Summary.
We study the 2D Ising model in a rectangular box Λ L of linear size O(L). We determine the exact asymptotic behaviour of the large deviations of the magnetization ∑ t∈ΛL σ(t) when L→∞ for values of the parameters of the model corresponding to the phase coexistence region, where the order parameter m * is strictly positive. We study in particular boundary effects due to an arbitrary real-valued boundary magnetic field. Using the self-duality of the model a large part of the analysis consists in deriving properties of the covariance function <σ(0)σ(t)>, as |t|→∞, at dual values of the parameters of the model. To do this analysis we establish new results about the high-temperature representation of the model. These results are valid for dimensions D≥2 and up to the critical temperature. They give a complete non-perturbative exposition of the high-temperature representation.
We then study the Gibbs measure conditioned by {|∑ t∈ΛL σ(t) −m|Λ L ||≤|Λ L |L − c}, with 0<c<1/4 and −m *<m<m *. We construct the continuum limit of the model and describe the limit by the solutions of a variational problem of isoperimetric type.
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Received: 17 October 1996 / In revised form: 7 March 1997
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Pfister, CE., Velenik, Y. Large deviations and continuum limit in the 2D Ising model. Probab Theory Relat Fields 109, 435–506 (1997). https://doi.org/10.1007/s004400050139
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DOI: https://doi.org/10.1007/s004400050139