It is known that the Kontsevich model is equivalent to the hermitian 1-matrix model by choosing a special relation between the moments of the external matrix (Kontsevich times) and the parameters of the potential of the hermitian 1-matrix model. This equivalence can easily be proven by topological recursion.
We will present the quartic analogue of the Kontsevich model with the action S=Tr(E \phi^2+\frac{\lambda}{4}\phi^4), where E is the external matrix and \lambda a real number. The Dyson-Schwinger equations (loop equations) will be shown, as well as some results for the first correlation functions. From these results the genus zero spectral curve of the model is identified, wich has a the symmetry x(z)=-y(-z). Furthermore, it turns out that the results and loop equations of the quartic model are structurally the same as from the hermitian 2-matrix model with a genus zero assumption for the spectral curve. The hermitian 2-matrix model satisfies a more general topological expansion.
This talk is partially based on https://arxiv.org/abs/1906.04600.
Joseph Ben Geloun
Fabien Vignes-Tourneret