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Many-body perturbation theory (MBPT) is widely used in quantum physics,chemistry, and materials science. At the heart of MBPT is the Feynmandiagrammatic expansion, which is, simply speaking, an elegant way of organizingthe combinatorially growing number of terms of a certain Taylor expansion. Inparticular, the construction of the `bold Feynman diagrammatic expansion'involves the partial resummation to infinite order of possibly divergent seriesof diagrams. This procedure demands investigation from both the combinatorial(perturbative) and the analytical (non-perturbative) viewpoints. In Part II ofthis two-part series, we approach the analytical investigation of the bolddiagrammatic expansion in the simplified setting of Gibbs measures (known asthe Euclidean lattice field theory in the physics literature). Usingnon-perturbative methods, we rigorously construct the Luttinger-Ward formalismfor the first time, and we prove that the bold diagrammatic series can beobtained directly via an asymptotic expansion of the Luttinger-Ward functional,circumventing the partial resummation technique. Moreover we prove that theDyson equation can be derived as the Euler-Lagrange equation associated with avariational problem involving the Luttinger-Ward functional. We also establisha number of key facts about the Luttinger-Ward functional, such as itstransformation rule, its form in the setting of the impurity problem, and itscontinuous extension to the boundary of the domain of physical Green'sfunctions.

Bold Feynman diagrams and the Luttinger-Ward formalism via Gibbs measures. Part II: Non-perturbative analysis