Formal multidimensional integrals, stuffed maps, and topological recursion

Borot, Gaëtan

Annales de l’Institut Henri Poincaré D, Tome 1 (2014) no. 2, pp. 225-264.

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 1d 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 0 ,ω 2 0) of the topological recursion is characterized: for ω 1 0, by a non-linear, non-local Riemann-Hilbert problem on the discontinuity locus Γ to determine; for ω 2 0, 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 0 being the generating series of disks, ω 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.

Classification : 05-XX, 15-XX, 30-XX
Mots-clés : Map enumeration, matrix models, 2D quantum gravity, loop equations, Tutte equation, topological recursion