96-343 C.-E. Pfister, Y. Velenik

Large Deviations and Continuum Limit in the 2D Ising model (281K, uuencoded gzipped PostScript) Jul 23, 96

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Abstract. We study the 2D Ising model in a rectangular box $\Lambda_L$ of linear size $O(L)$. We determine the exact asymptotic behaviour of the large deviations of the magnetization $\sum_{t\in\Lambda_L}\sigma(t)$ when $L\ra\infty$ 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 $\bra\,\sigma(0)\sigma(t)\,\ket$, as $\|t\|\ra\infty$, 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 $D\geq 2$ and up to the critical temperature. We then study the Gibbs measure conditionned by $\{\,|\,\sum_{t\in \Lambda_L}\sigma(t)-m|\Lambda_L|\,|\leq |\Lambda_L|L^{-c}\,\}$, 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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