Level set representation for the Gibbs states of the ferromagnetic Ising model (Q1178979)

We prove the existence of a real-valued random field with parameters in the \(d\)-dimensional cubic lattice, such that the distribution of the level set of this random field is a Gibbs state for the nearest neighbour ferromagnetic Ising model. Using this, we prove the continuity of the percolation probability with respect to the parameter \((\beta,h)\) in the uniqueness region except on the critical curve \(\Gamma_ c=\{(\beta,h_ c(\beta))\},\) where \(h_ c(\beta)\) is the critical level of the external field above which percolation takes place.