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Systematic Resummation of the Large N expansion of Vector Models: Application to the Hubbard model and $2 + 1 $ dimensional QED

We introduce a hierarchy of closed equations for charge density correlation functions in the Hubbard model and $2 + 1$ dimensional QED. Each step in the hierarchy can be considered a large $N$ truncation of an exact, but infinite set of equations relating all $k-$point charge correlators. $N$ is the number of fermion spin components. Each step in the hierarchy sums up an infinite number of large $N$ diagrams, including all diagrams up to some fixed order, for $k$ point functions with $k \leq K$. Higher point functions are replaced with their leading large $N$ behavior. The simplest truncation gives a closed nonlinear equation for the $2$ point function.