JournalsaihpdVol. 1, No. 2pp. 225–264

Formal multidimensional integrals, stuffed maps, and topological recursion

  • Gaëtan Borot

    Max-Planck-Institut für Mathematik, Bonn, Germany
Formal multidimensional integrals, stuffed maps, and topological recursion cover
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Abstract

We show that the large NN expansion in the multi-trace 1 formal hermitian matrix model is governed by a topological recursion with initial conditions. In terms of a 1dd 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 (ω10,ω20\omega_1^0, \omega_2^0) of the topological recursion is characterized: for ω10\omega_1^0, by a non-linear, non-local Riemann-Hilbert problem on the discontinuity locus Γ\Gamma to determine; for ω20\omega_2^0, by a related but linear, non-local Riemann-Hilbert problem on the discontinuity locus Γ\Gamma. In combinatorics, this model enumerates discrete surfaces (maps) whose elementary 2-cells can have any topology - ω10\omega_1^0 being the generating series of disks, ω20\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/NN 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. 225–264

DOI 10.4171/AIHPD/7