Convergence of density expansions of correlation functions and the Ornstein-Zernike equation

We prove convergence of the multi-body correlation function as a power series in the density. We work in the context of the cluster expansion in the canonical ensemble and we obtain bounds uniform in the volume and the number of particles. In the thermodynamic limit, the coefficients are characterized by sums over some class of two-connected graphs. We introduce the "direct correlation function" in the canonical ensemble and we prove that in the thermodynamic limit it is given by a convergent power series in the density with coefficients given by sums over some other class of two-connected graphs. Furthermore, it satisfies the Ornstein-Zernike equation from which quantified approximations can be derived.