# 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 $\lambda$.

open access link

@article{Grosse:2019jnv,
author = "Grosse, Harald and Hock, Alexander and Wulkenhaar,
Raimar",
title = "{Solution of all quartic matrix models}",
year = "2019",
eprint = "1906.04600",
archivePrefix = "arXiv",
primaryClass = "math-ph",
SLACcitation = "%%CITATION = ARXIV:1906.04600;%%"
}

Keywords:

*none*