Large deviations and continuum limit in the 2D Ising model |
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Authors: | C.-E. Pfister Y. Velenik |
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Affiliation: | (1) Département de Mathématiques, EPF-L, CH-1015 Lausanne, Switzerland e-mail: cpfister@eldp.epfl.ch; velenik@eldp.epfl.ch, CH |
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Abstract: | 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. Received: 17 October 1996 / In revised form: 7 March 1997 |
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Keywords: | AMS Subject Classification: (1991) 60F10 60G60 60K35 82B20 82B24 |
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