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arxiv: 2502.04129 · v2 · submitted 2025-02-06 · 🧮 math.PR · math-ph· math.MP

Discontinuous transition in 2D Potts: I. Order-Disorder Interface convergence

Moritz Dober , Alexander Glazman , S\'ebastien Ott This is my paper

Pith reviewed 2026-05-23 04:21 UTC · model grok-4.3

classification 🧮 math.PR math-phmath.MP
keywords Potts modelDobrushin interfaceorder-disorder transitionFK percolationBrownian bridgeOrnstein-Zernike asymptoticsAshkin-Teller modelphase separation
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The pith

The Dobrushin order-disorder interface in the q-state Potts model at its discontinuous transition is well-defined, fluctuates on scale sqrt(N), and converges to a Brownian bridge.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that in the two-dimensional q-state Potts model for q greater than 4, at the temperature of its discontinuous phase transition, the interface between an ordered phase and the disordered phase can be made rigorous. With Dobrushin boundary conditions on an N by N box, this interface has fluctuations of order the square root of N and, after rescaling by that factor, converges in law to a standard Brownian bridge. The identical statement holds for the corresponding FK-percolation model. A reader would care because the result supplies the first precise scaling limit for an interface in a lattice model whose transition is first-order, a setting where such limits have been harder to obtain than for continuous transitions.

Core claim

The Dobrushin order-disorder interface is a well-defined object, has √N fluctuations, and converges to a Brownian bridge under diffusive scaling. The same holds for the corresponding FK-percolation model for all q>4. Proofs rely on a coupling between FK-percolation, the six-vertex model, and the random-cluster representation of an Ashkin-Teller model, which maps the interface to a long subcritical cluster whose mixing properties permit an Ornstein-Zernike renewal analysis; along the way the paper derives Ornstein-Zernike asymptotics for the two-point function of the Ashkin-Teller model.

What carries the argument

The coupling that transfers the FK interface to a long subcritical cluster in the Ashkin-Teller random-cluster model, together with the Ornstein-Zernike renewal analysis of that cluster.

If this is right

  • The interface admits a renewal decomposition that yields exact tail estimates and mixing rates.
  • Ornstein-Zernike decay holds for the two-point function of the Ashkin-Teller model under the same coupling.
  • The convergence statement transfers directly to the FK-percolation representation for every q>4.
  • The companion order-order analysis produces a free layer of width sqrt(N) whose two boundaries converge to non-intersecting Brownian bridges.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same coupling technique may produce scaling limits for interfaces in other first-order models once an analogous subcritical cluster representation is available.
  • The Brownian-bridge limit invites comparison with the scaling limits known for critical interfaces in the Ising model, suggesting a possible universality class for certain discontinuous transitions.
  • The renewal picture could be used to study the probability that two or more such interfaces remain disjoint on large but finite boxes.

Load-bearing premise

The coupling between FK-percolation, the six-vertex model, and the Ashkin-Teller random-cluster representation exists and preserves the interface as a long subcritical cluster whose mixing properties support the renewal analysis.

What would settle it

Monte Carlo simulation on large grids in which the rescaled order-disorder interface path fails to converge in distribution to a Brownian bridge or exhibits fluctuations whose variance grows at a rate other than linear in N.

Figures

Figures reproduced from arXiv: 2502.04129 by Alexander Glazman, Moritz Dober, S\'ebastien Ott.

Figure 1
Figure 1. Figure 1: Sample of a 600x600 Potts model with 25 colours at Tc(25) with Dobrushin boundary conditions. Left: order-disorder interface; upper part has blue b.c., bottom has white b.c. (no colour favoured), colours are: blue for the first and interpolate between yellow and red for colours 2 to 25. Right: colours are: blue for the first, red for the second, and interpolate between yellow and orange for colours 3 to 25… view at source ↗
Figure 2
Figure 2. Figure 2: Left: wired-free Dobrushin boundary condition. Right: the graph G2. These works on subcritical clusters of the FK-percolation all take place in any di￾mensions. On Z 2 , planar duality allows to rewrite the interface of the Potts model at T < Tc(q) as a subcritical FK-percolation cluster conditioned to contain (0, 0) and (N, 0). Our contribution to this line of works is to derive a “renewal picture” for th… view at source ↗
Figure 3
Figure 3. Figure 3: Left: Tile associated to a mid-edge. Right: Tile centred at the middle of a horizontal primal edge (solid black) or its associated vertical dual edge (dashed black), with its two possible local loop configurations. (Z × Z≥0) \ Λn states the existence of a path in Z 2 with diagonal connectivity going from (k, y) to (Z × Z≥0) \Λn and consisting of vertices where σ¯ = 1. By Z 2 with diag￾onal connectivity we … view at source ↗
Figure 4
Figure 4. Figure 4: Dobrushin boundary condition for the six-vertex height function. the ATRC model at its self-dual line when U > J remarkably exhibits a unique Gibbs measure. This brings more symmetries that play a key role in the study of interfaces. This comes at some cost: the ATRC model is supported on pairs of edge config￾urations. Thus, the domain Markov property is significantly weaker than in the FK percolation defi… view at source ↗
Figure 5
Figure 5. Figure 5: Left: the sets Λ2,2 (solid) and Λ ′ 2,2 (hollow). Right: the inner tiles Ai 2,2 (white) and the boundary tiles ∂A2,2 (grey). connected. As a convention, sets of edges or of dual edges will be identified with the corresponding sets of mid-points whenever the meaning is clear from the context. Tiles. To each primal-dual pair of edges e, e∗ , associate a tile t given by the convex hull of their endpoints and … view at source ↗
Figure 6
Figure 6. Figure 6: Left: the graph K2,2. Center: the graph K∗ 2,2 (vertices surrounded by a circle are identified). Right: planar duality relation between their edges. • and let K∗ n,m be the graph obtained from (V∗E¯n,m , ∗E¯ n,m) by identifying the vertices in ∂ in L◦V∗E¯n,m . Connectivity events. Given Λ, ∆ ⊂ L• and a percolation configuration ω ∈ {0, 1} E • , we write Λ ω←→ ∆ for the event that Λ and ∆ are connected by a… view at source ↗
Figure 7
Figure 7. Figure 7: Tiles of the oriented loop model and their types and weights, and the mapping from oriented loop arcs to six-vertex edge orientations. 4.1. Different models and combinatorial mappings. This section provides an overview of the combinatorial objects that will be encountered, as well as a descrip￾tion of their relations. We first discuss oriented loop configurations, which serve as an intermediate step in the… view at source ↗
Figure 8
Figure 8. Figure 8: The six-vertex types for all representations at a tile corresponding to a horizontal primal edge e. Top: the spin at the left endpoint of e is fixed to be +. Bottom: the height at the left endpoint of e is fixed to be 0. and we impose σ•(i) = −σ◦(u) otherwise; see [PITH_FULL_IMAGE:figures/full_fig_p017_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Boundary conditions on oriented loops and on six-vertex spins. The edges drawn are those induced by the oriented loop boundary conditions. Borel sigma algebras. Define Q and Q′ respectively as the product measures on [0, 1]L and [0, 1]L⋄ . Finally, the coupling measure is defined by Ψ 1/0 G := FK1/0 G ⊗ Q ⊗ Q ′ . In Lemmata 4.1 and 4.2 below, we describe how to obtain the following two measures as marginal… view at source ↗
Figure 10
Figure 10. Figure 10: Left: A realisation of C = CvL (endpoints of solid edges) in K for n = m = 3, the dual ∗∂ edge L• C of its edge-boundary in L• (dashed edges), and the surrounding polygon P (grey). Right: a path in ∗∂ edge K C connecting (−n − 3 2 , 1 2 ) and (n + 3 2 , 1 2 ) on which σ◦ = +1 (red dashed edges), and a path in ∗∂ edge K C connecting (−n− 1 2 , − 1 2 ) and (n+ 1 2 , − 1 2 ) on which σ◦ = −1 (green dashed ed… view at source ↗
Figure 11
Figure 11. Figure 11: Left: part of the rotated square lattice L. Its faces are the tiles in L⋄. Center: part of the augmented lattice L. Right: L-domains given by the vertices strictly within simple circuits in L, L•, L◦, respectively. The lower left and right L-domains are even and odd, respectively. following relation (see [PITH_FULL_IMAGE:figures/full_fig_p025_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Left: a circuit in L• (thick black edges) and the corresponding L•-domain of the first kind (thin and thick black edges). Right: a circuit in L◦ (dashed edges) obtained by shifting the left one by (1/2, 1/2) and the corresponding L•-domain of the second kind (black edges). The tiles corre￾sponding to the edges of the domains are shaded grey, and the vertices of the corresponding L-domains are surrounded b… view at source ↗
Figure 13
Figure 13. Figure 13: Left: a path in Γ1 (black disks), the blue edges have their state forced to be (0, 1), the black disks in a green square imply the realization of Mai at that place. Right: a realization of Mai (blue edges are open in ωττ′, while the dashed red path represent the dual of edges closed in ωτ ). (3) If w ∈ Mai , and j ∈ Γl with ∥i − j∥∞ ≤ 2l+ 1, then the configuration obtained by swapping the value of w|EBL([… view at source ↗
Figure 14
Figure 14. Figure 14: Forward and backward cones, and the associated diamond. cone generated by the directions dual to t. The latter set is a line when ν is strictly convex. Let V ⊂ Z 2 . We will say that V is: • (t, δ)-forward-confined if there exists u ∈ V such that V ⊂ u + Y ◀ t,δ. When it exists, such a u is unique; we denote it by f(V ). • (t, δ)-backward-confined if there exists v ∈ V such that V ⊂ v + Y ▶ t,δ. When it e… view at source ↗
Figure 15
Figure 15. Figure 15: Backward, forward, and diamond confined marked graphs,with their displacement [PITH_FULL_IMAGE:figures/full_fig_p038_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Concatenation of confined graphs. Red dots are the marked vertices of the marked graphs. These sets can be seen as equivalence classes of general marked forward/backward/diamond confined graphs modulo translations. A general such graph, γ ′ L , γ′ R, γ′ , can then be re￾covered uniquely from an element γL, γR, γ of BL, BR, A by specifying the translation vector. 7.2. Main result of the section. Our main g… view at source ↗
Figure 17
Figure 17. Figure 17: From top to bottom: a realization of C0 (0 is in red); the cells of the associated coarse-graining; the connections implied by the presence of the cells; the skeleton (in black). Note that the wanted probabilities are zero whenever A ∩ ∆K does not contain a path going from 0 to ∂ in∆K. Proof. By the definition of U, the ratio weak mixing property (Theorem 4) and mono￾tonicity: Φ 1,1 [∆K∩A] ln(K) 2 (0 ∆K∩A… view at source ↗
Figure 18
Figure 18. Figure 18: Left: a cone-point that is not regular. Right: a regular cone￾point. Note than when Y ◀ t,δ contains exactly one element of {±e1, ±e2}, all cone-points are necessarily regular. The idea for going from Lemma 7.5 to the next lemma is simple: when v ↔ w, up to an exponentially small error, there must be at least c|v − w|/K cone-points of the skeleton by Lemma 7.5. Up to anther exponentially small error, a po… view at source ↗
Figure 19
Figure 19. Figure 19: From left to right. 1) Backward confined graph which is not irreducible: the graph can be written as the concatenation of two graphs so that the concatenation point is a regular cone-point. 2) The graph is not irreducible as its endpoints will not generate regular cone-points when concatenated. 3) The graph is not irreducible as it contains a regular cone￾point. 4) An irreducible graph. 7.6. Proof of Theo… view at source ↗
Figure 20
Figure 20. Figure 20: Top: A cluster with its irreducible decomposition. Bottom: the corresponding percolation events. Bold black edges are open in ωτ (hence also in ωττ′); golden edges are closed in ωτ but open in ωττ′; thin lines are the dual of edges that are closed in ωτ , but for which the state of ωττ′ is not prescribed. irreducible components: C0 = ηL ⊔ η1 ⊔ · · · ⊔ ηM ⊔ ηR, where M ≥ 1, (ηL, 0), (ηR, x), η1, . . . , ηM… view at source ↗
Figure 21
Figure 21. Figure 21: Black edges have state (1, 1), gold ones have state (0, 1), thin lines are dual to edges where ωτ is 0. Left: the percolation event one conditions on. Right: the boundary conditions it effectively imposes. DB0, DBR, DR0, DR with DL, D′ L supported on E▷ \ EΛM , DRB, DR supported on E◁ \ EΛM , and DL0, D′ L0 , DB0, DR0 supported on EΛM , such that B0 = DB0 ∩ DBR, {YF◁ = α} = DR0 ∩ DR, {YF▷ = ξ} = DL ∩ DL0,… view at source ↗
Figure 22
Figure 22. Figure 22: The grey grid represent the blocks of the block-percolation of Theorem 8. The orange squares have a strictly smaller than one probability to be open. The white squares have a probability as small as wanted to be open. The red regions represent E▷, Eϵ,×, E◁ [PITH_FULL_IMAGE:figures/full_fig_p048_22.png] view at source ↗
Figure 23
Figure 23. Figure 23: The grey grid represent the blocks of the block-percolation of Theorem 8. The orange squares have a strictly smaller than one probability to be open. The light blue squares have a probability as small as wanted to be open. By exponential decay in the blue blocks, a positive density of bottleneck have the property that the two orange sides of the blue square are not connected in the blue blocks. Then, with… view at source ↗
Figure 24
Figure 24. Figure 24: V¯ n,m (black circles), En,m (black lines), and E + b (red lines). for simplicity, but the analysis can easily be adapted to other directions at the cost of slightly heavier notations. For notational convenience, we define the relevant objects with explicit n, m dependency, but will only stress the m dependence, as the n one is obvious (n with therefore usually be omitted from the notation). Recall the se… view at source ↗
Figure 25
Figure 25. Figure 25: The open cluster Ccr (in black) and the edge forced to be closed by its presence (dual of the red). The blue points are where connections to Ccr have to arrive. In particular, for C a realisation of Ccr such that C ⊂ Λn¯1,c0n, and n large enough, Φm(∃x : d(x, C) ≥ K, x ↔ C | Ccr = C) ≤ X y∈L(C) X x∈Z2:d∞(x,y)≥K e −c|d∞(x,y)| ≤ e −cK/2 , as |L(C)| ≤ 2c0n, and c2 is large enough. In particular, Φm [PITH_FU… view at source ↗
Figure 26
Figure 26. Figure 26: The presence of x, y, z in CPtsπ/8 (Ccr) forces C to be included in the union of the blue diamonds/cones and of the orange rectangles. Enlarging the aperture of the cones at y allows to ensure that the orange rectangles are included in the cones started from y. Lemma 8.9. There are c5 > 0, c > 0 such that for any r ≥ c5, there is n0 ≥ 0 such that for any n ≥ n0, m ≥ c0n + ln2 (n), Φm [PITH_FULL_IMAGE:fig… view at source ↗
Figure 27
Figure 27. Figure 27: For C ∈ GCl, the blue stipes contain both ln(n) cone-points, the yellow stripes each contain at last one cone-point, WL(C), WR(C) are in the green stipes. The decoupling event D lives in the orange part. One can then uniquely decompose a cluster C ∈ GCl as C = vL + ηL ◦ η ◦ ηR with ηL ∈ B1 L , ηR ∈ B1 R, η ∈ A, X(ηL) = WL(C) − vL, X(ηR) = vR − WR(C). Now, notice that by the cone constraint, b(ηL) has degr… view at source ↗
Figure 28
Figure 28. Figure 28: Decoupling paths imposed by the event D. • A˜ R(ηR) is the event that edges of uR +ηR are open in ωττ′, edges of (uR +ηR) \ fe(uR + ηR) are open in ωτ , edges of uR + ∂ exηR are closed in ωτ , fe(uR + ηR) is closed in ωτ . As for Φm (see (52)), Φ(CvL = vL + ηL ◦ η ◦ ηR) = CJ,U Φ [PITH_FULL_IMAGE:figures/full_fig_p059_28.png] view at source ↗
Figure 29
Figure 29. Figure 29: For n = m = 3. Left: vertices of G (solid) and its dual (hollow) with “1/0” boundary conditions (black solid and dashed edges), the dual clus￾ter C ′ = C ′ (endpoints of dashed edges), and the lowermost path p [PITH_FULL_IMAGE:figures/full_fig_p066_29.png] view at source ↗
read the original abstract

We study a $q$-state Potts model on the square grid when $q>4$ at the point $T_c(q)$ of its (discontinous) transition. This model exhibits exactly $q+1$ extremal Gibbs measures: $q$ ordered (monochromatic) and one disordered (free). The current work deals with the Dobrushin order--disorder boundary conditions on a finite $N\times N$ box. Our main result is that this interface is a well-defined object, has $\sqrt{N}$ fluctuations, and converges to a Brownian bridge under diffusive scaling. The same holds also for the corresponding FK-percolation model for all $q>4$. Our proofs rely on a coupling between FK-percolation, the six-vertex model, and the random-cluster representation of an Ashkin--Teller model (ATRC), and on a detailed study of the latter. The coupling relates the interface in FK-percolation to a long subcritical cluster in the ATRC model. For this cluster we develop a ``renewal picture'' \`a la Ornstein-Zernike. This is based on fine mixing properties of the ATRC model that we establish using the link to the six-vertex model and its height function. Along the way, we derive various properties of the Ashkin-Teller model, such as Ornstein-Zernike asymptotics for its two-point function. In a companion work, we provide a detailed study of the Potts model under order-order Dobrushin conditions. We show emergence of a free layer of width $\sqrt{N}$ between the two ordered phases (wetting) and establish convergence of its boundaries to two Brownian bridges conditioned not to intersect.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript studies the Dobrushin order-disorder interface in the q-state Potts model on the square grid for q>4 at the critical temperature Tc(q) of its discontinuous transition. It proves that the interface is well-defined, exhibits √N fluctuations, and converges to a Brownian bridge under diffusive scaling. The same result is established for the corresponding FK-percolation model. The proofs are based on a coupling with the six-vertex model and the random-cluster representation of the Ashkin-Teller model (ATRC), developing a renewal picture using Ornstein-Zernike asymptotics derived from the six-vertex height function.

Significance. This work provides rigorous evidence for the scaling behavior of interfaces in 2D Potts models with first-order transitions, which has been a challenging area. The multi-model coupling approach and the detailed analysis of the ATRC model represent a significant technical achievement. If the results hold, they lay the groundwork for further studies, including the wetting phenomenon discussed in the companion paper. The absence of free parameters and the use of established models strengthen the result.

minor comments (2)
  1. [Abstract] Abstract: the phrase 'a detailed study of the latter' for the ATRC model leaves unclear which mixing and Ornstein-Zernike properties are proved here versus taken from prior six-vertex literature; a short sentence distinguishing the new contributions would improve readability.
  2. [Introduction] The introduction should include an explicit statement of the precise scaling limit (e.g., the topology on path space and the variance of the limiting Brownian bridge) already in the first paragraph, rather than deferring all details to later sections.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary and significance assessment of the manuscript, as well as the recommendation for minor revision. The referee's description of the main results on the Dobrushin order-disorder interface convergence in the q-Potts and FK models is accurate.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper's derivation for the Dobrushin order-disorder interface convergence relies on an external coupling to the six-vertex model and ATRC representation, followed by an Ornstein-Zernike analysis of mixing properties derived from the height function; these steps are presented as independent technical inputs rather than self-referential. The companion work citation addresses only the distinct order-order case and carries no load for the central claims on √N fluctuations or Brownian bridge convergence. No equation or premise reduces by construction to a fitted input, self-definition, or unverified self-citation chain, leaving the argument self-contained against the stated external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the existence of the stated multi-model coupling and on the mixing properties of the ATRC model that enable the renewal picture; these are domain assumptions drawn from connections to the six-vertex model.

axioms (2)
  • domain assumption A coupling exists between FK-percolation, the six-vertex model, and the ATRC model that maps the order-disorder interface to a long subcritical cluster.
    Invoked explicitly in the abstract as the foundation for the proofs.
  • domain assumption The ATRC model possesses fine mixing properties sufficient for an Ornstein-Zernike renewal analysis of its clusters.
    Derived via the link to the six-vertex height function, as stated in the abstract.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Discontinuous transition in 2D Potts: II. Order-Order Interface convergence

    math.PR 2026-04 unverdicted novelty 8.0

    At the discontinuous transition of the 2D q-Potts model for q>4, the order-order interface forms a disordered layer whose boundaries converge diffusively to a pair of non-intersecting Brownian motions.

Reference graph

Works this paper leans on

59 extracted references · 59 canonical work pages · cited by 1 Pith paper

  1. [1]

    Yacine Aoun, Moritz Dober, and Alexander Glazman, Phase diagram of the A shkin-- T eller model , Communications in Mathematical Physics 405 (2024), no. 2, 37

    work page 2024

  2. [2]

    D. B. Abraham and H. Kunz, Ornstein- Z ernike theory of classical fluids at low density , Phys. Rev. Lett. 39 (1977), no. 16, 1011--1014. 452115

    work page 1977

  3. [3]

    Alexander, Stability of the W ulff minimum and fluctuations in shape for large finite clusters in two-dimensional percolation , Probab

    Kenneth S. Alexander, Stability of the W ulff minimum and fluctuations in shape for large finite clusters in two-dimensional percolation , Probab. Theory Related Fields 91 (1992), no. 3-4, 507--532. 1151807 (93e:60191)

    work page 1992

  4. [4]

    Theory Related Fields 110 (1998), no

    , On weak mixing in lattice models, Probab. Theory Related Fields 110 (1998), no. 4, 441--471. 1626951 (99e:60211)

    work page 1998

  5. [5]

    Kenneth S Alexander, Power-law corrections to exponential decay of connectivites and correlations in lattice models, Annals of probability (2001), 92--122

    work page 2001

  6. [6]

    Alexander, Mixing properties and exponential decay for lattice systems in finite volumes, Ann

    Kenneth S. Alexander, Mixing properties and exponential decay for lattice systems in finite volumes, Ann. Probab. 32 (2004), no. 1A, 441--487. 2040789 (2004m:60217)

    work page 2004

  7. [7]

    Y. Aoun, S. Ott, and Y. Velenik, Ornstein-- Z ernike behavior for I sing models with infinite-range interactions , Annales de l'Institut Henri Poincare (B) Probabilites et statistiques 60 (2024), no. 1, 167--207

    work page 2024

  8. [8]

    Ashkin and E

    J. Ashkin and E. Teller, Statistics of two-dimensional lattices with four components, Physical Review 64 (1943), no. 5-6, 178--184

    work page 1943

  9. [9]

    Beffara and H

    V. Beffara and H. Duminil-Copin , Smirnov's fermionic observable away from criticality, Ann. Probab. 40 (2012), no. 6, 2667--2689. 3050513

    work page 2012

  10. [10]

    3, 397--406

    R J Baxter, S B Kelland, and F Y Wu, Equivalence of the P otts model or W hitney polynomial with an ice-type model , Journal of Physics A: Mathematical and General 9 (1976), no. 3, 397--406

    work page 1976

  11. [11]

    J. T. Chayes and L. Chayes, Ornstein- Z ernike behavior for self-avoiding walks at all noncritical temperatures , Comm. Math. Phys. 105 (1986), no. 2, 221--238. 849206

    work page 1986

  12. [12]

    3, 269--341

    Massimo Campanino, JT Chayes, and L Chayes, Gaussian fluctuations of connectivities in the subcritical regime of percolation, Probability theory and related fields 88 (1991), no. 3, 269--341

    work page 1991

  13. [13]

    Dmitry Chelkak, Hugo Duminil-Copin , Cl\' e ment Hongler, Antti Kemppainen, and Stanislav Smirnov, Convergence of I sing interfaces to S chramm's SLE curves , C. R. Math. Acad. Sci. Paris 352 (2014), no. 2, 157--161. 3151886

    work page 2014

  14. [14]

    Massimo Campanino and Dmitry Ioffe, Ornstein- Z ernike theory for the B ernoulli bond percolation on Z^d , Ann. Probab. 30 (2002), no. 2, 652--682. 1905854 (2003e:60216)

    work page 2002

  15. [15]

    Theory Related Fields 125 (2003), no

    Massimo Campanino, Dmitry Ioffe, and Yvan Velenik, Ornstein- Z ernike theory for finite range I sing models above T_c , Probab. Theory Related Fields 125 (2003), no. 3, 305--349. 1964456 (2005b:82016)

    work page 2003

  16. [16]

    , Fluctuation theory of connectivities for subcritical random cluster models, Ann. Probab. 36 (2008), no. 4, 1287--1321. 2435850 (2009m:60228)

    work page 2008

  17. [17]

    Chayes and J

    L. Chayes and J. Machta, Graphical representations and cluster algorithms i. discrete spin systems, Physica A: Statistical Mechanics and its Applications 239 (1997), no. 4, 542--601

    work page 1997

  18. [18]

    Dmitry Chelkak and Stanislav Smirnov, Discrete complex analysis on isoradial graphs, Adv. Math. 228 (2011), no. 3, 1590--1630

    work page 2011

  19. [19]

    Hugo Duminil-Copin, Lectures on the I sing and P otts models on the hypercubic lattice , PIMS-CRM Summer School in Probability, Springer, 2017, pp. 35--161

    work page 2017

  20. [20]

    Hugo Duminil-Copin, Alex M Karrila, Ioan Manolescu, and Mendes Oulamara, Delocalization of the height function of the six-vertex model, Journal of the European Mathematical Society (2024)

    work page 2024

  21. [21]

    54, SOC Mathematique France, 2021, pp

    Hugo Duminil-Copin , Maxime Gagnebin, Matan Harel, Ioan Manolescu, and Vincent Tassion, Discontinuity of the phase transition for the planar random-cluster and P otts models with q>4 , Annales Scientifiques de l'Ecole Normale Sup \'e rieure, vol. 54, SOC Mathematique France, 2021, pp. 1363--1413

    work page 2021

  22. [22]

    Hugo Duminil-Copin , Vladas Sidoravicius, and Vincent Tassion, Continuity of the phase transition for planar random-cluster and P otts models with 1 q 4 , Comm. Math. Phys. 349 (2017), no. 1, 47--107

    work page 2017

  23. [23]

    Edwards and Alan D

    Robert G. Edwards and Alan D. Sokal, Generalization of the F ortuin- K asteleyn- S wendsen- W ang representation and M onte C arlo algorithm , Phys. Rev. D (3) 38 (1988), no. 6, 2009--2012. 965465

    work page 1988

  24. [24]

    Fan, On critical properties of the A shkin- T eller model , Physics Letters A 39 (1972), no

    C. Fan, On critical properties of the A shkin- T eller model , Physics Letters A 39 (1972), no. 2, 136

    work page 1972

  25. [25]

    Chungpeng Fan, Remarks on the eight-vertex model and the ashkin-teller model of lattice statistics, Phys. Rev. Lett. 29 (1972), 158--160

    work page 1972

  26. [26]

    C. M. Fortuin and P. W. Kasteleyn, On the random-cluster model. I . I ntroduction and relation to other models , Physica 57 (1972), 536--564

    work page 1972

  27. [27]

    C. M. Fortuin, P. W. Kasteleyn, and J. Ginibre, Correlation inequalities on some partially ordered sets, Comm. Math. Phys. 22 (1971), 89--103

    work page 1971

  28. [28]

    a ggstr \

    Hans-Otto Georgii, Olle H \"a ggstr \"o m, and Christian Maes, The random geometry of equilibrium phases, Phase transitions and critical phenomena, V ol. 18, Phase Transit. Crit. Phenom., vol. 18, Academic Press, San Diego, CA, 2001, pp. 1--142. 2014387 (2004h:82022)

    work page 2001

  29. [29]

    Alexander Glazman and Piet Lammers, Delocalisation and continuity in 2D : loop O(2) , six-vertex, and random-cluster models , arXiv preprint arXiv:2306.01527 (2023)

    work page arXiv 2023

  30. [30]

    2, 209--256

    Alexander Glazman and Ioan Manolescu, Structure of G ibbs measures for planar FK -percolation and P otts models , Probability and Mathematical Physics 4 (2023), no. 2, 209--256

    work page 2023

  31. [31]

    Alexander Glazman and Ron Peled, On the transition between the disordered and antiferroelectric phases of the 6-vertex model, Electronic Journal of Probability 28 (2023), 1--53

    work page 2023

  32. [32]

    Grimmett, The random-cluster model, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol

    G. Grimmett, The random-cluster model, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 333, Springer-Verlag, Berlin, 2006

    work page 2006

  33. [33]

    B 868 (2013), no

    Yuan Huang, Youjin Deng, Jesper Lykke Jacobsen, and Jes\' u s Salas, The H intermann- M erlini- B axter- W u and the infinite-coupling-limit A shkin- T eller models , Nuclear Phys. B 868 (2013), no. 2, 492--538. 3006181

    work page 2013

  34. [34]

    Richard Holley, Remarks on the FKG inequalities , Comm. Math. Phys. 36 (1974), 227--231. 0341552 (49 \#6300)

    work page 1974

  35. [35]

    none, 1 -- 32

    Matan Harel and Yinon Spinka, Finitary codings for the random-cluster model and other infinite-range monotone models, Electronic Journal of Probability 27 (2022), no. none, 1 -- 32

    work page 2022

  36. [36]

    Ioffe, Ornstein- Z ernike behaviour and analyticity of shapes for self-avoiding walks on Z ^d , Markov Process

    D. Ioffe, Ornstein- Z ernike behaviour and analyticity of shapes for self-avoiding walks on Z ^d , Markov Process. Related Fields 4 (1998), no. 3, 323--350. 1670027 (2000d:60162)

    work page 1998

  37. [37]

    Ioffe, S

    D. Ioffe, S. Ott, S. Shlosman, and Y. Velenik, Critical prewetting in the 2d I sing model , Ann. of Prob. 50 (2022), no. 3, 1127--1172

    work page 2022

  38. [38]

    Kesten, Percolation theory for mathematicians, Progress in Probability and Statistics, vol

    H. Kesten, Percolation theory for mathematicians, Progress in Probability and Statistics, vol. 2, Birkh \"a user Boston, Mass., 1982. 692943 (84i:60145)

    work page 1982

  39. [39]

    Related Fields 10 (2004), no

    Yevgeniy Kovchegov, The B rownian bridge asymptotics in the subcritical phase of B ernoulli bond percolation model , Markov Process. Related Fields 10 (2004), no. 2, 327--344

    work page 2004

  40. [40]

    Kelly and S

    D.G. Kelly and S. Sherman, General G riffiths's inequality on correlation in I sing ferromagnets , J. Math. Phys. 9 (1968), 466--484

    work page 1968

  41. [41]

    Kadanoff and Franz J

    Leo P. Kadanoff and Franz J. Wegner, Some critical properties of the eight-vertex model, Phys. Rev. B 4 (1971), 3989--3993, 10.1103/PhysRevB.4.3989 https://link.aps.org/doi/10.1103/PhysRevB.4.3989

    work page doi:10.1103/physrevb.4.3989 1971

  42. [42]

    2, 1181--1205

    Marcin Lis, On delocalization in the six-vertex model, Communications in Mathematical Physics 383 (2021), no. 2, 1181--1205

    work page 2021

  43. [43]

    , Spins, percolation and height functions, Electronic Journal of Probability 27 (2022), 1--21

    work page 2022

  44. [44]

    Alain Messager, Salvador Miracle-Sole, Jean Ruiz, and Senya Shlosman, Interfaces in the potts model ii: Antonov's rule and rigidity of the order disorder interface, Communications in mathematical physics 140 (1991), 275--290

    work page 1991

  45. [45]

    Mittag and M

    L. Mittag and M. J. Stephen, Dual transformations in many-component I sing models , Journal of Mathematical Physics 12 (1971), no. 3, 441--450, doi:10.1063/1.1665606 https://doi.org/10.1063/1.1665606

    work page doi:10.1063/1.1665606 1971

  46. [46]

    Sebastien Ott, A new perspective on the equivalence between W eak and S trong S pacial M ixing in two dimensions , 2025

    work page 2025

  47. [47]

    S\' e bastien Ott and Yvan Velenik, Potts models with a defect line, Comm. Math. Phys. 362 (2018), no. 1, 55--106. 3833604

    work page 2018

  48. [48]

    Leonard S Ornstein and F Zernike, Accidental deviations of density and opalescence at the critical point of a single substance, Proc. Akad. Sci. 17 (1914), 793

    work page 1914

  49. [49]

    12, 2680--2684

    Linus Pauling, The structure and entropy of ice and of other crystals with some randomness of atomic arrangement, Journal of the American Chemical Society 57 (1935), no. 12, 2680--2684

    work page 1935

  50. [50]

    48(2), Cambridge Univ Press, 1952, pp

    Renfrey Burnard Potts, Some generalized order-disorder transformations, Proceedings of the Cambridge Philosophical Society, vol. 48(2), Cambridge Univ Press, 1952, pp. 106--109

    work page 1952

  51. [51]

    Pfister and Y

    C.-E. Pfister and Y. Velenik, Random-cluster representation of the A shkin- T eller model , J. Statist. Phys. 88 (1997), no. 5-6, 1295--1331. 1478070

    work page 1997

  52. [52]

    Ralph Tyrell Rockafellar, Convex analysis, Princeton University Press, Princeton, 1970

    work page 1970

  53. [53]

    3, 1977--1988

    Gourab Ray and Yinon Spinka, A short proof of the discontinuity of phase transition in the planar random-cluster model with q>4 , Communications in Mathematical Physics 378 (2020), no. 3, 1977--1988

    work page 2020

  54. [54]

    36, Birkhauser Verlag AG Viadukstrasse 40-44, PO Box 133, CH-4010 Basel, Switzerland, 1963, p

    Franz Rys, \"U ber ein zweidimensionales klassisches K onfigurationsmodell , Helvetica Physica Acta, vol. 36, Birkhauser Verlag AG Viadukstrasse 40-44, PO Box 133, CH-4010 Basel, Switzerland, 1963, p. 537

    work page 1963

  55. [55]

    Oded Schramm, Scaling limits of loop-erased random walks and uniform spanning trees, Israel J. Math. 118 (2000), 221--288

    work page 2000

  56. [56]

    Stanislav Smirnov, Conformal invariance in random cluster models. I . H olomorphic fermions in the I sing model , Ann. of Math. (2) 172 (2010), no. 2, 1435--1467

    work page 2010

  57. [57]

    Wegner, D uality relation between the A shkin- T eller and the eight-vertex model , Journal of physics

    F. Wegner, D uality relation between the A shkin- T eller and the eight-vertex model , Journal of physics. C, Solid state physics 5 (1972), no. 11, L131–L132

    work page 1972

  58. [58]

    Wu, Ising model with four-spin interactions, Phys

    F.W. Wu, Ising model with four-spin interactions, Phys. Rev. B 4 (1971), 2312--2314

    work page 1971

  59. [59]

    F Zernike, The clustering-tendency of the molecules in the critical state and the extinction of light caused thereby, Koninklijke Nederlandse Akademie van Wetenschappen Proceedings Series B Physical Sciences 18 (1916), 1520--1527

    work page 1916