Large deviations and continuum limit in the 2D Ising model

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 , 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∈Λ_L σ(t) −m|Λ_L| |≤|Λ_L| L^{−c} }, with 0<c<1/4 and −m *<m