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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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