A diagrammatic equation for oriented planar graphs

In this paper we derive a diagrammatic equation for the planar sector of square non-Hermitian random matrix models. Our fundamental equation is first obtained by a graph counting argument (inspired by the Polchinski equation in quantum field theory) and subsequently derived independently by a precise saddle point analysis of the corresponding random matrix integral. We solve the equation perturbatively for a generic model and conclude by exhibiting two duality properties of the perturbative solution.