Critical Large Deviations for Gaussian Fields in the Phase Transition Regime, I

Abstract

We investigate large deviations for the empirical distribution functional of a Gaussian random field on $\mathbb{R}^{\mathbb{Z}^d}, d \geq 3$, in the phase transition regime. We first prove that the specific entropy governs an $N^d$ volume order large deviation principle outside the Gibbsian class. Within the Gibbsian class we derive an $N^{d-2}$ capacity order large deviation principle with exact rate function, and we apply this result to the asymptotics of microcanonical ensembles. We also give a spins' profile description of the field and show that smooth profiles obey $N^{d-2}$ order large deviations, whereas discontinuous profiles obey $N^{d-1}$ surface order large deviations.