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# Annales de l’Institut Henri Poincaré D

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**Volume 1, Issue 2, 2014, pp. 225–264**

**DOI: 10.4171/AIHPD/7**

Published online: 2014-07-03

Formal multidimensional integrals, stuffed maps, and topological recursion

Gaëtan Borot^{[1]}(1) Max-Planck-Institut für Mathematik, Bonn, Germany

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.

*Keywords: *Map enumeration, matrix models, 2D quantum gravity, loop equations, Tutte equation, topological recursion

Borot Gaëtan: Formal multidimensional integrals, stuffed maps, and topological recursion. *Ann. Inst. Henri Poincaré Comb. Phys. Interact.* 1 (2014), 225-264. doi: 10.4171/AIHPD/7