Discrete Green's functions and functional determinants

A theorem is shown in which the elements of the inverse of a symmetric matrix F are constructed by Jacobi's formula using the derivative of the determinant detF with respect to its elements, and the determinant is defined by the partition function of a statistical field theory with interaction matrix F, generally Z = (detF)^-1/2. When the matrix F is the discrete Laplacian, the theorem is the essential connection between the discrete Green's function and two-point correlations of the lattice gas.