We showed that the (Gaertner-Ellis) logarithmic moment generating function for Born measures associated with KMS states of weakly interacting fermions on the lattice can be written as the limit of logarithms of Gaussian Berezin integrals. This representation can be used to prove analyticity (near the origin) of the generating function, yielding, in particular, a large deviation principle for the Born measures. The construction does not use translation invariance and also random background potentials can be considered. A possible and important extension of this result, in view of applications to the macroscopic behavior of the electric conductivity at nanoscales, in presence of interactions, will be discussed. Recent results on the macroscopic behavior at nanoscales for free fermions with disorder will be presented to motivate the study of large deviation principles for the Born measures of lattice fermions.
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