# Formal multidimensional integrals, stuffed maps, and topological recursion

### Gaëtan Borot

Max-Planck-Institut für Mathematik, Bonn, Germany

## 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 ($ω_{1},ω_{2}$) of the topological recursion is characterized: for $ω_{1}$, by a non-linear, non-local Riemann-Hilbert problem on the discontinuity locus $Γ$ to determine; for $ω_{2}$, by a related but linear, non-local Riemann-Hilbert problem on the discontinuity locus $Γ$. In combinatorics, this model enumerates discrete surfaces (maps) whose elementary 2-cells can have any topology - $ω_{1}$ being the generating series of disks, $ω_{2}$ 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.

## Cite this article

Gaëtan Borot, Formal multidimensional integrals, stuffed maps, and topological recursion. Ann. Inst. Henri Poincaré Comb. Phys. Interact. 1 (2014), no. 2, pp. 225–264

DOI 10.4171/AIHPD/7