We prove that the Gärtner–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.

中文翻译：

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弱相互作用费米子的大偏差：生成函数作为高斯 Berezin 积分和大 Pfaffian 的界限

我们证明了与晶格上弱相互作用费米子的 KMS 态相关的概率分布的 Gärtner-Ellis 生成函数可以写成高斯 Berezin 积分的对数极限。高斯积分的协方差被证明具有统一的 Pfaffian 界，并且在感兴趣的一般情况下是可求和的，包括以下系统不是翻译不变。因此，Berezin 积分表示可用于根据其参数的幂来获得生成函数的收敛扩展。Berezin 积分的对数展开式的推导和分析是本文第二部分的主题。这些技术结果也很有用，例如，在量子信息理论的背景下，在计算与费米子吉布斯态相关的相对熵密度，以及弱相互作用费米子系统的量子法向涨落理论中。