Convergence of Ising interfaces to Schrammʼs SLE curves

Chelkak, Dmitry; Duminil-Copin, Hugo; Hongler, Clément; Kemppainen, Antti; Smirnov, Stanislav

doi:10.1016/j.crma.2013.12.002

Cited by 131 publications

(142 citation statements)

References 23 publications

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“…Thus, it follows from (4) and Lemma 2.12c that if E and E are 0-and 1-clusters of ω, respectively, then Int(E) ∩ Int(E ) ↓ = ∅. On the other hand, Int(E c ) = ∪ Int(E) over all c-clusters E of ω in γ, as follows from (5) and (6). We therefore conclude that Int(E 0 ) ∩ Int(E 1 ) ↓ = ∅.…”

Section: 4mentioning

confidence: 90%

Exponential Decay of Loop Lengths in the Loop O(n) Model with Large n

Duminil-Copin

Peled

Samotij

et al. 2016

Commun. Math. Phys.

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Abstract. The loop O(n) model is a model for a random collection of non-intersecting loops on the hexagonal lattice, which is believed to be in the same universality class as the spin O(n) model. It has been conjectured that both the spin and the loop O(n) models exhibit exponential decay of correlations when n > 2. We verify this for the loop O(n) model with large parameter n, showing that long loops are exponentially unlikely to occur, uniformly in the edge weight x. Our proof provides further detail on the structure of typical configurations in this regime. Putting appropriate boundary conditions, when nx 6 is sufficiently small, the model is in a dilute, disordered phase in which each vertex is unlikely to be surrounded by any loops, whereas when nx 6 is sufficiently large, the model is in a dense, ordered phase which is a small perturbation of one of the three ground states.

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Section: 4mentioning

confidence: 90%

Exponential Decay of Loop Lengths in the Loop O(n) Model with Large n

Duminil-Copin

Peled

Samotij

et al. 2016

Commun. Math. Phys.

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“…The scaling limit of the (static) two-dimensional Ising model with nearest-neighbor interactions is now well understood; see [8,12,13,53] and references therein. We may call this limit the (static, critical) continuous Ising model.…”

Section: Introductionmentioning

confidence: 99%

Convergence of the Two‐Dimensional Dynamic Ising‐Kac Model to

Mourrat

Weber

2016

Comm Pure Appl Math

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The Ising-Kac model is a variant of the ferromagnetic Ising model in which each spin variable interacts with all spins in a neighborhood of radius 1 for 1 around its base point. We study the Glauber dynamics for this model on a discrete two-dimensional torus Z 2 =.2N C 1/Z 2 for a system size N 1 and for an inverse temperature close to the critical value of the mean field model. We show that the suitably rescaled coarse-grained spin field converges in distribution to the solution of a nonlinear stochastic partial differential equation.This equation is the dynamic version of theˆ4 2 quantum field theory, which is formally given by a reaction-diffusion equation driven by an additive space-time white noise. It is well-known that in two spatial dimensions such equations are distribution valued and a Wick renormalization has to be performed in order to define the nonlinear term. Formally, this renormalization corresponds to adding an infinite mass term to the equation. We show that this need for renormalization for the limiting equation is reflected in the discrete system by a shift of the critical temperature away from its mean field value.

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“…In two dimensions and at the critical point, the scaling limit geometry of the boundaries of such clusters is known (see [7][8][9][10]26]) or conjectured (see [14,27]) to be described by some member of the one-parameter family of Schramm-Loewner evolutions (SLE κ with κ > 0) and related conformal loop ensembles (CLE κ with 8/3 < κ < 8). What makes SLEs and CLEs natural candidates is their conformal invariance, a property expected of the scaling limit of two-dimensional statistical mechanical models at the critical point.…”

Section: Introductionmentioning

confidence: 99%

Random walk loop soups and conformal loop ensembles

Brug¹,

Camia

Lis

2015

Probab. Theory Relat. Fields

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The random walk loop soup is a Poissonian ensemble of lattice loops; it has been extensively studied because of its connections to the discrete Gaussian free field, but was originally introduced by Lawler and Trujillo Ferreras as a discrete version of the Brownian loop soup of Lawler and Werner, a conformally invariant Poissonian ensemble of planar loops with deep connections to conformal loop ensembles (CLEs) and the Schramm-Loewner evolution (SLE). Lawler and Trujillo Ferreras showed that, roughly speaking, in the continuum scaling limit, "large" lattice loops from the random walk loop soup converge to "large" loops from the Brownian loop soup. Their results, however, do not extend to clusters of loops, which are interesting because the connection between Brownian loop soup and CLE goes via cluster boundaries. In this paper, we study the scaling limit of clusters of "large" lattice loops, showing that they converge to Brownian loop soup clusters. In particular, our results imply that the collection of outer boundaries of outermost clusters composed of "large" lattice loops converges to CLE. B Tim van de Brug

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Convergence of Ising interfaces to Schrammʼs SLE curves

Abstract: Abstract. We show how to combine our earlier results to deduce strong convergence of the interfaces in the planar critical Ising model and its random-cluster representation to Schramm's SLE curves with parameter κ = 3 and κ = 16/3 respectively.

Cited by 131 publications

References 23 publications

Exponential Decay of Loop Lengths in the Loop O(n) Model with Large n

Exponential Decay of Loop Lengths in the Loop O(n) Model with Large n

Convergence of the Two‐Dimensional Dynamic Ising‐Kac Model to

Random walk loop soups and conformal loop ensembles

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