Formal multidimensional integrals, stuffed maps, and topological recursion (Q407409)

Summary: 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 \((\omega_1^0, \omega_2^0)\) 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.