Friday, July 19, 2013

1307.4957 (Gaëtan Borot)

Formal multidimensional integrals, stuffed maps, and topological

Gaëtan Borot
We show that the large N expansion in the multi-trace 1 formal hermitian matrix model is governed by the Eynard-Orantin topological recursion. 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 (W10,W20) of the Eynard-Orantin recursion is characterized: for W10, by a non-linear, non-local Riemann-Hilbert problem on the discontinuity locus C to determine ; for W20, by a related but linear, non-local Riemann-Hilbert problem on the discontinuity locus C. In combinatorics, this model enumerates discrete surfaces (maps) whose elementary 2-cells can have any topology. In particular, by substitution one may consider maps whose elementary cells are themselves maps, for which we propose the name "stuffed maps".
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