  Many-body perturbation theory (MBPT) is widely used in quantum physics,
chemistry, and materials science. At the heart of MBPT is the Feynman
diagrammatic expansion, which is, simply speaking, an elegant way of organizing
the combinatorially growing number of terms of a certain Taylor expansion. In
particular, the construction of the `bold Feynman diagrammatic expansion'
involves the partial resummation to infinite order of possibly divergent series
of diagrams. This procedure demands investigation from both the combinatorial
(perturbative) and the analytical (non-perturbative) viewpoints. In Part I of
this two-part series, we illustrate how the combinatorial properties of Feynman
diagrams in MBPT can be studied in the simplified setting of Gibbs measures
(known as the Euclidean lattice field theory in the physics literature) and
provide a self-contained explanation of Feynman diagrams in this setting. We
prove the combinatorial validity of the bold diagrammatic expansion, with
methods generalizable to several types of field theories and interactions. Our
treatment simplifies the presentation and numerical study of known techniques
in MBPT such as the self-consistent Hartree-Fock approximation (HF), the
second-order Green's function approximation (GF2), and the GW approximation.
The bold diagrams are closely related to the Luttinger-Ward (LW) formalism,
which was proposed in 1960 but whose analytic properties have not been
rigorously established. The analytical study of the LW formalism in the setting
of Gibbs measures will be the topic of Part II.
