a rectangle is complementary to the event of a vertical closed crossing of the dual rectangle. This duality, observed classically by Sykes and Essam, relates percolation on GG to percolation on G∗G^\ast with transformed parameters, and suggests that for a planar graph isomorphic to its own dual — a self-dual lattice — the critical point should be the self-dual value. Proving this 'self-duality predicts criticality' principle in generality is a long-standing challenge: it is accessible for lattices with enough symmetry (Kesten's theorem proving that the bond percolation threshold of the square lattice is 1/21/2) but hard without.

Scullard, Ziff, Chayes–Lei and Wu developed a far-reaching generalization. One considers a periodic planar arrangement of polygon 'generators' (triangles, quadrilaterals) whose corner vertices form the vertices of a 'hyperlattice'; the state of each generator is an independent random partition of its corners (which corners the configuration connects to which), and different generators are independent. Bollobás and Riordan gave this a precise formulation and proved (their Theorem 2.1) that whenever such a model is self-dual — meaning the dual hyperlattice model, with parameters transformed by the duality map, is isomorphic to the original — then any parameter point strictly dominating the self-dual one percolates and any strictly dominated point exhibits exponential decay: the self-dual point is critical. This rigorously confirms the Scullard–Ziff criterion for independent percolation on a wide class of lattices, including bow-tie, checkerboard and martini-type families, with inhomogeneous (position-dependent) parameters.

The random-cluster or Fortuin–Kasteleyn (FK) model is the natural dependent counterpart. On a finite graph it is the probability measure on subgraphs ω\omega in which the weight of ω\omega is proportional to ∏e∈ωpe∏e∉ω(1−pe) qk(ω)\prod_{e\in\omega}p_e\prod_{e\notin\omega}(1-p_e)\,q^{k(\omega)}, where k(ω)k(\omega) is the number of connected components and q>0q>0 is the cluster weight; boundary conditions are 'wired' (boundary vertices forced into one component) or 'free'. For integer qq this is the FK representation of the ferromagnetic qq-state Potts model, so the Potts critical point coincides with the FK critical point. For q≥1q\ge 1 the model satisfies positive association (the FKG lattice condition), but its edge states are globally dependent — the measure couples disjoint regions — so the product-measure sharp-threshold machinery underlying the Bollobás–Riordan proof breaks down. Chayes and Lei described the duality criterion for random-cluster models on the triangular lattice (with a sketch proof of criticality under an extra assumption), and Wu derived closed-form self-dual critical frontiers for Potts models on triangular-type and kagome-type lattices; neither supplies a proof of criticality, and Bollobás–Riordan state the extension as an open problem.

For the random-cluster model, criticality at the self-dual point is rigorously known only in settings with symmetry or integrability: on the square lattice Z2\mathbb Z^2 for q≥1q\ge 1 (Beffara–Duminil-Copin, with the order of the transition on Z2\mathbb Z^2 settled for 1≤q≤41\le q\le 4 and for q>4q>4), and on isoradial graphs (graphs that admit a planar embedding in which every face is inscribed in a circle of common radius) with Z-invariant weights satisfying the Yang–Baxter (star–triangle) relation for q≤4q\le 4 (Beffara–Duminil-Copin–Smirnov), where the parafermionic observable (a lattice analogue of a holomorphic function) provides exact solvability. The standard route to such results is a Russo–Seymour–Welsh (RSW) bound: a uniform lower bound, uniform in scale, for crossing probabilities of large rectangles at the candidate critical point. The most general RSW theorems presently available — Duminil-Copin–Tassion's renormalization of crossing probabilities for the planar random-cluster model, and Köhler-Schindler–Tassion's RSW theorem for arbitrary planar percolation processes invariant under the full symmetry group of Z2\mathbb Z^2 that satisfy positive association — require translations together with rotations and reflections. General self-dual polygon configurations, whose appeal is precisely that they need no symmetry beyond periodicity, do not satisfy this hypothesis; the Köhler-Schindler–Tassion paper explicitly files the Bollobás–Riordan setting under 'Bernoulli percolation on non-symmetric lattices', i.e. outside its scope. Whether planar self-duality alone forces criticality for these dependent measures is the open question.

Problem Statement

Prove the random-cluster/Potts analogue of Theorem 2.1 of Bollobás–Riordan's 'Percolation on self-dual polygon configurations'. Let HH be a planar self-dual polygon (hyperlattice) configuration in the Bollobás–Riordan class — a periodic plane hyperlattice built from polygon generators, isomorphic to its dual — and consider the bond random-cluster (Fortuin–Kasteleyn) model with cluster weight q≥1q\ge 1 on the planar graph associated with the configuration, with edge parameters to which the Chayes–Lei/Wu/Scullard–Ziff duality criterion applies (for integer qq, equivalently the ferromagnetic qq-state Potts model). Let S\mathcal S be the self-dual manifold of parameter vectors fixed by the duality transformation, and order parameter vectors by the source's domination order: one parameter vector strictly dominates another if it assigns strictly larger probability to every non-trivial connection (increasing) event in every generator. Establish that: (1) every parameter vector strictly dominated by some point of S\mathcal S has connection probabilities decaying exponentially with graph distance (strictly below the critical manifold the free and wired infinite-volume measures coincide, so this claim is unaffected by the choice of boundary condition); and (2) every parameter vector strictly dominating some point of S\mathcal S percolates (an infinite open cluster exists for wired boundary conditions). Equivalently: prove that the critical manifold of the random-cluster model on every such configuration coincides with the self-dual manifold predicted by the duality criterion, at the same level of generality at which Bollobás–Riordan's Theorem 2.1 settles the independent case; the behavior exactly on S\mathcal S (e.g. power-law bounds on the one-arm probability) is a stronger, separate question, as in the source. An answer must delimit exactly the class of configurations and parameters for which it proves the statement; a proof valid only for a proper subclass does not by itself resolve the general problem posed here. A rigorous refutation answers the question in the negative and is equally admissible: an explicit configuration of the class, together with a proof that criticality fails there — for instance, exponential decay of wired connection probabilities at a parameter vector strictly above S\mathcal S, or percolation at a parameter vector strictly below S\mathcal S. Behavior exactly on S\mathcal S does not refute (1)–(2): percolation at a single point of S\mathcal S alone is inconclusive, since a critical point may be percolative, and exponential decay of the free measure at a point of S\mathcal S — as occurs, for example, at the self-dual point of the random-cluster model on Z2\mathbb Z^2 with q>4q>4 — is likewise consistent with S\mathcal S being the critical manifold.

References

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