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.
Publié le : 1993-10-14
Classification:
Large deviations,
random fields,
Gaussian processes,
statistical mechanics,
60F10,
60G60,
60G15,
60K35
@article{1176989003,
author = {Bolthausen, Erwin and Deuschel, Jean-Dominique},
title = {Critical Large Deviations for Gaussian Fields in the Phase Transition Regime, I},
journal = {Ann. Probab.},
volume = {21},
number = {4},
year = {1993},
pages = { 1876-1920},
language = {en},
url = {http://dml.mathdoc.fr/item/1176989003}
}
Bolthausen, Erwin; Deuschel, Jean-Dominique. Critical Large Deviations for Gaussian Fields in the Phase Transition Regime, I. Ann. Probab., Tome 21 (1993) no. 4, pp. 1876-1920. http://gdmltest.u-ga.fr/item/1176989003/