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