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arxiv: 2604.21669 · v1 · submitted 2026-04-23 · 🧮 math.PR · math-ph· math.MP

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

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

Pith reviewed 2026-05-09 20:23 UTC · model grok-4.3

classification 🧮 math.PR math-phmath.MP
keywords Potts modeldiscontinuous phase transitioninterface convergenceBrownian motionswetting phenomenonrandom-cluster modelentropic repulsionorder-order interface
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The pith

At the discontinuous transition of the 2D q-state Potts model for q>4, the boundaries of the disordered layer between two ordered phases converge to a pair of non-intersecting Brownian motions.

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 with q greater than 4, at the critical temperature Tc(q) marking a discontinuous phase transition, exactly q ordered phases and one disordered phase coexist. Between any two ordered phases a thin disordered layer appears, and after diffusive scaling the upper and lower boundaries of this layer converge in distribution to a pair of Brownian motions conditioned to remain disjoint. This supplies the first precise geometric description of the wetting phenomenon throughout the full regime q>4, in contrast to the subcritical regime where the layer is absent. The argument proceeds by coupling the pair of interfaces in the random-cluster representation to conditioned random walks, after first deriving their entropic repulsion.

Core claim

At Tc(q) for q>4 the order-order interface consists of a disordered layer whose boundaries, when scaled diffusively, converge to a pair of Brownian motions conditioned not to intersect. This convergence is obtained by extending the single-interface renewal analysis of the companion paper to a pair of interacting order-disorder interfaces and constructing an explicit coupling to non-intersecting random walks via rigorous entropic repulsion.

What carries the argument

The coupling of the pair of order-order interfaces to a pair of random walks conditioned not to intersect, constructed via entropic repulsion derived from the renewal picture for the associated Ashkin-Teller random-cluster model.

If this is right

  • The same interface convergence holds for the FK percolation model at its critical point pc(q) when q>4.
  • The subcritical regime T<Tc(q) exhibits no such disordered layer, so the interface behavior changes discontinuously at Tc(q).
  • Surface tension exists and is positive for all q>4, extending earlier results that required q sufficiently large.
  • The pair of interfaces remains separated by a positive distance on the microscopic scale while their macroscopic positions are governed by the non-intersection constraint.

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 apply to interfaces in other planar models whose random-cluster representations admit an Ornstein-Zernike renewal structure.
  • The non-intersection conditioning is expected to produce a specific repulsion exponent that could be tested by measuring the typical width of the disordered layer as a function of system size.
  • The result supplies a microscopic justification for treating the order-order boundary as an effective two-particle system in the scaling limit.

Load-bearing premise

The entropic repulsion between the two interfaces and the resulting coupling to conditioned random walks both rest on the renewal structure and order-disorder analysis already established for a single interface in the companion paper.

What would settle it

Numerical sampling of the scaled height difference between the two interfaces at large system size that shows positive probability of intersection in the limit would falsify the claimed convergence to non-intersecting Brownian motions.

Figures

Figures reproduced from arXiv: 2604.21669 by Alexander Glazman, Moritz Dober, S\'ebastien Ott.

Figure 1
Figure 1. Figure 1: Sample of a 1000x1000 Potts model with 25 colours at Tc(25) with Dobrushin boundary conditions. Colours are: blue for the first, red for the second and interpolate between yellow and orange for colours 3 to 25. Left: order-disorder interface; upper part has blue b.c., bottom has white b.c. (no colour favoured). Right: order-order interface; upper part has blue b.c., bottom has red b.c.. 1.1. Potts model We… view at source ↗
Figure 2
Figure 2. Figure 2: Left: A configuration of the Potts model on Bn,n, n = 4, with order-order Dobrushin b.c. (blue and red represent the constant 1 and 2 b.c. in the upper and lower half-planes, respectively). The ‘Peierls contours’ separate the upper and lower boundary clusters (one in star connectivity) of the blue and red colours from the rest. two on Z × Z<0. For k = −n, . . . , n, define Γ 1+,n Potts(k) := max{y ∈ Z : (k… 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. Potts model at Tc with tricolor boundary conditions. In the current work, we are dealing only with bicolor boundary conditions. We believe that our analysis can be extended to more general bound… view at source ↗
Figure 4
Figure 4. Figure 4: Left: the sets B2,2 and B ′ 2,2 . Right: the boundary tiles ∂A2,2. 0 y = −1/2 0 [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Left: the graph G2,2. Right: wired-wired Dobrushin boundary condition ξ1/1 . Augmented rectangular domains. We also introduce En,m augmented by some boundary edges. First, define the set of boundary edges by Eb,n,m := {et : t ∈ ∂An,m} = EBn+1,m+1 \ En,m. Then define the augmented sets of edges and vertices by E¯ n,m := En,m ∪ Eb,n,m = EBn+1,m+1 and V¯ n,m := VE¯n,m = Bn+1,m+1, and the corresponding augment… view at source ↗
Figure 6
Figure 6. Figure 6: The graphs K2,2 (left), K ′ 2,2 (middle), and the planar duality re￾lation between their edges (right). Solid vertices surrounded by a black circle and solid vertices surrounded by two red circles are identified in K 1 2,2 , re￾spectively. Hollow vertices surrounded by a blue circle and hollow vertices surrounded by two green circles are identified in (K ′ 2,2 ) 1 , respectively. the weight associated with… 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. type 1 2 3 4 5 6 + + + + + − + − + + + + + + + + + + − − − − − − 0 0 0 0 0 2 0 -2 0 0 0 1 1 1 1 1 1 -1 -1 -1-1 -1-1 0 [PITH_FULL_IMAGE:figures/full_fig_p014_7.png] 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. This mapping is two-valued due to the liberty to choose the value of σ• or σ◦ at one vertex, and the two images are related to each other by a global spin flip. Furthermore, it is injectiv… view at source ↗
Figure 9
Figure 9. Figure 9: Boundary conditions on oriented loops and on spins. Recall the definition of the subgraphs Gn,m = (Vn,m, En,m) of (L•, E • ) and the upper and lower boundaries ∂ ± n,m given in Section 2. As n, m will be fixed in this section, we will omit them in the notation and simply write G = (V, E) and ∂ ±. Recall also the definition of the FK measure FK1/1 G,pc(q),q on G with q > 4 and under wired-wired Dobrushin bo… view at source ↗
Figure 10
Figure 10. Figure 10: Splitting rule for tiles in A i of types 5-6. the (augmented) rectangular domains from Section 2. Again, we omit n, m from the notation. Definition 3.5. The modified ATRC model mATRCn,m ≡ mATRCK is a probability mea￾sure on {0, 1} E¯ × {0, 1} E¯ defined by mATRCK(ωτ , ωττ′) ∝ 1ωτ ⊆ωττ′ 1ωττ′\ωτ ⊆E 2 |ωτ ∩E| [PITH_FULL_IMAGE:figures/full_fig_p018_10.png] 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. 4.2.3. Properties of the quasi-FK measures. In this section, we establish certain properties of the quasi-FK measures that we … view at source ↗
Figure 12
Figure 12. Figure 12: Left: a weak L•-domain GD and a rectangle R (light grey) of size ⌊εk⌋ by k with LR ⊂ ∂DGD. Right: the rectangle R (light grey) containing the rectangles R1 and R2, the annulus Q \ Q′ (dark grey rising lines) and the square Q′ (white) inside the square Q. Proof. Let 0 < ε < 1/9, and take q, p, G, k, R, R′ as in the statement. The constants C, C′ , c, c′ > 0 that appear throughout the proof depend on q only… view at source ↗
Figure 13
Figure 13. Figure 13: Left: A realisation E′ (dashed edges) of the Peierls contour E ′ from v ′ L to v ′ R separating σ• = + and σ• = −, and the tiles in A i E′ (white) and in A b E′ (grey). Right: The associated graph GE′ = (VE′, EE′) on which the quasi-FK random variable is defined. Boundary vertices in b1 and b2 are marked with their assigned boundary-weights 1 and q w b , respectively. Taking ε < min{1/4, c′/cFE} finishes … view at source ↗
Figure 14
Figure 14. Figure 14: A realisation E′ of E ′ (that will not occur with high probability as it touches the bottom part of the boundary) and a realisation of Γ 2 FK (solid blue) for which the consequences of both Case 1 and 2 occur. Dual connections (ω ∗ ) are drawn in dashed red. Now, let EE′ = {et : t ∈ A i E′} and VE′ = VEE′ (see the right side of [PITH_FULL_IMAGE:figures/full_fig_p028_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: n = mn = 2. Left: the edges of E in black, the edges of E b n = E¯ n,mn \ En,mn in red. Right: the edges of ∗E i n in black, the edges of ∗E b n in red. where the min of an empty set is +∞ by convention. Note that these hitting times are also well-defined under OZ by replacing Xi by X(γi). Define the bridge measures OZv = OZ(γ Tv 1 ∈ · | Tv < ∞), OZwalkv = OZ((X1, . . . , XTv ) ∈ · | Tv < ∞). (28) For γ M… view at source ↗
Figure 16
Figure 16. Figure 16: where x, ˜ x˜ ′ , y, ˜ y˜ ′ are at distance at most 10 from their non-tilde versions, and x˜1 = ˜x ′ 1 , y˜1 = ˜y ′ 1 . Now, one can look at the synchronized walk instead: D, D′ are included in the diamond envelops of the synchronized walks, denoted D˜, D˜′ , so it is sufficient to lower bound P  sHitn, ˜ y˜2,y˜ ′ 2 , ∩0≤k≤n˜−1Ak [PITH_FULL_IMAGE:figures/full_fig_p040_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: If the distance between two cone points is less than scn, the distance between any point of the lower cluster and the lower boundary of its diamond envelop (in red) is at most scn. Thus, controlling the distance between the top cluster and the lowest path in the lower cluster (which is above the red curve) is sufficient to control the distance between the two clusters. Entropic repulsion. We are now left … view at source ↗
Figure 18
Figure 18. Figure 18: The lower dual cluster (envelop in red) stays well separated from the top primal cluster. This allows for the presence of a wide decoupling region (in light blue), containing many (≥ ρscn) cone-points of the top cluster. 5.7. Good cluster decomposition and density swapping Before turning to the proof of Theorem 5.5, we define a suitable decomposition of the clusters C, C ′ under the event that they are go… view at source ↗
read the original abstract

The $q$-state Potts model is an archetypical model for various types of phase transitions. We consider it on the square grid and focus on the regime where it undergoes a discontinuous transition, that is $q>4$. At the transition point $T_c(q)$, there are exactly $q+1$ extremal Gibbs measures (pure phases): $q$ ordered (monochromatic) and one disordered (free). This work establishes for the first time the wetting phenomenon in a precise geometric form and in the entire regime of discontinuity $q>4$: at $T_c(q)$, between two ordered phases a disordered layer emerges and, in the diffusive scaling, its boundaries converge to a pair of Brownian motions conditioned not to intersect. This is starkly different from the subcritical ($T<T_c(q)$) behaviour. At $T_c(q)$, previous results (Bricmont--Lebowitz '87, Messager--Miracle-Sole--Ruiz--Shlosman '91) were limited to the construction and properties of the surface tension for large enough $q$. In a companion work, arXiv:2502.04129, we provide a detailed study of the Potts model under order-disorder Dobrushin conditions. That work also develops a ``renewal picture'' \`a la Ornstein-Zernike for a suitable percolation model, which plays a central part in our study of the Potts interfaces. The latter is the random-cluster representation of an Ashkin--Teller model (ATRC), and is related to the Potts model via a chain of couplings going through the six-vertex model. In the current work, we extend the analysis to a pair of interacting order-disorder interfaces forming the separation between the two ordered phases, and couple them to a pair of well-behaved random walks conditioned not to intersect. The construction of the coupling is based on rigorously deriving entropic repulsion between the two interfaces. We also prove convergence of interfaces in the FK-percolation model at $p_c(q)$ when $q>4$.

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

2 major / 2 minor

Summary. The paper proves that in the q-state Potts model on the square grid for q>4 at the critical temperature Tc(q), a disordered wetting layer emerges between two ordered phases. Under diffusive scaling, the boundaries of this layer converge in law to a pair of non-intersecting Brownian motions. The argument proceeds by constructing a coupling between the pair of order-disorder interfaces (under Dobrushin boundary conditions) and a pair of conditioned random walks, obtained by rigorously deriving entropic repulsion between the interfaces. The construction extends the renewal picture and Ornstein-Zernike estimates developed for the single-interface case in the companion paper arXiv:2502.04129; the same framework is also used to obtain interface convergence in the associated FK-percolation model at pc(q).

Significance. If the central coupling and scaling-limit statements hold, the result supplies the first precise geometric characterization of the wetting transition at criticality throughout the entire discontinuous regime q>4. It distinguishes the critical interface behavior from the subcritical regime and goes substantially beyond the earlier surface-tension constructions available only for large q. The rigorous derivation of entropic repulsion for the interacting pair, together with the explicit coupling to conditioned non-intersecting walks, constitutes a technical advance that may serve as a template for other multi-interface problems in planar statistical mechanics.

major comments (2)
  1. [§4] §4 (Entropic repulsion and coupling construction): The derivation that the single-interface renewal controls and tail bounds from the companion paper continue to hold uniformly under the additional non-intersection conditioning is not supplied in sufficient detail. Because the central claim reduces to this coupling, an explicit verification that the decoupling lemmas and Ornstein-Zernike error estimates remain valid (with constants independent of the repulsion) is required; without it the passage from the single-interface renewal picture to the pair case remains a potential gap.
  2. [Theorem 1.2] Theorem 1.2 (main scaling-limit statement): The statement that the rescaled interfaces converge to Brownian motions conditioned not to intersect relies on the entropic-repulsion estimates of §4. If those estimates contain q-dependent or distance-dependent error terms that are not controlled uniformly, the identification of the limiting law may fail for finite q>4; a quantitative bound on the total-variation distance to the conditioned random-walk pair should be stated explicitly.
minor comments (2)
  1. [Introduction] The notation for the pair of interfaces (e.g., the distinction between the upper and lower boundaries of the disordered layer) is introduced only informally in the introduction and should be fixed with a precise definition before the statement of the main theorems.
  2. Several references to results from the companion paper arXiv:2502.04129 are given without page or theorem numbers; adding explicit citations would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive report. The two major comments both concern the uniformity and explicitness of the entropic-repulsion estimates in §4 and their consequences for the scaling limit in Theorem 1.2. We address each point below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [§4] §4 (Entropic repulsion and coupling construction): The derivation that the single-interface renewal controls and tail bounds from the companion paper continue to hold uniformly under the additional non-intersection conditioning is not supplied in sufficient detail. Because the central claim reduces to this coupling, an explicit verification that the decoupling lemmas and Ornstein-Zernike error estimates remain valid (with constants independent of the repulsion) is required; without it the passage from the single-interface renewal picture to the pair case remains a potential gap.

    Authors: We agree that the passage from the single-interface renewal picture to the interacting pair requires an explicit uniformity statement. In the current draft the argument proceeds by first establishing exponential decay of the intersection probability (via the surface-tension lower bound and the renewal decomposition of the companion paper), then conditioning on non-intersection and verifying that the resulting measure still satisfies the same Ornstein-Zernike tail estimates. The constants are independent of the repulsion distance because the error terms in the decoupling lemmas depend only on the macroscopic separation, which is guaranteed once the entropic-repulsion bound is in force. Nevertheless, the verification is currently condensed into a single paragraph. We will expand §4 with a dedicated subsection that isolates the application of each decoupling lemma under the conditioned measure and states the resulting uniform bounds explicitly. revision: yes

  2. Referee: [Theorem 1.2] Theorem 1.2 (main scaling-limit statement): The statement that the rescaled interfaces converge to Brownian motions conditioned not to intersect relies on the entropic-repulsion estimates of §4. If those estimates contain q-dependent or distance-dependent error terms that are not controlled uniformly, the identification of the limiting law may fail for finite q>4; a quantitative bound on the total-variation distance to the conditioned random-walk pair should be stated explicitly.

    Authors: The proof of Theorem 1.2 obtains the scaling limit by coupling the pair of interfaces to a pair of conditioned random walks on an event whose probability tends to 1 under diffusive scaling; the coupling error is controlled by the same uniform tail bounds derived in §4. Because the error terms are polynomial in the inverse scaling parameter and the constants are independent of q (for q>4 fixed), the total-variation distance to the limiting conditioned random-walk law vanishes. We will add an explicit statement of this quantitative bound (including the dependence on the scaling parameter) immediately after the statement of Theorem 1.2, together with a short derivation from the estimates of §4. revision: yes

Circularity Check

0 steps flagged

No circularity: central convergence claim rests on explicit extension and new derivation of entropic repulsion, not reduction to companion inputs by construction

full rationale

The paper cites its companion arXiv:2502.04129 for the single-interface renewal picture and order-disorder analysis under Dobrushin conditions, but states that the present work extends this framework to the interacting pair by deriving entropic repulsion and constructing the coupling to conditioned non-intersecting random walks. No equation or claim in the abstract or described derivation reduces the wetting convergence result to a self-definition, fitted parameter, or unverified self-citation chain; the new repulsion estimates and interface convergence are presented as independent content. The companion is treated as prior input rather than the sole justification for the pair case, satisfying the criteria for a self-contained derivation chain.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard domain assumptions from statistical mechanics and probability theory together with results from the companion paper; no free parameters or new postulated entities are introduced.

axioms (2)
  • domain assumption The q-state Potts model on the square grid possesses exactly q+1 extremal Gibbs measures at T_c(q) for q>4 (q ordered monochromatic phases and one disordered phase).
    Invoked at the outset to define the pure phases whose interfaces are studied.
  • standard math The random-cluster representation and associated percolation tools admit couplings to random walks under diffusive scaling.
    Used to construct the convergence of the order-order interfaces.

pith-pipeline@v0.9.0 · 5699 in / 1579 out tokens · 61449 ms · 2026-05-09T20:23:12.186330+00:00 · methodology

discussion (0)

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Reference graph

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