Exact large deviation bounds up to \(T_ c\) for the Ising model in two dimensions
Consider the Ising model on \(Z^2\). Let \(\Lambda\) be a square box of volume \(L^2\). Let \(P^\beta_{\Lambda, +}\) be the corresponding Gibbs measure on \(\Lambda\) with boundary conditions and inverse temperature \(\beta\). Define the block-spin magnetization \(X_\Lambda\) by \(X_\Lambda = {1 \over |\Lambda |} \sum_{x \in \Lambda} \sigma_x\), where \(\sigma_x = \pm 1\). Since \(P^\beta_+\) is ergodic, \(X_\Lambda\) converges to \(m^*\) \(P^\beta_+\)-a.s. The main result of the present paper is the following Theorem. Let \(m \in [-m^*, m^*]\) and \(\beta > \beta_c\). Then \[ \limsup_{L \to \infty} {1 \over L} \log P^\beta_{\Lambda, +} (X_\Lambda \leq m) \leq -2 \bigl( \lambda_F \alpha (m) \bigr)^{1/2}, \] where \(\alpha (m) = (m^*- m)/2 m^*\) and \(\lambda_F\) is the volume of the Wulff shape, generated by the surface tension function \(F\) of the model. This result combined with the lower large deviation bounds for \(X_\Lambda\), obtained by the author [J. Stat. Phys. 74, 411-432 (1994)], gives exact large deviation bounds for the \(2D\) Ising model.
scientific article; zbMATH DE number 784420