Solution of all quartic matrix models

Abstract

We consider the quartic analogue of the Kontsevich model, which is defined by a measure $\exp (-N\mathrm{Tr}(E\hat{I}{¦}^2+(\hat{I}»/4)\hat{I}{¦}^4))d\hat{I}¦$ on Hermitian $N\times N$-matrices, where $E$ is any positive matrix and $\hat{I}»$ 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 $\le 4$. 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 $\ddot{I}{‰}_{g,n}$ 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 $\hat{I}»\hat{I}{¦}_4^4$-model on noncommutative Moyal space does not have a triviality problem.

31 pages, LaTeX. v2: A problem in v1 specific to dimension 4 is now solved. v3: representation of 2-point function as rational function is added. v4: completely rewritten