Second order large deviation estimates for ferromagnetic systems in the phase coexistence region

Abstract

We consider thed-dimensional Ising model with ferromagnetic nearest neighbor interaction at inverse temperature β. Let$M_\Lambda = |\Lambda |^{ - 1} \sum\limits_{i \in \Lambda } {\sigma _i }$ be the magnetization inside ad-dimensional hyper cube Λ, μ₊ be the+Gibbs state andm*(β) be the spontaneous magnetization. For β such thatm*(β)>0 we find a sufficient condition (easily verified to hold for large β) for μ₊({M _Λ∈[a,b]}) to decay exponentially with |Λ|^(d−1)/d when −m*<b<m*, −1≦a<b. Ford=2 this sufficient condition is the exponential decay of a connectivity function. We also prove a partial converse to this result, obtain a sharper result for the magnetization ond−1 dimensional cross sections of the model and prove a similar result ford=2, −m*<a<b<m*, and β large, when free boundary conditions are chosen outside Λ.