  We consider the zero-dimensional O(N) vector model as a simple example to
calculate n-point correlation functions using perturbation theory, the large-N
expansion, and the functional renormalization group (FRG). Comparing our
findings with exact results, we show that perturbation theory breaks down for
moderate interactions for all N, as one should expect. While the
interaction-induced shift of the free energy and the self-energy are well
described by the large-N expansion even for small N, this is not the case for
higher-order correlation functions. However, using the FRG in its one-particle
irreducible formalism, we see that very few running couplings suffice to get
accurate results for arbitrary N in the strong coupling regime, outperforming
the large-N expansion for small N. We further remark on how the derivative
expansion, a well-known approximation strategy for the FRG, reduces to an exact
method for the zero-dimensional O(N) vector model.
