Ising Model: Local Spin Correlations and Conformal Invariance

Clément Hongler, Sung Chul Park
Springer, 2019
Article

We study the 2-dimensional Ising model at critical temperature on a simply connected subset of the square grid Z2. The scaling limit of the critical Ising model is conjectured to be described by Conformal Field Theory; in particular, there is expected to be a precise correspondence between local lattice fields of the Ising model and the local fields of Conformal Field Theory. Towards the proof of this correspondence, we analyze arbitrary spin pattern probabilities (probabilities of finite spin configurations occurring at the origin), explicitly obtain their infinite-volume limits, and prove their conformal covariance at the first (non-trivial) order. We formulate these probabilities in terms of discrete fermionic observables, enabling the study of their scaling limits. This generalizes results of Hongler (Conformal invariance of Ising model correlations. Ph.D. thesis, [Hon10]), Hongler and Smirnov (Acta Math 211(2):191-225, [HoSm13]), Chelkak, Hongler, and Izyurov (Ann. Math. 181(3), 1087-1138, [CHI15]) to one-point functions of any local spin correlations. We introduce a collection of tools which allow one to exactly and explicitly translate any spin pattern probability (and hence any lattice local field correlation) in terms of discrete complex analysis quantities. The proof requires working with multipoint lattice spinors with monodromy (including construction of explicit formulae in the full plane), and refined analysis near their source points to prove convergence to the appropriate continuous conformally covariant functions.

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Clément Hongler, Sung Chul Park
Springer, 2019
Article

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https://infoscience.epfl.ch/record/266627?ln=fr

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Conformal Invariance of Spin Pattern Probabilities in the Planar Ising Model

Clément Hongler, Sung Chul Park

We study the 2-dimensional Ising model at critical temperature on a smooth simply-connected graph Ω.We rigorously prove the conformal invariance of arbitrary spin-pattern probabilities centered at a point a and derive formulas to compute the probabilities as functions of the conformal map from Ω to the unit disk. Our methods extend those of [Hon10] and [CHI13] which proved conformal invariance of energy densities and spin correlations for points fixed far apart from each other. We use discrete complex analysis techniques and construct a discrete multipoint fermionic observable that takes values related to pattern probabilities in the planar Ising model. Refined analysis of the convergence of the discrete observable to a continuous, conformally covariant function completes the result.

2015

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Ising Model and Field Theory: Lattice Local Fields and Massive Scaling Limit

Sung Chul Park

This thesis is devoted to the study of the local fields in the Ising model. The scaling limit of the critical Ising model is conjecturally described by Conformal Field Theory. The explicit predictions for the building blocks of the continuum theory (spin and energy density) have been rigorously established [HoSm13, CHI15]. We study how the field-theoretic description of these random fields extends beyond the critical regime of the model. Concretely, the thesis consists of two parts: The first part studies the behaviour of lattice local fields in the critical Ising model. A lattice local field is a function of a finite number of spins at microscopic distances from a given point. We study one-point functions of these fields (in particular, their asymptotics under scaling limit and conformal invariance). Our analysis, based on discrete complex analysis methods, results in explicit computations which are of interest in applications (e.g. [HKV17]). The second part considers the behaviour of the massive spin field. In the subcritical massive scaling limit regime first considered by Wu, McCoy, Tracy, and Barouch [WMTB76], we show that the correlations of the massive spin field in a bounded domain have a scaling limit. Furthermore, to this end we generalise the notions and methods of discrete complex analysis in the critical case to the massive regime, and give a new derivation of the formula for the two-point correlation in the full plane in terms of a Painlevé III transcendent.

EPFL2019

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Conformal Invariance of Spin Correlations in the Planar Ising Model

Clément Hongler

We rigorously prove the existence and the conformal invariance of scaling limits of the magnetization and multi-point spin correlations in the critical Ising model on arbitrary simply connected planar domains. This solves a number of conjectures coming from the physical and the mathematical literature. The proof relies on convergence results for discreteholomorphic spinor observables and probabilistic techniques.

2015

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