Perturbative and Geometric Analysis of the Quartic Kontsevich Model

The analogue of Kontsevich's matrix Airy function, with the cubic potential $\operatorname{Tr}\big(\Phi^3\big)$ replaced by a quartic term $\operatorname{Tr}\big(\Phi^4\big)$ same covariance, provides toy model for quantum field theory in which all correlation functions can be computed exactly and explicitly. In this paper we show that distinguished polynomials functions, themselves given quickly growing series Feynman ribbon graphs, sum up to much simpler highly structured expressions. These expressions are deeply connected meromorphic forms conjectured obey blobbed topological recursion. Moreover, how exact solutions permit explore critical phenomena Kontsevich model.