Abstract
We study the 2-dimensional Ising model at critical temperature on a simply connected subset {Omega_{delta}} of the square grid {deltamathbb{Z}^{2}}. 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.
Highlights
The 2D Ising model is one of the most studied models of statistical mechanics
In its simplest formulation it consists of a random assignment of ±1 spins σx to the faces of the square grid Z2 (Fig. 1); the spins tend to align with their neighbors; the probability of a configuration is proportional to e−β H(σ ) where the energy H (σ ) = − i∼ j σi σ j sums over pairs of adjacent faces; alignment strength is controlled by the parameter β > 0, usually identified with the inverse temperature
Of particular physical interest is the phase transition at the critical point βc: for β < βc the system is disordered at large scales while for β > βc a long-range ferromagnetic order arises
Summary
The 2D Ising model is one of the most studied models of statistical mechanics. In its simplest formulation it consists of a random assignment of ±1 spins σx to the faces of (subgraphs of) the square grid Z2 (Fig. 1); the spins tend to align with their neighbors; the probability of a configuration is proportional to e−β H(σ ) where the energy H (σ ) = − i∼ j σi σ j sums over pairs of adjacent faces; alignment strength is controlled by the parameter β > 0, usually identified with the inverse temperature. Theorems 1.1–1.2 give the infinite-volume limits, and first-order CFT corrections of the one point function of any lattice local field ΦδF (x) in terms of those of spin-symmetric and spin-antisymmetric pattern fields, whose one-point functions can be obtained explicitly We believe extending this to multi-field correlations of fields with scaling dimension D ≤ 1 should carry over from [Hon10,CHI15]. Explicit computations Explicit calculation of infinite-volume limits and finitesize corrections of pattern probabilities in the critical Ising model is of general interest, and may be useful for the program of Application 1.3.2 The lower-case versions of the spin-fermions above again are their continuous counterparts, and when there is a † superscript, that denotes the difference of the bounded-domain and full-plane spin-fermions
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