  We prove that for various impurity models, in both classical and quantum
settings, the self-energy matrix is a sparse matrix with a sparsity pattern
determined by the impurity sites. In the quantum setting, such a sparsity
pattern has been known since Feynman. Indeed, it underlies several numerical
methods for solving impurity problems, as well as many approaches to more
general quantum many-body problems, such as the dynamical mean field theory.
The sparsity pattern is easily motivated by a formal perturbative expansion
using Feynman diagrams. However, to the extent of our knowledge, a rigorous
proof has not appeared in the literature. In the classical setting, analogous
considerations lead to a perhaps less-known result, i.e., that the precision
matrix of a Gibbs measure of a certain kind differs only by a sparse matrix
from the precision matrix of a corresponding Gaussian measure. Our argument for
this result mainly involves elementary algebraic manipulations and is in
particular non-perturbative. Nonetheless, the proof can be robustly adapted to
various settings of interest in physics, including quantum systems (both
fermionic and bosonic) at zero and finite temperature, non-equilibrium systems,
and superconducting systems.
