# Solution of all quartic matrix models

June 11, 2019

We consider the quartic analogue of the Kontsevich model, which is defined by
a measure $\exp(-N\,\mathrm{Tr}(E\Phi^2+(\lambda/4)\Phi^4)) d\Phi$ on Hermitean
$N \times N$-matrices, where $E$ is any positive matrix and $\lambda$ a scalar.
We prove that the two-point function admits an explicit solution formula in
terms of the roots of a meromorphic function $J$ constructed from the spectrum
of $E$. Structures which appear in this solution can be assembled into complex
curves.
We also solve the large-$N$ limit to unbounded operators $E$. The
renormalised two-point function is given by an integral formula involving a
regularisation of $J$. We prove triviality of the renormalised four-dimensional
quartic matrix model for all positive $\lambda$.

Keywords:

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