Formal multidimensional integrals, stuffed maps, and topological recursion

Abstract

We show that the large $N$ expansion in the multi-trace 1 formal hermitian matrix model is governed by a topological recursion with initial conditions. In terms of a 1$d$ gas of eigenvalues, this model includes – on top of the squared Vandermonde – multilinear interactions of any order between the eigenvalues. In this problem, the initial data (${\omega }_1^0,{\omega }_2^0$) of the topological recursion is characterized: for ${\omega }_1^0$, by a non-linear, non-local Riemann-Hilbert problem on the discontinuity locus $\Gamma$ to determine; for ${\omega }_2^0$, by a related but linear, non-local Riemann-Hilbert problem on the discontinuity locus $\Gamma$. In combinatorics, this model enumerates discrete surfaces (maps) whose elementary 2-cells can have any topology - ${\omega }_1^0$ being the generating series of disks, ${\omega }_2^0$ that of cylinders. In particular, by substitution one may consider maps whose elementary cells are themselves maps, for which we propose the name ”stuffed maps”. In a sense, our results complete the program of the ”moment method” initiated in the 90s to compute the formal 1/$N$ in the one hermitian matrix model.