The energy density in the planar Ising model

Hongler, Clément; Smirnov, Stanislav

doi:10.1007/s11511-013-0102-1

Cited by 83 publications

(163 citation statements)

References 28 publications

(48 reference statements)

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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%

“…If a is on the top boundary , a ∈ I * N , similar statements hold. The parafermionic observables can be defined similarly in any square lattice domain [HoSm10b]. At the critical point, β = β c , one can treat scaling limits as follows.…”

Section: Operator Insertions In the Ising Model Transfer Matrix Formamentioning

confidence: 99%

“…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%

“…One can then show that the solutions of discrete RBVPs converge to the solutions of continuous RBVPs, which are conformally covariant [Smi06,Smi10a,Hon10a,HoSm10b,ChSm09,ChIz11,CHI12].…”

Section: Introductionmentioning

confidence: 99%

“…Conversely, the s-holomorphic parafermionic observables of [Smi10a,ChSm09,HoSm10b,Hon10a] are indeed correlation functions of the fermion operators. …”

Section: Introductionmentioning

confidence: 99%

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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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Section: Introductionmentioning

confidence: 93%

Section: Operator Insertions In the Ising Model Transfer Matrix Formamentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…One can then show that the solutions of discrete RBVPs converge to the solutions of continuous RBVPs, which are conformally covariant [Smi06,Smi10a,Hon10a,HoSm10b,ChSm09,ChIz11,CHI12].…”

Section: Introductionmentioning

confidence: 99%

“…Conversely, the s-holomorphic parafermionic observables of [Smi10a,ChSm09,HoSm10b,Hon10a] are indeed correlation functions of the fermion operators. …”

Section: Introductionmentioning

confidence: 99%

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Discrete holomorphicity and Ising model operator formalism

Hongler¹,

Kytölä²

2015

Analysis, Complex Geometry, and Mathematical Physics

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Exponential Decay for the Near‐Critical Scaling Limit of the Planar Ising Model

Camia

Jian

Newman

2020

Comm Pure Appl Math

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We consider the Ising model at its critical temperature with external magnetic field ha 15=8 on the square lattice with lattice spacing a. We show that the truncated two-point function in this model decays exponentially with a rate independent of a as a # 0. As a consequence, we show exponential decay in the near-critical scaling limit Euclidean magnetization field. For the lattice model with a D 1, the mass (inverse correlation length) is of order h 8=15 as h # 0;for the Euclidean field, it equals exactly C h 8=15 for some C . Although there has been much progress in the study of critical scaling limits, results on nearcritical models are far fewer due to the lack of conformal invariance away from the critical point. Our arguments combine lattice and continuum FK representations, including coupled conformal loop and measure ensembles, showing that such ensembles can be useful even in the study of near-critical scaling limits. Thus we provide the first substantial application of measure ensembles.

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Slice regular functions of several octonionic variables

Ren

Yang

2020

Math Methods in App Sciences

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A slice theory of several octonionic variables is introduced in the article as a generalization of the holomorphic theory of several complex variables. Our new trick is to focus our attentions on some subset of O n with the same complex structures. In our setting, the Bochner-Martinelli formula and the Hartogs extension theorems are established for slice functions of several octonionic variables. KEYWORDS Bochner-Martinelli formula, Hartogs phenomena, octonions, slice regular functions MSC CLASSIFICATION 30G35; 31B10Math Meth Appl Sci. 2020;43:6031-6042.wileyonlinelibrary.com/journal/mma

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The energy density in the planar Ising model

Cited by 83 publications

References 28 publications

Discrete holomorphicity and Ising model operator formalism

Discrete holomorphicity and Ising model operator formalism

Exponential Decay for the Near‐Critical Scaling Limit of the Planar Ising Model

Slice regular functions of several octonionic variables

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