Large Deviations in Weakly Interacting Fermions -- Generating Functions as Gaussian Berezin Integrals and Bounds on Large Pfaffians

TL;DR

This paper demonstrates that generating functions for weakly interacting fermions can be expressed as limits of Gaussian Berezin integrals with bounded Pfaffians, enabling convergent expansions useful in quantum information and fluctuation analysis.

Contribution

It introduces a novel representation of the G"{a}rtner--Ellis generating function as Gaussian Berezin integrals with uniform Pfaffian bounds, applicable even to non-translation invariant systems.

Findings

Gaussian Berezin integrals represent the generating function.

Pfaffian bounds are uniform and summable.

Expansions of logarithms of Berezin integrals are convergent.

Abstract

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