Logarithmic variance for the height function of square-ice

Duminil-Copin, Hugo; Harel, Matan; Laslier, Benoît; Raoufi, Aran; Ray, Gourab

doi:10.48550/arxiv.1911.00092

Cited by 9 publications

(35 citation statements)

References 29 publications

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“…For the result on the square ice, we also refer to [10]. It was later shown that conditional on this delocalisation, the delocalisation is logarithmic: the previously mentioned dichotomy result [14]. These results have recently been extended to the more general six-vertex model for a = b = 1 and 1 ≤ c ≤ 2, see [15].…”

Section: Introductionmentioning

confidence: 83%

“…Second, we demonstrate that the absolute value of the height function satisfies the FKG lattice condition whenever nonnegative boundary conditions are imposed (Theorem 2.8). The property holds true for both the solid-on-solid model as well as for the discrete Gaussian model, and may be of independent interest given the recent successes that it has had in the context of the square ice and the six-vertex model in proving logarithmic delocalisation [14,15]. The general strategy of these articles stems from an older work [17] where the FKG lattice condition is used to prove a dichotomy for the quantitative behaviour at and off criticality in the random-cluster model.…”

Section: Introductionmentioning

confidence: 92%

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Delocalisation and absolute-value-FKG in the solid-on-solid model

Lammers¹,

Ott²

2021

Preprint

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The solid-on-solid model is a model of height functions, introduced to study the interface separating the + and − phase in the Ising model. The planar solidon-solid model thus corresponds to the three-dimensional Ising model. Delocalisation of this model at high temperature and at zero slope was first derived by Fröhlich and Spencer, in parallel to proving the Kosterlitz-Thouless phase transition. The first main result of this article consists of a simple, alternative proof for delocalisation of the solid-on-solid model. In fact, the argument is more general: it works on any planar graph-not just the square lattice-and implies that the interface delocalises at any slope rather than exclusively at the zero slope. The second main result is that the absolute value of the height function in this model satisfies the FKG lattice condition. This property is believed to be intimately linked to the (quantitative) understanding of delocalisation, given the recent successes in the context of the square ice and (more generally) the six-vertex model. The new FKG inequality is shown to hold true for both the solid-on-solid model as well as for the discrete Gaussian model, which implies immediately that the two notions of delocalisation, namely delocalisation in finite volume and delocalisation of shift-invariant Gibbs measures, coincide.

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Section: Introductionmentioning

confidence: 83%

Section: Introductionmentioning

confidence: 92%

Delocalisation and absolute-value-FKG in the solid-on-solid model

Lammers¹,

Ott²

2021

Preprint

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“…The work [31] extends this result to prove that the variance is of order log(L). It is further conjectured that the scaling limit of h L is the continuum Gaussian free field, and that the level lines of h L (see Figure 3) scale to the Conformal Loop Ensemble (CLE) with parameter κ = 4.…”

Section: Height Functionmentioning

confidence: 67%

“…which is a form of correlation decay. While (19) does not distinguish between the disordered and critical cases (in the sense discussed in Section 1.2), recent work of Duminil-Copin-Harel-Laslier-Raoufi-Ray [31] can be used to obtain a rate of convergence in (19),…”

Section: Constant Boundary Conditionsmentioning

confidence: 99%

“…Proof of Lemma 2.7. Let (f, g) be as in the lemma and ( f , g) be defined by (31) so that (f, g) and ( f , g) have the same distribution. In particular, g has the same distribution as g. Thus, f and g form a coupling of µ 1 and µ 2 which, by its definition and the assumption (29) satisfies that f ≥ g everywhere, almost surely.…”

Section: Disagreement Percolation and Cluster Swappingmentioning

confidence: 99%

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Three lectures on random proper colorings of $\mathbb{Z}^d$

Peled¹,

Spinka²

2020

Preprint

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A proper q-coloring of a graph is an assignment of one of q colors to each vertex of the graph so that adjacent vertices are colored differently. Sample uniformly among all proper q-colorings of a large discrete cube in the integer lattice Z d . Does the random coloring obtained exhibit any large-scale structure? Does it have fast decay of correlations? We discuss these questions and the way their answers depend on the dimension d and the number of colors q. The questions are motivated by statistical physics (anti-ferromagnetic materials, square ice), combinatorics (proper colorings, independent sets) and the study of random Lipschitz functions on a lattice. The discussion introduces a diverse set of tools, useful for this purpose and for other problems, including spatial mixing, entropy and coupling methods, Gibbs measures and their classification and refined contour analysis.

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On the Six-Vertex Model’s Free Energy

Duminil-Copin

Kozłowski

Krachun

et al. 2022

Commun. Math. Phys.

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In this paper, we provide new proofs of the existence and the condensation of Bethe roots for the Bethe Ansatz equation associated with the six-vertex model with periodic boundary conditions and an arbitrary density of up arrows (per line) in the regime $$\Delta <1$$ Δ < 1 . As an application, we provide a short, fully rigorous computation of the free energy of the six-vertex model on the torus, as well as an asymptotic expansion of the six-vertex partition functions when the density of up arrows approaches 1/2. This latter result is at the base of a number of recent results, in particular the rigorous proof of continuity/discontinuity of the phase transition of the random-cluster model, the localization/delocalization behaviour of the six-vertex height function when $$a=b=1$$ a = b = 1 and $$c\ge 1$$ c ≥ 1 , and the rotational invariance of the six-vertex model and the Fortuin–Kasteleyn percolation.

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Logarithmic variance for the height function of square-ice

Cited by 9 publications

References 29 publications

Delocalisation and absolute-value-FKG in the solid-on-solid model

Delocalisation and absolute-value-FKG in the solid-on-solid model

Three lectures on random proper colorings of $\mathbb{Z}^d$

On the Six-Vertex Model’s Free Energy

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