We show that the large expansion in the multi-trace 1 formal hermitian matrix model is governed by a topological recursion with initial conditions. In terms of a 1 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 () of the topological recursion is characterized: for , by a non-linear, non-local Riemann-Hilbert problem on the discontinuity locus to determine; for , by a related but linear, non-local Riemann-Hilbert problem on the discontinuity locus . In combinatorics, this model enumerates discrete surfaces (maps) whose elementary 2-cells can have any topology - being the generating series of disks, 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/ 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. 225–264DOI 10.4171/AIHPD/7