Conformal invariance of spin correlations in~the~planar Ising model

Челкак, Д. С.; Hongler, Clément; Izyurov, Konstantin

doi:10.4007/annals.2015.181.3.5

Cited by 119 publications

(300 citation statements)

References 43 publications

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“…Recalling that v δ = δ 2 , relation (3.33), that was recently proved by Chelkak, Hongler and Izyurov [CHI12], yields immediately that for every n ∈ N and distinct x 1 , . .…”

Section: Proof For Random Field Ising Modelmentioning

confidence: 81%

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Polynomial chaos and scaling limits of disordered systems

Caravenna¹,

Sun²,

Zygouras³

2016

J. Eur. Math. Soc.

140

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Abstract. Inspired by recent work of Alberts, Khanin and Quastel [AKQ14a], we formulate general conditions ensuring that a sequence of multi-linear polynomials of independent random variables (called polynomial chaos expansions) converges to a limiting random variable, given by a Wiener chaos expansion over the d-dimensional white noise. A key ingredient in our approach is a Lindeberg principle for polynomial chaos expansions, which extends earlier work of Mossel, O'Donnell and Oleszkiewicz [MOO10]. These results provide a unified framework to study the continuum and weak disorder scaling limits of statistical mechanics systems that are disorder relevant, including the disordered pinning model, the (long-range) directed polymer model in dimension 1 + 1, and the two-dimensional random field Ising model. This gives a new perspective in the study of disorder relevance, and leads to interesting new continuum models that warrant further studies.

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“…Recalling that v δ = δ 2 , relation (3.33), that was recently proved by Chelkak, Hongler and Izyurov [CHI12], yields immediately that for every n ∈ N and distinct x 1 , . .…”

Section: Proof For Random Field Ising Modelmentioning

confidence: 81%

“…The "effective dimension" for this model is therefore d eff := 1+ [CGN12,CGN13], using breakthrough results on the scaling limit of correlation functions of the critical twodimensional Ising model, determined in [CHI12].…”

Section: Continuum Limits Of Disordered Systemsmentioning

confidence: 99%

Polynomial chaos and scaling limits of disordered systems

Caravenna¹,

Sun²,

Zygouras³

2016

J. Eur. Math. Soc.

140

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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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“…A number of results in this framework has been obtained in recent years: the convergence in the scaling limit has been shown for parafermionic observables [Smi06,Smi10a,ChSm09], for the energy correlations [HoSm10b,Hon10a] and for the spin correlations [ChIz11,CHI12]. These scaling limit results for correlations rely, for a large part, on discrete complex analysis.…”

Section: Introductionmentioning

confidence: 93%

“…The convergence of s-holomorphic observables is in particular the main tool to establish convergence of Ising interfaces to SLE [Smi06,CDHKS12,HoKy11], and to prove conformal invariance of the energy and the spin correlations [HoSm10b,Hon10a,CHI12].…”

Section: Introductionmentioning

confidence: 99%

Discrete holomorphicity and Ising model operator formalism

Hongler¹,

Kytölä²

2015

Analysis, Complex Geometry, and Mathematical Physics

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Abstract. We explore the connection between the transfer matrix formalism and discrete complex analysis approach to the two dimensional Ising model.We construct a discrete analytic continuation matrix, analyze its spectrum and establish a direct connection with the critical Ising transfer matrix. We show that the lattice fermion operators of the transfer matrix formalism satisfy, as operators, discrete holomorphicity, and we show that their correlation functions are Ising parafermionic observables. We extend these correspondences also to outside the critical point.We show that critical Ising correlations can be computed with operators on discrete Cauchy data spaces, which encode the geometry and operator insertions in a manner analogous to the quantum states in the transfer matrix formalism.

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Conformal invariance of spin correlations in~the~planar Ising model

Cited by 119 publications

References 43 publications

Polynomial chaos and scaling limits of disordered systems

Polynomial chaos and scaling limits of disordered systems

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

Discrete holomorphicity and Ising model operator formalism

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