Solution of all quartic matrix models

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 Hermitian ${N}\times{N}$-matrices, where $E$ is any positive matrix and $\lambda$ a scalar. It was previously established that the large-$N$ limit of the second moment (the planar two-point function) satisfies a non-linear integral equation. By employing tools from complex analysis, in particular the Lagrange-B\"urmann inversion formula, we identify the exact solution of this non-linear problem, both for finite $N$ and for a large-${N}$ limit to unbounded operators $E$ of spectral dimension $\leq 4$. For finite $N$, the two-point function is a rational function evaluated at the preimages of another rational function $R$ constructed from the spectrum of $E$. Subsequent work has constructed from this formula a family $\omega_{g,n}$ of meromorphic differentials which obey blobbed topological recursion. For unbounded operators $E$, the renormalised two-point function is given by an integral formula involving a regularisation of $R$. This allowed a proof, in subsequent work, that the $\lambda\Phi^4_4$-model on noncommutative Moyal space does not have a triviality problem.