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.
Citation
Erwin Bolthausen. Jean-Dominique Deuschel. "Critical Large Deviations for Gaussian Fields in the Phase Transition Regime, I." Ann. Probab. 21 (4) 1876 - 1920, October, 1993. https://doi.org/10.1214/aop/1176989003
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