Loop Vertex Representation for Cumulants, Part I: Bounds on Free Energy with Sources

In this paper we construct cumulants for stable random matrix models with single trace interactions of arbitrarily high even order. We obtain explicit and convergent expansions for it and we prove that it is an analytic function inside a cardioid domain in the complex plane. We also prove their Borel-LeRoy summability at the origin of the coupling constant. Our proof is uniform in the external variables.