Formal multidimensional integrals, stuffed maps, and topological recursion

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.