  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 II of
this two-part series, we approach the analytical investigation of the bold
diagrammatic expansion in the simplified setting of Gibbs measures (known as
the Euclidean lattice field theory in the physics literature). Using
non-perturbative methods, we rigorously construct the Luttinger-Ward formalism
for the first time, and we prove that the bold diagrammatic series can be
obtained directly via an asymptotic expansion of the Luttinger-Ward functional,
circumventing the partial resummation technique. Moreover we prove that the
Dyson equation can be derived as the Euler-Lagrange equation associated with a
variational problem involving the Luttinger-Ward functional. We also establish
a number of key facts about the Luttinger-Ward functional, such as its
transformation rule, its form in the setting of the impurity problem, and its
continuous extension to the boundary of the domain of physical Green's
functions.
