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arxiv: 2604.24743 · v1 · submitted 2026-04-27 · 🧮 math.PR · math-ph· math.MP

Recognition: unknown

The impact of disorder and non-convex interactions on delocalisation of height functions

Paul Dario , Diederik van Engelenburg , Christophe Garban
Authors on Pith no claims yet

Pith reviewed 2026-05-08 01:46 UTC · model grok-4.3

classification 🧮 math.PR math-phmath.MP
keywords quenched disorderBKT transitionroughening transitionheight functionsVillain modelXY modelpercolationannealed interactions
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The pith

Quenched disorder preserves the BKT and roughening transitions in XY, Villain, and height function models.

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

The paper studies the XY model, Villain model, and their dual height functions under specific forms of quenched disorder. It shows that the Berezinskii-Kosterlitz-Thouless transition persists on the infinite cluster of supercritical Bernoulli percolation, extending prior results from the XY model to the Villain model. The roughening transition in the dual integer-valued height models survives when neighboring heights are forced equal independently with probability below one half. For annealed interactions of the form absolute gradient to the p, with p between zero and two, the height functions exhibit a quantified rough phase. These results establish robustness of the transitions against the given disorders and non-elliptic potentials.

Core claim

The central discovery is that the BKT phase transition and order-disorder transition hold for the Villain model on supercritical percolation clusters, while the roughening transition persists in the XY height function and integer-valued Gaussian free field under dual disorder that enforces equal neighboring heights with probability less than 1/2. For annealed Gaussian interactions, including all potentials |∇h|^p with p in (0,2], a quantified rough phase is proved to exist.

What carries the argument

The key machinery consists of percolation-cluster reductions for the spin models combined with duality arguments for the height functions under enforced equalities, which control fluctuations and establish delocalisation bounds.

If this is right

  • The Villain model on the infinite percolation cluster undergoes the BKT transition at the same critical temperature as the clean model.
  • The XY height function and integer-valued GFF exhibit unbounded fluctuations above the roughening threshold under the dual disorder.
  • All annealed potentials |∇h|^p for p ≤ 2 produce a rough phase with explicit lower bounds on height variance.
  • The order-disorder transition in dimensions three and higher continues to hold under the percolation disorder for both XY and Villain models.

Where Pith is reading between the lines

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

  • The same percolation-cluster technique may extend to other short-range disorders that preserve a unique infinite cluster with positive density.
  • Quantitative rough-phase estimates could be checked numerically for p = 1 or p = 1.5 to test the uniformity in p.
  • The results suggest that delocalisation in these models is stable against moderate random constraints but may fail for stronger disorders that fragment the lattice.
  • Connections to interface models in random media could be explored by replacing the Gaussian annealed part with other convex or non-convex potentials.

Load-bearing premise

The disorder must be exactly supercritical Bernoulli percolation for the spin models or independent neighbor-height equalities with probability below 1/2 for the dual models, and the interactions are the standard or annealed forms without further modifications.

What would settle it

A direct computation or simulation showing that height variance remains bounded (localisation) for the dual model when neighbor equalities are enforced with probability 0.4 would disprove persistence of the roughening transition.

Figures

Figures reproduced from arXiv: 2604.24743 by Christophe Garban, Diederik van Engelenburg, Paul Dario.

Figure 1.1
Figure 1.1. Figure 1.1: An illustration of the phase transition for integer-valued height functions. view at source ↗
Figure 1.2
Figure 1.2. Figure 1.2: A two-dimensional shift-invariant graph. view at source ↗
Figure 1.3
Figure 1.3. Figure 1.3: The XY/Villain model: two spins are added on each edge of view at source ↗
Figure 2.1
Figure 2.1. Figure 2.1: The graph Z d , the extended graph Z d n (with n = 4) view at source ↗
Figure 2.2
Figure 2.2. Figure 2.2: The multigraph Z d n−mult (with n = 3). In the rest of this article, we will identify the vertices of Z d with the corresponding vertices of the extended lattice Z d n and of the multigraph Z d n−mult. We emphasize that two vertices x, y ∈ Z d which are neighbours on Z d are not neighbour on Z d n . We write x ∼n y to refer to pair of vertices which are neighbours in Z d n . 2.2 Angles An angle is an ele… view at source ↗
Figure 2.3
Figure 2.3. Figure 2.3: A site percolation configuration on the extended lattice. view at source ↗
Figure 3.1
Figure 3.1. Figure 3.1: The three spin systems of Section 3.1.1: the XY models on the extended lattice Z d n with n = 4 (on the left) and the XY/Villain model (on the right). • The third one is a spin system defined on the extended graph Z d 3 where the interaction on the edges connected to Z d is the one of an XY model with inverse temperature β1 and the interaction along the other edges is the one of a Villain model with inve… view at source ↗
Figure 3.2
Figure 3.2. Figure 3.2: The Villain and XY/Villain models as a limit of XY models. view at source ↗
Figure 3.3
Figure 3.3. Figure 3.3: Splitting an edge in the Villain model: an edge with conductance view at source ↗
Figure 3.4
Figure 3.4. Figure 3.4: An illustration of the proof of Lemma 3.13. The Villain model is approximated by a chain of XY models. Each edge of the chain is split into k = 2 edges with reduced conductances. Two parallel edges are then generated by reducing the value of the coupling contants from infinity to 0. By the Ginibre inequality, this operation reduces the two-point functions. Lemma 3.13 (Splitting an edge of the Villain mod… view at source ↗
Figure 4.1
Figure 4.1. Figure 4.1: The first two models of XY height functions introduced in Section view at source ↗
Figure 4.2
Figure 4.2. Figure 4.2: The Z-XY/GFF height function on the multigraph Z 2 3−mult. • The second one is an XY height function multigraph Z 2 n−mult with n ≥ 2 and with two inverse temperatures β1, nβ2 ≥ 0 (see view at source ↗
Figure 4.3
Figure 4.3. Figure 4.3: The convergence of the XY height function toward the integer-valued Gaussian free field view at source ↗
Figure 4.4
Figure 4.4. Figure 4.4: The box Λ g L (n) with L = 1 and n = 2. All the edges on the boundary of the box and the vertex 0 are connected to the ghost (in the case L = 1, all the vertices are connected to the ghost). 4.2.1. Duality between spin systems and height functions. We first recall some of the results of [vEL25]. We will only collect here the results of [vEL25] which are needed in order to establish Wells’ inequality for … view at source ↗
Figure 4.5
Figure 4.5. Figure 4.5: A spanning tree of the box Λ g L (n) (with L = 1 and n = 2). and Ω div−S 1 ΛL :=    θ ∈ Ω S 1 ΛL : ∀x ∈ ΛL ∪ {g}, X ⃗e∈E⃗x(Λ g L (n)) θ⃗e = 0 mod 2π    . (4.10) • We consider an (arbitrary) spanning tree T ⊆ Λ g L (n) and set Ω S 1 T := n θ : E⃗ (Λg L (n)) \ E⃗ (T) → [0, 2π) ; ∀⃗e ∈ E⃗ (Λg L (n)), θ−⃗e = −θ⃗e mod 2π o . Note that this space can be identified with the space of functions {θ : E(Λ… view at source ↗
Figure 4.6
Figure 4.6. Figure 4.6: An illustration of the lattice Z 2 (with dots and unbroken lines) together with the dual lattice (with crosses and dotted lines) 4.5 Delocalisation for the integer-valued Gaussian free field on a supercritical perco￾lation In this section, we implement the renormalisation argument used to prove the existence of a phase transition for the integer-valued height functions on any supercritical percolation (a… view at source ↗
Figure 4.7
Figure 4.7. Figure 4.7: A representation of the percolation configurations view at source ↗
Figure 4.8
Figure 4.8. Figure 4.8: An illustration of a good box: on the figure, two rectangles are drawn and they are view at source ↗
Figure 4.9
Figure 4.9. Figure 4.9: An illustration for the model introduced in Definition view at source ↗
Figure 4.10
Figure 4.10. Figure 4.10: An example of the site percolation on the renormalised lattice (closed sites are drawn view at source ↗
Figure 4.11
Figure 4.11. Figure 4.11: The collection of edges C ∗ of the dual lattice (Z 2 ) ∗ view at source ↗
Figure 4.12
Figure 4.12. Figure 4.12: The collection of edges of Z 2 which are dual to the edges of C ∗ . 53 view at source ↗
Figure 4.13
Figure 4.13. Figure 4.13: Reducing the values of the conductance of the edges which are not in view at source ↗
Figure 4.14
Figure 4.14. Figure 4.14: An example of rooted graph equipped with conductances view at source ↗
Figure 4.15
Figure 4.15. Figure 4.15: Identification of vertices: on this graph, the two vertices view at source ↗
Figure 4.16
Figure 4.16. Figure 4.16: Addition of vertices on an edge: this operation reduces the variance of the height of view at source ↗
Figure 4.17
Figure 4.17. Figure 4.17: The graph surgery used in the proof of Theorem view at source ↗
Figure 4.18
Figure 4.18. Figure 4.18: The graph surgery used in the proof of Theorem view at source ↗
read the original abstract

We study the behaviour of four spins systems (the XY model, the Villain model, the XY height function and the integer-valued Gaussian free field) in the presence of a non-elliptic quenched disorder. In the article [DG25], it was shown that the phase transitions of the XY model (the Berezinskii-Kosterlitz-Thouless phase transition in $d = 2$ and the order/disorder phase transition when $d \geq 3$) persist on the infinite cluster of a supercritical Bernoulli percolation. A first objective of this article is to extend these results to the Villain model. Our second objective is to analyse, for $d=2$, how the corresponding dual integer-valued height function models behave in the presence of a dual quenched disorder. These dual models are respectively the XY height function and the integer-valued Gaussian free field. Without disorder, these models are known to exhibit a phase transition in two dimensions called the roughening transition [FS81, Lam22b]. We show that this phase transition persists when the quenched disorder is given by enforcing $\varphi(x) = \varphi(y)$ independently with probability $\bar{p} < 1/2$ for neighboring sites $x, y$. Finally, we apply our methods to integer-valued height functions with annealed Gaussian interactions and prove the existence of a (quantified) rough phase. This includes all potentials of the form $|\nabla h|^p$ for $p \in (0, 2]$, recovering recent results of [OS25].

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 / 3 minor

Summary. The manuscript extends prior results on the BKT phase transition of the XY model under supercritical Bernoulli percolation disorder to the Villain model. In d=2 it shows persistence of the roughening transition for the dual XY height function and integer-valued GFF under independent bond-enforcement disorder (ϕ(x)=ϕ(y) with probability p̄<1/2). It further establishes a quantified rough phase for integer-valued height functions with annealed interactions, including all potentials of the form |∇h|^p for p∈(0,2], recovering and extending results of OS25.

Significance. If the central claims hold, the work is a solid incremental advance in the rigorous analysis of disordered lattice models. It demonstrates robustness of the BKT and roughening transitions to the specified quenched disorders by adapting techniques from DG25, FS81, Lam22b and OS25, and supplies explicit quantification of the rough phase for a family of non-convex annealed interactions. These are concrete, falsifiable statements that build directly on the cited literature without introducing new internal inconsistencies.

minor comments (3)
  1. Abstract: the phrase 'four spins systems' is imprecise; the Villain model is listed separately from the XY model and the two height-function models, so the enumeration should be clarified for accuracy.
  2. Section on dual models: the precise range of p̄ (strictly less than 1/2) and the independence assumption on the enforcement events should be restated explicitly when the dual disorder is introduced, to make the statement of the roughening-transition persistence self-contained.
  3. Section on annealed interactions: the quantified bounds on the rough phase for |∇h|^p (p∈(0,2]) should be compared briefly with the corresponding statements in OS25 so that the improvement or recovery is transparent to the reader.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading, accurate summary of our results, and recommendation for minor revision. The assessment that the work represents a solid incremental advance is appreciated.

Circularity Check

0 steps flagged

Minor self-citation of prior work by overlapping authors, but central claims are independent extensions

full rationale

The paper extends the BKT and roughening persistence results from the cited [DG25] (by two of the present authors) to the Villain model and to dual height-function models under specific dual disorder, while also treating annealed non-convex interactions and recovering [OS25]. These extensions rely on standard model definitions and adaptations of existing techniques rather than any derivation that reduces by construction to a fitted parameter, self-definition, or unverified self-citation chain. No load-bearing step collapses to its own inputs; the work is self-contained against the external benchmarks of the cited prior results.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard background from probability theory, percolation, and statistical mechanics. No free parameters are introduced or fitted. No new entities are postulated.

axioms (2)
  • standard math Standard definitions and properties of the XY model, Villain model, and integer-valued Gaussian free field on Z^d hold as in the cited literature.
    Invoked implicitly when extending results from [DG25] and [FS81].
  • domain assumption Supercritical Bernoulli percolation on Z^d has a unique infinite cluster with positive density.
    Used for the first objective on the infinite cluster.

pith-pipeline@v0.9.0 · 5591 in / 1486 out tokens · 61690 ms · 2026-05-08T01:46:45.437986+00:00 · methodology

discussion (0)

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Reference graph

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