Order and location of the self-dual phase transition in the three-body random-cluster model on the triangular lattice

Chayes and Lei (2006) introduced a random-cluster model on the triangular lattice with three-body interactions: each up-pointing triangle is restricted to five states (empty, one of three single edges, or full) with weight proportional to $q^{c(ω)} s^{|s(ω)|} a^{|a(ω)|} e^{|e(ω)|}$ , where $q ≥ 1$ is the cluster weight. Its dual parameters satisfy $s^*/e^* = qs/a$ and $a^*/e^* = q^2 e/a$ , so the self-dual locus is $a = qe$ ; in the region $ae ≥ 2s^2$ the measures have positive correlations and any bond-density discontinuity must lie on the self-dual curve. The open problem, stated explicitly in the source, is to determine rigorously the order and location of the transition at self-duality: prove discontinuity for large $q$ , determine the rigorous dividing value of $q$ (conjecturally 4, but possibly reduced for three-body interactions), and prove that continuous transitions occur exactly on the self-dual curve.

Background

The Fortuin–Kasteleyn random-cluster representation connects $q$ -state Potts models to dependent bond percolation with cluster weight $q$ , and for the standard nearest-neighbor model on planar lattices the interplay of duality and positive correlations now yields a complete picture: the phase transition is continuous for $1 ≤ q ≤ 4$ and discontinuous for $q > 4$ . None of that machinery applies directly to the model studied here. Chayes and Lei, 'Random Cluster Models on the Triangular Lattice' (Journal of Statistical Physics 122:647–670, 2006), analyze a random-cluster model with genuine three-body interactions, defined as follows. On the triangular lattice, a bond configuration $ω$ occupies edges only in patterns that leave each up-pointing triangle in one of five states: empty ( $E$ ), exactly one occupied edge ( $S_1$ , $S_2$ , $S_3$ — with a common weight $s$ in the isotropic case), or all three edges occupied ( $A$ ). The weight of a configuration in a finite volume is $W(ω) ∝ q^{c(ω)} · s^{|s(ω)|} · a^{|a(ω)|} · e^{|e(ω)|}$ , where $c(ω)$ is the number of connected components (boundary conditions enter only here), $|s(ω)|$ , $|a(ω)|$ , $|e(ω)|$ count triangles in the single-edge, full, and empty states respectively, the nonnegative parameters satisfy $a + 3s + e = 1$ , and the cluster weight satisfies $q ≥ 1$ . Infinite-volume measures are extracted by the usual wired and free boundary-condition limits (all boundary vertices identified into one, respectively left disconnected). For integer $q ≥ 2$ this is the graphical representation of a Potts spin model with two- and three-site interactions, $−βH = Σ_{up triangles} [J(δ_{σ_x σ_y} + δ_{σ_y σ_z} + δ_{σ_x σ_z}) + κ δ_{σ_x σ_y σ_z}]$ with $J > 0$ and $κ$ of either sign, where the sum runs over the up-pointing triangles of the lattice and $δ$ denotes the Kronecker delta between adjacent spins $σ$ . Planar duality maps the parameters to $(a^*, s^*, e^*)$ with $s^*/e^* = qs/a$ and $a^*/e^* = q^2 e/a$ (Theorem 3.1 of the source, matching equation (15) of Wu and Lin 1980), so the self-dual locus is the curve $a = qe$ . The key analytic region is $q ≥ 1$ together with $ae ≥ 2s^2$ : there the measures satisfy the FKG (Fortuin–Kasteleyn–Ginibre) lattice condition, i.e. have positive correlations (Theorem 3.4), and this region is invariant under duality — a property that fails for the full eight-state representation, whose positivity region ( $g ≥ 0$ in the spin parametrization, where $A = 3S^2 + g(1+S)^3$ , $S = e^J − 1$ , $g = e^κ − 1$ ) is not mapped into itself by duality. Within the positive-correlation region the source proves: (i) Proposition 3.10 — any discontinuity of the bond density must occur on the self-dual curve $a = qe$ ; (ii) Proposition 3.9 — at any self-dual point that is a continuity point of the bond density, the percolation probability vanishes and correlation functions admit power-law lower bounds. For $q = 1$ (percolation) the corresponding Section 2 of the source already proves that the transition occurs at the self-dual point with critical behavior, under the analogous positivity condition (Theorem 2.5). In the realm of positive correlations, percolation is synonymous with magnetization, so the bond-density/percolation questions below coincide with the order-of-transition questions for the associated Potts spin model. What remains open, explicitly recorded in the source's introduction, Section 3.4, and conclusion: the order of the transition at self-dual points for $q > 1$ , the rigorous dividing value of $q$ (the exact-solution analysis of Baxter, Temperley and Ashley 1978 suggests $q = 4$ as on the square lattice, but the paper itself notes evidence for a reduced dividing value in a related square-lattice three-body model), a rigorous proof of discontinuity for large $q$ (expected to require adapting reflection-positivity methods, since 'certain modifications will be needed'), and the fact that continuous transitions occur exactly on the self-dual line rather than merely exhibiting criticality there. The later literature (Duminil-Copin–Sidoravicius–Tassion 2016; Duminil-Copin–Gagnebin–Harel–Manolescu–Tassion 2021; Ray–Spinka 2020; Duminil-Copin–Li–Manolescu 2018) settles the continuous/discontinuous dichotomy only for the standard nearest-neighbor model, on the square lattice and isoradial graphs; its proofs use structure absent here — a duality-invariant ferromagnetic region in the full state space and, in places, integrable inputs — and the source notes that the dual of the ferromagnetic region is not ferromagnetic in the full eight-state space, and that BK-type inequalities do not apply in the present Potts setting.

Problem Statement

Setting. On the infinite triangular lattice, consider the random-cluster model of Chayes–Lei: bond configurations $ω$ whose restriction to each up-pointing triangle is one of five states — empty ( $E$ ), exactly one occupied edge ( $S_1$ , $S_2$ , $S_3$ ), or all three edges occupied ( $A$ ) — with finite-volume weight $W(ω) ∝ q^{c(ω)} · s^{|s(ω)|} · a^{|a(ω)|} · e^{|e(ω)|}$ , where $c(ω)$ is the number of connected components (this is where boundary conditions enter) and $|s(ω)|$ , $|a(ω)|$ , $|e(ω)|$ count up-pointing triangles in the single-edge states $S_1, S_2, S_3$ , the full state $A$ , and the empty state $E$ , respectively. The parameters satisfy $a, s, e ≥ 0$ and $a + 3s + e = 1$ , and the cluster weight satisfies $q ≥ 1$ . Infinite-volume measures are obtained by the usual wired and free boundary-condition limits. The dual parameters satisfy $s^*/e^* = qs/a$ and $a^*/e^* = q^2 e/a$ ; the self-dual locus is $a = qe$ . Throughout, work in the isotropic case ( $s_1 = s_2 = s_3 = s$ ) and inside the positive-correlation region $q ≥ 1$ , $ae ≥ 2s^2$ , where the FKG lattice condition holds, any bond-density discontinuity must lie on $a = qe$ (Proposition 3.10 of the source), and at self-dual continuity points the percolation probability — the probability that the origin belongs to an infinite occupied cluster — vanishes, with power-law lower bounds on correlations (Proposition 3.9). In this region percolation coincides with magnetization of the associated Potts spin model with two- and three-site interactions.

Problem. Determine the order and location of the percolation/magnetization phase transition at self-duality, specifically:

(1) Order. For each $q ≥ 1$ , decide whether the bond density (the infinite-volume expected fraction of occupied edges) is continuous or discontinuous at the self-dual curve $a = qe$ , as the parameters traverse any FKG-monotone curve (a curve in parameter space along which the measure is monotone increasing) within the region that crosses that locus. In particular, prove that for all sufficiently large $q$ the transition is discontinuous — a jump in bond density and coexistence of multiple infinite-volume states at self-duality.

(2) Dividing value. Determine the rigorous dividing value $q_c$ for this model: the threshold such that the transition is continuous for $1 ≤ q < q_c$ and discontinuous for $q > q_c$ (or prove that no single threshold exists, exhibiting the correct classification as a function of $q$ — and, at fixed $q$ , of the self-dual point, if the order of the transition is not the same at every point of the locus). The Baxter–Temperley–Ashley exact-solution argument suggests $q_c = 4$ , as in the nearest-neighbor model, but this is unproved for the three-body model and the source itself records evidence for a reduced dividing value in a related square-lattice three-body model; the value must be derived for this model, not imported. The already-settled endpoint $q = 1$ (percolation, continuous) is excluded from what needs proof.

(3) Location. Prove that whenever the transition is continuous it occurs exactly on the self-dual curve $a = qe$ — i.e., that there is no percolation strictly below self-duality in the continuous regime, so the percolation threshold and the self-dual crossing coincide for every FKG-monotone curve in the region — or else rigorously locate where continuous transitions occur.

Completion boundary. The problem is solved when the order of the transition at self-duality is determined for every $q ≥ 1$ in the region — equivalently, (1) and (2) are settled with whatever value or structure of $q_c$ is true — together with (3) in the continuous cases. A proof of discontinuity for large $q$ alone, or of continuity on an interval of $q$ alone, is substantial partial progress but not a completion.

Scope note (per the source's own remarks). The source additionally poses extension directions that are source-faithful follow-on problems but lie outside this problem's completion boundary: extending correlation control beyond $ae ≥ 2s^2$ to the entire self-dual curve; anisotropic triangle parameters (unequal single-edge weights $s_1, s_2, s_3$ ); and necessary-and-sufficient duality-invariant positivity criteria for the full eight-state representation including double-edge configurations.

References

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