# Critical large deviations for Gaussian fields in the phase transition regime. I

Bolthausen, E; Deuschel, J-D (1993). Critical large deviations for Gaussian fields in the phase transition regime. I. The Annals of Probability, 21(4):1876-1920.

## 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.

## 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.

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Item Type: Journal Article, refereed, original work 07 Faculty of Science > Institute of Mathematics 510 Mathematics Large deviations; random fields; Gaussian processes; statistical mechanics English 1993 20 May 2010 15:25 05 Apr 2016 13:28 Institute of Mathematical Statistics 0091-1798 https://doi.org/10.1214/aop/1176989003 http://www.ams.org/mathscinet-getitem?mr=1245293http://www.zentralblatt-math.org/zmath/en/search/?q=an:0801.60018