## Critical large deviations for Gaussian fields in the phase transition regime. I.(English)Zbl 0801.60018

The authors analyze large deviations for the empirical distributions of $$Z^ d$$-indexed stationary Gaussian random fields with $$d \geq 3$$. The fields are assumed to have the covariance structure given by the Green function of an irreducible transient random walk. The volume normalization $$(n^ d)$$ is shown to govern large deviations outside of the Gibbsian class, and the weak large deviation principle is established. Precise asymptotics with normalization $$n^{d-2}$$ is obtained within the Gibbsian class. The rate function in this case is given by the Dirichlet form of the diffusion associated with the covariance operator. The authors also give a spins’ profile description of the field and show that smooth profiles obey large deviations with normalization $$n^{d-2}$$, whereas discontinuous profiles obey large deviations with normalization $$n^{d-1}$$.

### MSC:

 60F10 Large deviations 60G60 Random fields 60G15 Gaussian processes 60K35 Interacting random processes; statistical mechanics type models; percolation theory
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