Solution of all quartic matrix models

We consider the quartic analogue of the Kontsevich model, which is defined by a measure $\exp (-N\mathrm{Tr}(E{\Phi }^2+(\lambda /4){\Phi }^4\left)\right)d\Phi$ on Hermitian $N\times N$-matrices, where E is any positive matrix and λ 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-Bürmann 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 ≤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.