Analyticity results for the cumulants in a random matrix model (Q2352773)

It is now well known that cumulants and correlation functions in random matrices theory can be formally reconstructed as a \(\frac{1}{N}\) series whose coefficients obeys the so-called topological property. However, except in the case of strictly convex potentials, little is known about the convergence of the series expansion which very often diverges. The purpose of this article is to discuss the convergence of the \(\frac{1}{N}\) series expansion and to provide an alternative expansion with better convergence properties in a specific case in relation with graphs enumeration. The main result of the paper is that the \(\frac{1}{N}\) expansion although Borel summable at \(N=\infty\) is not convergent since \(N=\infty\) lies precisely on the border of the convergence domain. The alternative expansion (in a parameter \(\lambda\) specific of the model) is proved to be convergent and connection between both series expansions are provided. Though dependent on the structure of the specific model presented in the paper, the proofs regarding analyticity of the various series expansions may provide interesting ideas to develop a general framework to solve the long-standing problem of convergence issues in random matrix theory.