Large deviations and variational principle for harmonic crystals

We consider massless Gaussian fields with covariance related to the Green f unction of a long range random walk on Z(d). These are viewed as Gibbs meas ures for a linear-quadratic interaction. We establish thermodynamic identit ies and prove a version of Gibbs' variational principle, showing that trans lation invariant Gibbs measures are characterized as minimizers of the rela tive entropy density. We then study the large deviations of the empirical f ield of a Gibbs measure. We show that a weak large deviation principle hold s at the volume order, with rate given by the relative entropy density.