Large deviations in weakly interacting fermions: Generating functions as Gaussian Berezin integrals and bounds on large Pfaffians

We prove that the G\"{a}rtner--Ellis generating function of probability distributions associated with KMS states weakly interacting fermions on lattice can be written as limit logarithms Gaussian Berezin integrals. The covariances integrals are shown to have a uniform Pfaffian bound and summable in general cases interest, including systems not translation invariant. integral representation thus used obtain convergent expansions terms powers its parameter. derivation analysis subject second part present work. Such technical results also useful, for instance, context quantum information theory, computation relative entropy densities fermionic Gibbs states, theory normal fluctuations fermion systems.